Kjll, Knlknkn; Lnl
Chapter 6 Resource Masters
Consumable Workbooks
Many of the worksheets contained in the Chapter Resource Masters booklets are available as consumable workbooks. Study Guide and Intervention Workbook Skills Practice Workbook Practice Workbook 0-07-828029-X 0-07-828023-0 0-07-828024-9
ANSWERS FOR WORKBOOKS The answers for Chapter 6 of these workbooks can be found in the back of this Chapter Resource Masters booklet.
Glencoe/McGraw-Hill
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher. Send all inquiries to: The McGraw-Hill Companies 8787 Orion Place Columbus, OH 43240-4027 ISBN: 0-07-828009-5 1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02 Algebra 2 Chapter 6 Resource Masters
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii Lesson 6-1
Study Guide and Intervention . . . . . . . . 313–314 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 315 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 Reading to Learn Mathematics . . . . . . . . . . 317 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 318
Lesson 6-6
Study Guide and Intervention . . . . . . . . 343–344 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 345 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 Reading to Learn Mathematics . . . . . . . . . . 347 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 348
Lesson 6-7
Study Guide and Intervention . . . . . . . . 349–350 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 351 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Reading to Learn Mathematics . . . . . . . . . . 353 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 354
Lesson 6-2
Study Guide and Intervention . . . . . . . . 319–320 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 321 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Reading to Learn Mathematics . . . . . . . . . . 323 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 324
Chapter 6 Assessment
Chapter 6 Test, Form 1 . . . . . . . . . . . . 355–356 Chapter 6 Test, Form 2A . . . . . . . . . . . 357–358 Chapter 6 Test, Form 2B . . . . . . . . . . . 359–360 Chapter 6 Test, Form 2C . . . . . . . . . . . 361–362 Chapter 6 Test, Form 2D . . . . . . . . . . . 363–364 Chapter 6 Test, Form 3 . . . . . . . . . . . . 365–366 Chapter 6 Open-Ended Assessment . . . . . . 367 Chapter 6 Vocabulary Test/Review . . . . . . . 368 Chapter 6 Quizzes 1 & 2 . . . . . . . . . . . . . . . 369 Chapter 6 Quizzes 3 & 4 . . . . . . . . . . . . . . . 370 Chapter 6 Mid-Chapter Test . . . . . . . . . . . . 371 Chapter 6 Cumulative Review . . . . . . . . . . . 372 Chapter 6 Standardized Test Practice . . 373–374 Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1 ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A32
Lesson 6-3
Study Guide and Intervention . . . . . . . . 325–326 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 327 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 Reading to Learn Mathematics . . . . . . . . . . 329 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 330
Lesson 6-4
Study Guide and Intervention . . . . . . . . 331–332 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 333 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 Reading to Learn Mathematics . . . . . . . . . . 335 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 336
Lesson 6-5
Study Guide and Intervention . . . . . . . . 337–338 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 339 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 Reading to Learn Mathematics . . . . . . . . . . 341 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 342
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Glencoe Algebra 2
Teacher’s Guide to Using the Chapter 6 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resources you use most often. The Chapter 6 Resource Masters includes the core materials needed for Chapter 6. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet. All of the materials found in this booklet are included for viewing and printing in the Algebra 2 TeacherWorks CD-ROM. Pages vii–viii include a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar.
Vocabulary Builder
There is one master for each lesson. These problems more closely follow the structure of the Practice and Apply section of the Student Edition exercises. These exercises are of average difficulty.
Practice
WHEN TO USE These provide additional practice options or may be used as homework for second day teaching of the lesson.
WHEN TO USE Give these pages to
students before beginning Lesson 6-1. Encourage them to add these pages to their Algebra 2 Study Notebook. Remind them to add definitions and examples as they complete each lesson.
Reading to Learn Mathematics
Study Guide and Intervention
Each lesson in Algebra 2 addresses two objectives. There is one Study Guide and Intervention master for each objective.
WHEN TO USE Use these masters as
One master is included for each lesson. The first section of each master asks questions about the opening paragraph of the lesson in the Student Edition. Additional questions ask students to interpret the context of and relationships among terms in the lesson. Finally, students are asked to summarize what they have learned using various representation techniques.
reteaching activities for students who need additional reinforcement. These pages can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent. There is one master for each lesson. These provide computational practice at a basic level. used with students who have weaker mathematics backgrounds or need additional reinforcement.
WHEN TO USE This master can be used
as a study tool when presenting the lesson or as an informal reading assessment after presenting the lesson. It is also a helpful tool for ELL (English Language Learner) students. There is one extension master for each lesson. These activities may extend the concepts in the lesson, offer an historical or multicultural look at the concepts, or widen students’ perspectives on the mathematics they are learning. These are not written exclusively for honors students, but are accessible for use with all levels of students.
Skills Practice
Enrichment
WHEN TO USE These masters can be
WHEN TO USE These may be used as
extra credit, short-term projects, or as activities for days when class periods are shortened.
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
iv
Assessment Options
The assessment masters in the Chapter 6 Resource Masters offer a wide range of assessment tools for intermediate and final assessment. The following lists describe each assessment master and its intended use.
Intermediate Assessment
• Four free-response quizzes are included to offer assessment at appropriate intervals in the chapter. • A Mid-Chapter Test provides an option to assess the first half of the chapter. It is composed of both multiple-choice and free-response questions.
Chapter Assessment
CHAPTER TESTS
• Form 1 contains multiple-choice questions and is intended for use with basic level students. • Forms 2A and 2B contain multiple-choice questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. • Forms 2C and 2D are composed of freeresponse questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Grids with axes are provided for questions assessing graphing skills. • Form 3 is an advanced level test with free-response questions. Grids without axes are provided for questions assessing graphing skills. All of the above tests include a freeresponse Bonus question. • The Open-Ended Assessment includes performance assessment tasks that are suitable for all students. A scoring rubric is included for evaluation guidelines. Sample answers are provided for assessment. • A Vocabulary Test, suitable for all students, includes a list of the vocabulary words in the chapter and ten questions assessing students’ knowledge of those terms. This can also be used in conjunction with one of the chapter tests or as a review worksheet.
Continuing Assessment
• The Cumulative Review provides students an opportunity to reinforce and retain skills as they proceed through their study of Algebra 2. It can also be used as a test. This master includes free-response questions. • The Standardized Test Practice offers continuing review of algebra concepts in various formats, which may appear on the standardized tests that they may encounter. This practice includes multiplechoice, grid-in, and quantitativecomparison questions. Bubble-in and grid-in answer sections are provided on the master.
Answers
• Page A1 is an answer sheet for the Standardized Test Practice questions that appear in the Student Edition on pages 342–343. This improves students’ familiarity with the answer formats they may encounter in test taking. • The answers for the lesson-by-lesson masters are provided as reduced pages with answers appearing in red. • Full-size answer keys are provided for the assessment masters in this booklet.
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Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6
Reading to Learn Mathematics
Vocabulary Builder
Vocabulary Builder
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 6. As you study the chapter, complete each term’s definition or description. Remember to add the page number where you found the term. Add these pages to your Algebra Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
axis of symmetry
completing the square
constant term
discriminant dihs·KRIH·muh·nuhnt linear term
maximum value
minimum value
parabola puh·RA·buh·luh quadratic equation kwah·DRA·tihk Quadratic Formula
! " # " $
! " # " $
! " " # " " $
(continued on the next page) Glencoe/McGraw-Hill
©
vii
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6
Reading to Learn Mathematics
Vocabulary Builder
Vocabulary Term Found on Page
(continued)
Definition/Description/Example
quadratic function
quadratic inequality
quadratic term
roots
Square Root Property
vertex
vertex form
Zero Product Property
zeros
©
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viii
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-1
Study Guide and Intervention
Graphing Quadratic Functions
Graph Quadratic Functions
Quadratic Function Graph of a Quadratic Function A function defined by an equation of the form f (x) b x-coordinate of vertex: 2a
ax 2
bx
c, where a
0 b ; 2a
A parabola with these characteristics: y intercept: c ; axis of symmetry: x
( 3) x or 3 . The x-coordinate of the vertex is 3 . 2(1) 2 2 3 Next make a table of values for x near . 2 x 0 1
3 2
a
1, b
3, and c
5, so the y-intercept is 5. The equation of the axis of symmetry is
x2 02 12
3 2 2
3x 3(0) 3(1)
3 3 2
5 5 5 5 5 5
f(x) 5 3
11 4
(x, f(x)) (0, 5) (1, 3)
3 11 , 2 4
f (x )
2 3
22 32
3(2) 3(3)
3 5
(2, 3) (3, 5)
O
x
Exercises
For Exercises 1–3, complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. 1. f(x) x2 6x 8 2. f(x) x2 2x 2 3. f(x) 2x2 4x 3
8, x x f (x) 3 1
3,
2 0
12 8 4 (x )
3
1 3 4 0
2, x x f (x) 1 3
4 –8 –4 O –4
1,
0 2 f (x )
1
2 2 1 1
3, x x f (x)
1, 1
1 1 f (x )
12
0 3
2 3
3 9
4
8x
8 4 –4 O 4 8
–8
–4
O –4
4
x
–8
x
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313
Glencoe Algebra 2
Lesson 6-1
Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex for the graph of f(x) x2 3x 5. Use this information to graph the function.
Example
NAME ______________________________________________ DATE
____________ PERIOD _____
6-1
Study Guide and Intervention
Graphing Quadratic Functions
(continued)
Maximum and Minimum Values
Maximum or Minimum Value of a Quadratic Function
The y-coordinate of the vertex of a quadratic function is the maximum or minimum value of the function.
The graph of f(x ) ax 2 bx c, where a 0, opens up and has a minimum when a 0. The graph opens down and has a maximum when a 0.
Determine whether each function has a maximum or minimum value. Then find the maximum or minimum value of each function. a. f(x) 3x 2 6x 7 For this function, a 3 and b 6. Since a 0, the graph opens up, and the function has a minimum value. The minimum value is the y-coordinate of the vertex. The x-coordinate of the b 6 vertex is 1.
2a 2(3)
Example
b. f(x) 100 2x x 2 For this function, a 1 and b 2. Since a 0, the graph opens down, and the function has a maximum value. The maximum value is the y-coordinate of the vertex. The x-coordinate of the vertex b 2 is 1.
2a 2( 1)
Evaluate the function at x 1 to find the minimum value. f(1) 3(1)2 6(1) 7 4, so the minimum value of the function is 4.
Evaluate the function at x 1 to find the maximum value. f( 1) 100 2( 1) ( 1)2 101, so the minimum value of the function is 101.
Exercises
Determine whether each function has a maximum or minimum value. Then find the maximum or minimum value of each function. 1. f(x) 2x2 x 10 2. f(x) x2 4x 7 3. f(x) 3x2 3x 1
min., 9
4. f(x)
7
4x x2
min.,
5. f(x) x2
11
7x 11
min.,
6. f(x)
1 x2 6x 4
16
max., 20
7. f(x) x2 5x 2
min.,
8. f(x) 20
5
6x x2
max., 5
9. f(x) 4x2 x 3
min.,
10. f(x)
17 x2 4x 10
max., 29
11. f(x) x2 10x 5
min., 2 15
12. f(x) 6x2 12x 21
max., 14
13. f(x) 25x2 100x 350
min.,
14. f(x)
20
0.5x2 0.3x 1.4
max., 27
15. f(x) x2 2 x 4
8
min., 250
©
min.,
1.445
314
max.,
7 31
Glencoe Algebra 2
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
6-1
Skills Practice
Graphing Quadratic Functions
For each quadratic function, find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. 1. f(x) 3x2 2. f(x) x2 1 3. f(x) x2 6x 15
0; x
4. f(x)
0; 0
2x2 11
1; x
5. f(x)
0; 0 x2 10x 5
15; x
6. f(x) 2x2
3; 3
8x 7
11; x
0; 0
5; x
5; 5
7; x
2; 2
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. 7. f(x) 2x2 8. f(x) x2 4x 2 0 4 4 6 9. f(x) x2 0 6x 2 0 3 8 4 6 8
0; x x f (x)
0; 0
2 8 1 0 2 0 f (x )
4; x
1 2 2 8 x
2; 2
2 0 4 16
8; x x f (x) 8 f (x )
3; 3
1 0
f (x) 16 4 f (x )
16
O
x
12 8 4 O –2 O 2 4 6x
x
Determine whether each function has a maximum or a minimum value. Then find the maximum or minimum value of each function. 10. f(x) 6x2 11. f(x) 8x2 12. f(x) x2 2x
min.; 0
13. f(x) x2 2x 15
max.; 0
14. f(x) x2 4x 1
min.;
15. f(x) x2
1
2x 3
min.; 14
16. f(x) 2x2 4x 3
max.; 3
17. f(x) 3x2 12x 3
min.;
18. f(x)
4
2x2 4x 1
max.;
1
min.;
9
315
min.;
1
©
Glencoe/McGraw-Hill
Glencoe Algebra 2
Lesson 6-1
NAME ______________________________________________ DATE
____________ PERIOD _____
6-1
Practice
(Average)
Graphing Quadratic Functions
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. 1. f(x) x2 0 8x 15 4 6 8 2. f(x) x2 6 4 4x 12 3. f(x) 2x2 2x 1 2 5
15; x x 4; 4
2 1 3 15
12; x x 2;
2
2 0 2
1; x x f (x) 5
0.5; 0.5
1 0 0.5 1 1 0.5 1
f (x) 15 3
16 12 8 4 O 2 4
f (x) 0 12 16 12 0 f (x )
16 12 8 4
f (x )
6
8x –6 –4 –2
O
2x
Determine whether each function has a maximum or a minimum value. Then find the maximum or minimum value of each function. 4. f(x) x2 2x 8 5. f(x) x2 6x 14 6. v(x) x2 14x 57
min.;
7. f(x)
9
2x2 4x 6
min.; 5
8. f(x) x2 4x 1
max.;
9. f(x)
8
2 2 x 3
8x
24
min.;
8
max.; 3
max.; 0
10. GRAVITATION From 4 feet above a swimming pool, Susan throws a ball upward with a velocity of 32 feet per second. The height h(t) of the ball t seconds after Susan throws it is given by h(t) 16t2 32t 4. Find the maximum height reached by the ball and the time that this height is reached. 20 ft; 1 s 11. HEALTH CLUBS Last year, the SportsTime Athletic Club charged $20 to participate in an aerobics class. Seventy people attended the classes. The club wants to increase the class price this year. They expect to lose one customer for each $1 increase in the price. a. What price should the club charge to maximize the income from the aerobics classes?
$45
b. What is the maximum income the SportsTime Athletic Club can expect to make?
$2025
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Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-1
Reading to Learn Mathematics
Graphing Quadratic Functions
How can income from a rock concert be maximized? Read the introduction to Lesson 6-1 at the top of page 286 in your textbook. • Based on the graph in your textbook, for what ticket price is the income the greatest? $40 • Use the graph to estimate the maximum income. about $72,000
Pre-Activity
Reading the Lesson
5x is the
linear 4
.
term, and 3 is the 4 ax2 x 3x2, a
b. For the quadratic function f(x) c 2. Consider the quadratic function f(x) a. The graph of this function is a b. The y-intercept is
3
,b
1
, and
bx
c, where a .
0.
parabola
.
c x
c. The axis of symmetry is the line d. If a 0, then the graph opens
b 2a
. and the function has a
upward
minimum
e. If a
value.
0, then the graph opens
downward
and the function has a
maximum
value.
(–2, 4)
3. Refer to the graph at the right as you complete the following sentences. a. The curve is called a b. The line x
f (x )
parabola
.
O (0, –1)
2 is called the axis of symmetry .
x
c. The point ( 2, 4) is called the
vertex
1), 1 is
.
d. Because the graph contains the point (0, the
y-intercept
.
Helping You Remember
4. How can you remember the way to use the x2 term of a quadratic function to tell whether the function has a maximum or a minimum value? Sample answer:
Remember that the graph of f(x) x 2 (with a 0) is a U-shaped curve that opens up and has a minimum. The graph of g(x) x 2 (with a 0) is just the opposite. It opens down and has a maximum.
317
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
Lesson 6-1
1. a. For the quadratic function f(x)
2x2
5x
3, 2x2 is the
quadratic constant term.
term,
NAME ______________________________________________ DATE
____________ PERIOD _____
6-1
Enrichment
Finding the Axis of Symmetry of a Parabola
As you know, if f(x) ax2 bx c is a quadratic function, the values of x b b2 2a 4ac b . 2a
that make f(x) equal to zero are
and
b
b2 2a f(x) 4ac
.
The average of these two number values is value when x
The function f(x) has its maximum or minimum b . Since the axis of symmetry 2a
O
b x = – –– 2a
f(x) = ax 2 + bx + c
of the graph of f (x) passes through the point where the maximum or minimum occurs, the axis of symmetry has the equation x b . 2a
x b b (– –– , f (– –– (( 2a 2a
Example
Use x x
10 2(5) b . 2a
Find the vertex and axis of symmetry for f(x)
5x 2
10x
7.
1
The x-coordinate of the vertex is 5x2 7 10x 12 7.
1.
Substitute x 1 in f(x) 2 f( 1) 5( 1) 10( 1) The vertex is ( 1, 12). The axis of symmetry is x
b , or x 2a
1.
Find the vertex and axis of symmetry for the graph of each function using x 1. f(x) x2 b . 2a
4x
8
2. g(x)
4x2
8x
3
3. y
x2
8x
3
4. f(x)
2x2
6x
5
5. A(x)
x2
12x
36
6. k(x)
2x2
2x
6
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318
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-2
Study Guide and Intervention
Solving Quadratic Equations by Graphing
Solve Quadratic Equations
Quadratic Equation Roots of a Quadratic Equation A quadratic equation has the form ax 2 bx c 0, where a 0. solution(s) of the equation, or the zero(s) of the related quadratic function
The zeros of a quadratic function are the x-intercepts of its graph. Therefore, finding the x-intercepts is one way of solving the related quadratic equation.
Example
Solve x2
x
6 x2
0 by graphing. x 6.
1 , and the equation of the 2 1 . 2
O
Graph the related function f(x)
f (x ) x
b The x-coordinate of the vertex is 2a 1 axis of symmetry is x . 2
Make a table of values using x-values around x f(x) 1 6
1 2
0 6
1 4
2 0
6
1 4
From the table and the graph, we can see that the zeros of the function are 2 and
3.
Exercises
Solve each equation by graphing. 1. x2 2x 8
O
0 2, f (x ) x
4
2. x2
4x
O
5 f (x )
0 5,
1
3. x2
5x
4 f (x )
0 1, 4
x
O
x
4. x2
10x f (x )
21
0
5. x2
4x
6
0 f (x )
6. 4x2
4x
1 f (x )
0
O
x
O
x
O
x
3, 7
©
no real solutions
319
1
Glencoe Algebra 2
Glencoe/McGraw-Hill
Lesson 6-2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-2
Study Guide and Intervention
(continued)
Solving Quadratic Equations by Graphing
Estimate Solutions Example
2 2(1) x f (x) 1 1 0 2
Often, you may not be able to find exact solutions to quadratic equations by graphing. But you can use the graph to estimate solutions.
Solve x 2 2x 2 0 by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. The equation of the axis of symmetry of the related function is x 1, so the vertex has x-coordinate 1. Make a table of values.
1 3 2 2 3 1
O
f (x )
x
The x-intercepts of the graph are between 2 and 3 and between 0 and 1. So one solution is between 2 and 3, and the other solution is between 0 and 1.
Exercises
Solve the equations by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 1. x2 4x 2 0 2. x2 6x 6 0 3. x2 4x 2 0
between 0 and 1; between 3 and 4 f (x )
between between
2 and 5 and f (x )
1; 4
between between
1 and 0; 4 and 3 f (x )
O
O
x
x
O
x
4.
x2
2x
4
0
5. 2x2
12x
17
0
6.
1 2 x 2
x
5 2
0
between 3 and 4; between 2 and 1 f (x )
between 2 and 3; between 3 and 4 f (x )
between 2 and between 3 and 4 f (x )
1;
O O
x
x
O
x
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320
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-2
Skills Practice
Solving Quadratic Equations By Graphing
Use the related graph of each equation to determine its solutions. 1. x2 2x 3 f (x )
0
2.
x2 f (x )
6x x2 6x
9
9 O
0 f (x ) x
3. 3x2
4x
3 f (x )
0
O
x
f (x ) f (x ) x2
2x 3
3x 2
4x
3 O
x
3, 1
3
no real solutions
4. x2
6x
5
0
5.
x2
2x
4
0
6. x2
6x
4
0
1, 5 f (x )
no real solutions f (x )
O
between 0 and 1; between 5 and 6 f (x )
x
O
x
O
x
Use a quadratic equation to find two real numbers that satisfy each situation, or show that no such numbers exist. 7. Their sum is 4, and their product is 0. 8. Their sum is 0, and their product is 36.
x2
4x
0; 0, f (x )
4
x2
36
36 24
0;
6, 6
f (x )
O
x
–12 –6
12 O 6 12 x
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321
Glencoe Algebra 2
Lesson 6-2
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
NAME ______________________________________________ DATE
____________ PERIOD _____
6-2
Practice
(Average)
Solving Quadratic Equations By Graphing
Use the related graph of each equation to determine its solutions. 1. 3x2 f (x )
3
3x 2 3
0 f (x )
2. 3x2
x f (x )
3
0
3. x2
3x
2 f (x )
0
O
x f (x )
O 3x 2
x
3
f (x )
x2
3x
2 O
x
x
1, 1
no real solutions
1, 2
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 4. 2x2 6x 5 0 5. x2 10x 24 0 6. 2x2 x 6 0
between 0 and 1; between 4 and 3
12 8 4 –6 –4 –2 O
6,
4 f (x )
between 2
2 and
1,
f (x )
x
O
x
Use a quadratic equation to find two real numbers that satisfy each situation, or show that no such numbers exist. 7. Their sum is 1, and their product is f (x )
6.
8. Their sum is 5, and their product is 8.
x2 x 3, 2
6
0;
x 2 5x 8 0; no such real numbers exist
O
x
For Exercises 9 and 10, use the formula h(t) v0t 16t 2, where h(t) is the height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. 9. BASEBALL Marta throws a baseball with an initial upward velocity of 60 feet per second. Ignoring Marta’s height, how long after she releases the ball will it hit the ground? 3.75 s 10. VOLCANOES A volcanic eruption blasts a boulder upward with an initial velocity of 240 feet per second. How long will it take the boulder to hit the ground if it lands at the same elevation from which it was ejected? 15 s
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Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-2
Reading to Learn Mathematics
Solving Quadratic Equations by Graphing
How does a quadratic function model a free-fall ride? Read the introduction to Lesson 6-2 at the top of page 294 in your textbook. Write a quadratic function that describes the height of a ball t seconds after 16t 2 125 it is dropped from a height of 125 feet. h(t)
Pre-Activity
Reading the Lesson
1. The graph of the quadratic function f(x) x2 x 6 is shown at the right. Use the graph to find the solutions of the quadratic equation x2 x 6 0. 2 and 3 y O
x
2. Sketch a graph to illustrate each situation. a. A parabola that opens b. A parabola that opens downward and represents a upward and represents a quadratic function with two quadratic function with real zeros, both of which are exactly one real zero. The negative numbers. zero is a positive number. y y
c. A parabola that opens downward and represents a quadratic function with no real zeros. y O
x
O
x
O
x
Helping You Remember
3. Think of a memory aid that can help you recall what is meant by the zeros of a quadratic function.
Sample answer: The basic facts about a subject are sometimes called the ABCs. In the case of zeros, the ABCs are the XYZs, because the zeros are the x-values that make the y-values equal to zero.
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Glencoe/McGraw-Hill
323
Glencoe Algebra 2
Lesson 6-2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-2
Enrichment
Graphing Absolute Value Equations
You can solve absolute value equations in much the same way you solved quadratic equations. Graph the related absolute value function for each equation using a graphing calculator. Then use the ZERO feature in the CALC menu to find its real solutions, if any. Recall that solutions are points where the graph intersects the x-axis. For each equation, make a sketch of the related graph and find the solutions rounded to the nearest hundredth. 1. | x 5| 0 2. | 4x 3| 5 0 3. | x 7| 0
5
No solutions
7
4. | x
3|
8
0
5.
|x
9, 3
3|
6
0
6. | x
2|
3
0
11, 5
1, 5
7. | 3x
4|
2
2,
2 3
8. | x
12 |
10
9. | x |
3
0
22,
2
3, 3
10. Explain how solving absolute value equations algebraically and finding zeros of absolute value functions graphically are related.
Sample answer: values of x when solving algebraically are the x-intercepts (or zeros) of the function when graphed.
Glencoe/McGraw-Hill
©
324
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Study Guide and Intervention
Solving Quadratic Equations by Factoring
When you use factoring to solve a quadratic equation,
0, then either a 0 or b 0, or both a and b 0.
Solve Equations by Factoring you use the following property.
Zero Product Property
For any real numbers a and b, if ab
Solve each equation by factoring. a. 15x b. 4x2 5x 3x2 15x Original equation 4x2 3x2 15x 0 Subtract 15x from both sides. 4x2 5x 3x(x 5) 0 Factor the binomial. (4x 7)(x 3x 0 or x 5 0 Zero Product Property 4x 7 0 x 0 or x 5 Solve each equation. x 3x2 The solution set is {0, 5}.
Example
21 5x 21 3) or
7 or 4
21 0 0 x
Original equation Subtract 21 from both sides. Factor the trinomial.
3 x
0 3
7 ,3 . 4
Zero Product Property Solve each equation.
The solution set is
Exercises
Solve each equation by factoring. 1. 6x2 2x 0 2. x2 7x 3. 20x2 25x
0,
4. 6x2
1
7x
{0, 7}
5. 6x2 27x 0
0,
6. 12x2
5
8x 0
0,
7. x2
7 x 30 0
0,
8. 2x2
9 x 3 0
0,
9. x2
2
14x 33 0
{5,
10. 4x2
6}
27x 7 0
3
,
1
29x 10 0
{ 11,
12. 6x2 5x
3}
4 0
11. 3x2
1
, ,
7
8x 1 0
10,
14. 5x2
1
12 0 15. 2x2
1 4
,
13. 12x2
28x
250x
5000
0
1 1
16. 2x2 11x 40 0
2
, 11,
6
21x 11 0
{100, 25}
18. 3x2 2x 21 0
17. 2x2
8,
19. 8x2
5
14x 3 0
1
2 0
7
, , ,5
3
17x 12 0
20. 6x2
11x
21. 5x2
3 1
,
2,
25x 12 0 23. 12x2
1
18x 6 0
3
4
36x 5 0
22. 12x2
24. 7x2
4
,
3
1
,
1
325
1
©
Glencoe/McGraw-Hill
Glencoe Algebra 2
Lesson 6-3
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Study Guide and Intervention
q)
(continued)
Solving Quadratic Equations by Factoring
Write Quadratic Equations
(x p)(x To write a quadratic equation with roots p and q, let 0. Then multiply using FOIL.
Example
in the form a. 3, 5 (x p)(x q) (x 3)[x ( 5)] (x 3)(x 5) x2 2x 15 The equation x2 3 and 5.
ax2
Write a quadratic equation with the given roots. Write the equation bx c 0. 0 0 0 0 2x b.
Write the pattern. Replace p with 3, q with Simplify. Use FOIL. 5.
7 1 , 8 3
(x x
7 8
p)(x x x
(3x 3 7)(3x 24
q)
1 3 1 3 1) 1)
0 0 0 0 24 0 0 13x 7 0 has
15
0 has roots
(8x
x
7) 8 24 (8x
7 8
24x2
13x
7
The equation 24x2 roots
7 1 and . 8 3
Exercises
Write a quadratic equation with the given roots. Write the equation in the form ax2 bx c 0. 1. 3, 4 2. 8, 2 3. 1, 9
x2
4. 5
x 10x
12 25
0 0
x2
5. 10, 7
10x 17x
16 70
0
6.
x2 x2
9. 7,
3 4
10x 13x
9 30
0 0
2, 15
x2
7.
1 ,5 3 3x 2 2 5 5x 2 2 , 3 2 3
x2
8. 2,
2 3 3x 2 4 , 9 9x 2 5 , 4 1 2
0
14x
5
0
11.
8x
1
4
0
4x 2
12. 9,
1 6 6x 2 3 1 , 7 5
25x
21
0
10. 3,
17x
6
0
14.
13x
4
0
15.
55x
9
0
13.
9x 2
16.
7 7 , 8 2 16x 2
4
0
17.
8x 2
1 3 , 2 4 8x 2
6x
5
0
18.
35x 2
1 1 , 8 6
22x
3
0
42x
49
10x
326
3
0
48x 2
14x
1
0
©
Glencoe/McGraw-Hill
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Skills Practice
Solving Quadratic Equations by Factoring
Solve each equation by factoring. 1. x2 64 { 8, 8} 2. x2 100 0 {10,
10}
3. x2
3x
2
0 {1, 2}
4. x2
4x
3
0 {1, 3}
5. x2
2x
3
0 {1,
3}
6. x2
3x
10
0 {5,
2}
7. x2
6x
5
0 {1, 5}
8. x2
9x
0 {0, 9}
9.
x2
6x
0 {0, 6}
10. x2
6x
8
0 { 2,
4}
11. x2
5x {0,
5}
12. x2
14x
49
0 {7}
13. x2
6
5x {2, 3}
14. x2
18x
81 { 9}
17. 4x2
5x
6
0
3
,
2
18. 3x2
13x
10
0
2
,5
Write a quadratic equation with the given roots. Write the equation in the form ax2 bx c 0, where a, b, and c are integers. 19. 1, 4 x 2
5x 7x
4
0 10 3 0 0
20. 6,
9 x2
3x 7x 0 2x
54
0
21.
2,
1 , 3
5 x2
22. 0, 7 x 2
1 3 , 2 4
23.
3 3x 2
10x
24.
8x 2
3
0
25. Find two consecutive integers whose product is 272. 16, 17
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327
Glencoe Algebra 2
Lesson 6-3
15. x2
4x
21 { 3, 7}
16. 2x2
5x
3
0
1
,
3
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Practice
(Average)
Solving Quadratic Equations by Factoring
Solve each equation by factoring. 1. x2 4. x2 7. x2 10. 10x2 12. x2 14. 36x2 16. 3x2 18. 3x2 20. 3x2 4x 6x 4x 12 8 0 {6, 0 {2, 4}
2}
2. x2 5. x2 8. 7x2
16x 3x
64 2
0 {8} 0 { 2,
3. x2
20x 9x 25
100 14
0 {10} 0 {2, 7}
1} 6. x2
9. x2
0 {0, 4} 9x 0,
4x 0,
4
2x 35x 8x
10x {5}
9
11. x2 13. 5x2 15. 2x2
99 { 9, 11} 60 90 0 {3, 4} 0 {9,
12x 25 2x 24x 8x
36 { 6}
5
1
,
0
5 1
5}
,
1 3}
17. 6x2 19. 15x2 21. 6x2
9x 0, 19x 5x 6
3
6 0
45
0 { 5,
3 2
,
2
4 2,
2
3
,
Write a quadratic equation with the given roots. Write the equation in the form ax2 bx c 0, where a, b, and c are integers. 22. 7, 2 23. 0, 3 24. 5, 8
x2
25. 7,
9x
8
14 56
0
26.
x2
6,
3x
3
0 18 0
x2
27. 3, 4
3x x
7 2
40 12
0 0
x2
28. 1,
1 2 2x 2 1 , 3
15x
0
29.
x2
1 ,2 3 3x 2 1 3 3x 2
9x
x2
30. 0,
3x
3
1
0
7x
2
0
33.
2x 2
2 , 3
7x
4 5
0
31.
32. 4,
3x 2 24, 26
8x
3
0
13x
4
0
15x 2
22x
8
0
34. NUMBER THEORY Find two consecutive even positive integers whose product is 624. 35. NUMBER THEORY Find two consecutive odd positive integers whose product is 323.
17, 19
36. GEOMETRY The length of a rectangle is 2 feet more than its width. Find the dimensions of the rectangle if its area is 63 square feet. 7 ft by 9 ft 37. PHOTOGRAPHY The length and width of a 6-inch by 8-inch photograph are reduced by the same amount to make a new photograph whose area is half that of the original. By how many inches will the dimensions of the photograph have to be reduced? 2 in.
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Glencoe/McGraw-Hill
328
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Reading to Learn Mathematics
Solving Quadratic Equations by Factoring
How is the Zero Product Property used in geometry? Read the introduction to Lesson 6-3 at the top of page 301 in your textbook. What does the expression x(x 5) mean in this situation?
Pre-Activity
It represents the area of the rectangle, since the area is the product of the width and length.
Reading the Lesson
1. The solution of a quadratic equation by factoring is shown below. Give the reason for each step of the solution. x2 x2 (x x 10x 3)(x 3 x 10x 21 7) 0 0 7 x 0 7 21
Original equation
Add 21 to each side. Factor the trinomial. Zero Product Property Solve each equation.
.
0 or x 3
The solution set is
{3, 7}
Marla (x 7)(x x2 2x
5) 35
0 0
Rosa (x 7)(x x2 2x
5) 35
0 0
Larry (x 7)(x x2 2x
5) 35
0 0
Who is correct? Rosa Explain the errors in the other two students’ work.
Sample answer: Marla used the wrong factors. Larry used the correct factors but multiplied them incorrectly.
Helping You Remember
3. A good way to remember a concept is to represent it in more than one way. Describe an algebraic way and a graphical way to recognize a quadratic equation that has a double root.
Sample answer: Algebraic: Write the equation in the standard form ax 2 bx c 0 and examine the trinomial. If it is a perfect square trinomial, the quadratic function has a double root. Graphical: Graph the related quadratic function. If the parabola has exactly one x-intercept, then the equation has a double root.
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329
Glencoe Algebra 2
Lesson 6-3
2. On an algebra quiz, students were asked to write a quadratic equation with 7 and 5 as its roots. The work that three students in the class wrote on their papers is shown below.
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Enrichment
Euler’s Formula for Prime Numbers
Many mathematicians have searched for a formula that would generate prime numbers. One such formula was proposed by Euler and uses a quadratic polynomial, x2 x 41. Find the values of x2 x 41 for the given values of x. State whether each value of the polynomial is or is not a prime number. 1. x 0 2. x 1 3. x 2
4. x
3
5. x
4
6. x
5
7. x
6
8. x
17
9. x
28
10. x
29
11. x
30
12. x
35
13. Does the formula produce all prime numbers greater than 40? Give examples in your answer.
14. Euler’s formula produces primes for many values of x, but it does not work for all of them. Find the first value of x for which the formula fails. (Hint: Try multiples of ten.)
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330
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Study Guide and Intervention
Completing the Square
Square Root Property Use the following property to solve a quadratic equation that is in the form “perfect square trinomial constant.”
Square Root Property For any real number x if x 2 n, then x n.
Example
a. x2 x2 x
Solve each equation by using the Square Root Property. b. 4x2 20x 25 32 8x 16 25 4x2 20x 25 8x 16 25 32 2 2 (x 4) (2x 5) 25 32 4 2x 5 25 or x 4 25 32 or 2x 5 x 5 4 9 or x 5 4 1 2x 5 4 2 or 2x 5 1}. x
5 4 2 2 5
32 4 2
The solution set is {9,
The solution set is
4 2 . 2
Exercises
Solve each equation by using the Square Root Property. 1. x2 18x 81 49 2. x2 20x 100 64 3. 4x2 4x 1 16
{2, 16}
{ 2,
18}
3
,
5
4. 36x2
12x
1
18
5. 9x2
12x
4
4
6. 25x2
40x
16
28
1
3 2
0, 4
4
2 7
7. 4x2
28x
49
64
8. 16x2
24x
9
81
9. 100x2
60x
9
121
15
,
1
3
,
3
{ 0.8, 1.4}
10. 25x2
20x
4
75
11. 36x2
48x
16
12
12. 25x2
30x
9
96
2
5 3
2
3
3
4 6
©
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331
Glencoe Algebra 2
Lesson 6-4
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Study Guide and Intervention
Completing the Square
(continued)
Complete the Square x2 1. Find b . 2
bx, follow these steps.
To complete the square for a quadratic expression of the form b . 2
!
2. Square
!
3. Add
b 2 to x2 2
bx.
Find the value of 2 c that makes x 22x c a perfect square trinomial. Then write the trinomial as the square of a binomial. Step 1 b 22; b 2
Example 1
Solve 2x2 completing the square. 2x2
2x2
Example 2
8x
8x 2
8x
24
0 by
24
24
0
0 2
Original equation Divide each side by 2. x2 4x 12 is not a perfect square.
4 2 2
11
x2 x2 121,
Step 2 112 121 Step 3 c 121 The trinomial is x2 22x which can be written as (x 11)2.
4x 12 x2 4x 4x 4 (x 2)2
0 12 12
Add 12 to each side.
4
Since
4, add 4 to each side.
16 x 2 4 x 6 or x 2 The solution set is {6,
Factor the square. Square Root Property Solve each equation.
2}.
Exercises
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 1. x2 10x c 2. x2 60x c 3. x2 3x c
25; (x
4. x2 3.2x
5)2 c 900; (x
5. x2
1 x 2
30)2 c 9
6. x2
; x
2.5x
3 2 c 2.56; (x
1.6)2
1
; x
1 2
1.5625; (x
1.25)2
Solve each equation by completing the square. 7. y2 4y 5 0 8. x2 8x 65 0 9. s2 10s 21 0
1, 5
10. 2x2 3x 1 0
5, 13
11. 2x2 13x 7 0
3, 7
12. 25x2 40x 9 0
1,
13. x2
1
4x 1 0 14. y2
1
,7
12y 4 0
1
15. t2
,
9
3t 8 0
2
©
3
6
4 2
332
3 2
41
Glencoe/McGraw-Hill
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Skills Practice
Completing the Square
Solve each equation by using the Square Root Property. 1. x2 8x 16 1 3, 5 2. x2 4x 4 1
1,
3
3. x2
12x
36
25
1,
11 2
4. 4x2
4x
1
9
1, 2 5 8 15
5. x2
4x
4
2
2 7
6. x2
2x
1
5 1
7. x2
6x
9
7 3
8. x2
16x
64
15
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 9. x2 10x c 25; (x
5)2 12)2
9 2
10. x2
14x
c 49; (x
7)2
5 2
11. x2
24x
c 144; (x
12. x2
5x
c
25
; x
13. x2
9x
c
81
; x
14. x2
x
c
1
; x
1 2
Solve each equation by completing the square. 15. x2 13x 36 0 4, 9 16. x2 3x 0 0,
3
0 2
17. x2
x
6
0 2,
3 4, 1
3 2 33
18. x2
4x
13
17 , 1
13 2
19. 2x2
7x
4
0
20. 3x2
2x
1
0
1
21. x2
3x
6
0
22. x2
x
3
0
1
23. x2
11
i
11
24. x2
2x
4
0 1
i
3
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
333
Lesson 6-4
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Practice
(Average)
Completing the Square
Solve each equation by using the Square Root Property. 1. x2 8x 16 1 2. x2 6x 9 1 3. x2 10x 25 16
5,
4. x2
3
14x 49 9
4,
5. 4x2
2
12x 9 4 6. x2
9, 4
1 2 9. 9x2
1
8x 16 8
4, 10
7. x2 6x 9 5 8. x2
1
,
2x
5
2 2
6x 1 2
3
5
1
2
1 3
2
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 10. x2 12x c 11. x2 20x c 12. x2 11x c
36; (x
13. x2 0.8x
6)2 c 100; (x
14. x2 2.2x
10)2 c 121
15. x2
; x
0.36x
11 2 c 0.16; (x
16. x2
5 x 6
0.4)2 c 1.21; (x
17. x2
1 x 4
1.1)2 c 0.0324; (x
18. x2
5 x 3
0.18)2
c
25
; x
5 2
1
; x
1 2
25
; x
5 2
Solve each equation by completing the square. 19. x2 22. x2 6x 18 8x 8 9x 3 0 0
4,
2 20. 3x2
23. x2
x 14x
2 19 5
0
2
0
,
1
21. 3x2 24. x2
5x 16x
2 7
0 1, 0 0
2
6, 3
25. 2x2
7
26. x2 x
30
0
8
27. 2x2 10x
71
5
4
28. x2
22 2
3x 6 0
1
29. 2x2
21 2
5x 6 0
5
30. 7x2
15 2
6x 2 0
3
i 2
15
5
i 4
23
3 7
i
5
31. GEOMETRY When the dimensions of a cube are reduced by 4 inches on each side, the surface area of the new cube is 864 square inches. What were the dimensions of the original cube? 16 in. by 16 in. by 16 in. 32. INVESTMENTS The amount of money A in an account in which P dollars is invested for 2 years is given by the formula A P(1 r)2, where r is the interest rate compounded annually. If an investment of $800 in the account grows to $882 in two years, at what interest rate was it invested? 5%
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334
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Reading to Learn Mathematics
Completing the Square
How can you find the time it takes an accelerating race car to reach the finish line? Read the introduction to Lesson 6-4 at the top of page 306 in your textbook. Explain what it means to say that the driver accelerates at a constant rate of 8 feet per second square.
Pre-Activity
If the driver is traveling at a certain speed at a particular moment, then one second later, the driver is traveling 8 feet per second faster.
Reading the Lesson
1. Give the reason for each step in the following solution of an equation by using the Square Root Property. x2 12x (x x x x 6 x 36 6)2 6 6 6 x 9 9 3 81 81 81
Original equation
Factor the perfect square trinomial. Square Root Property 81 9 Rewrite as two equations. Solve each equation.
9 or x 15
2. Explain how to find the constant that must be added to make a binomial into a perfect square trinomial.
Sample answer: Find half of the coefficient of the linear term and square it.
3. a. What is the first step in solving the equation 3x2 6x 5x 5 by completing the square? 12 0 by completing the
Divide the equation by 3.
b. What is the first step in solving the equation x2 square? Add 12 to each side.
Helping You Remember
4. How can you use the rules for squaring a binomial to help you remember the procedure for changing a binomial into a perfect square trinomial?
One of the rules for squaring a binomial is (x y) 2 x 2 2xy y 2. In completing the square, you are starting with x 2 bx and need to find y 2. This shows you that b 2y, so y b . That is why you must take half of
the coefficient and square it to get the constant that must be added to complete the square.
©
Glencoe/McGraw-Hill
335
Glencoe Algebra 2
Lesson 6-4
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Enrichment
The Golden Quadratic Equations
A golden rectangle has the property that its length can be written as a b, where a is the width of the rectangle and a a b
divided into a square and a smaller golden rectangle, as shown.
a
b
a . Any golden rectangle can be b a a
The proportion used to define golden rectangles can be used to derive two quadratic equations. These are sometimes called golden quadratic equations. Solve each problem.
a
b
1. In the proportion for the golden rectangle, let a equal 1. Write the resulting quadratic equation and solve for b.
2. In the proportion, let b equal 1. Write the resulting quadratic equation and solve for a.
3. Describe the difference between the two golden quadratic equations you found in exercises 1 and 2.
4. Show that the positive solutions of the two equations in exercises 1 and 2 are reciprocals.
5. Use the Pythagorean Theorem to find a radical expression for the diagonal of a golden rectangle when a 1.
6. Find a radical expression for the diagonal of a golden rectangle when b
1.
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336
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-5
Study Guide and Intervention
The Quadratic Formula and the Discriminant
Quadratic Formula The Quadratic Formula can be used to solve any quadratic equation once it is written in the form ax2 bx c 0.
Quadratic Formula The solutions of ax 2 bx c 0, with a 0, are given by x b b2 2a 4ac
.
Example x b ( 5) 5 5 2 2 9 81 b2 2a
Solve x2 x2
4ac ( 5)2 2(1)
5x 5x
14 by using the Quadratic Formula. 14 0.
Rewrite the equation as
Quadratic Formula
4(1)( 14)
Replace a with 1, b with
5, and c with
14.
Simplify.
7 or
2 2 and 7.
The solutions are
Exercises
Solve each equation by using the Quadratic Formula. 1. x2 2x 35 0 2. x2 10x 24 0 3. x2 11x 24 0
5,
4. 4x2
7
19x 5 0
4,
5. 14x2
6
9x 1 0
3, 8
6. 2x2 x 15 0
1
,
5
5x 2
1
8. 2y2
, y 1
3,
15 0 9. 3x2
5
7. 3x2
16x
16
0
2, 1
10. 8x2 6x 9 0
5
11. r2
,
3
3r 5 2 25
4, 4
0 12. x2 10x 50 0
13. x2
6x
23
0
14. 4x2
12x
63
0
15. x2
6x
21
0
3
©
4 2
3
6 2
3
2i
3
Glencoe Algebra 2
Glencoe/McGraw-Hill
337
Lesson 6-5
3 3
,
2 1
,
5
5 3
NAME ______________________________________________ DATE
____________ PERIOD _____
6-5
Study Guide and Intervention
(continued)
The Quadratic Formula and the Discriminant
Roots and the Discriminant
Discriminant The expression under the radical sign, b2 the discriminant. 4ac, in the Quadratic Formula is called
Roots of a Quadratic Equation
Discriminant b2 b2 b2 b2 4ac 4ac 4ac 4ac 0 and a perfect square 0, but not a perfect square 0 0 Type and Number of Roots 2 rational roots 2 irrational roots 1 rational root 2 complex roots
Find the value of the discriminant for each equation. Then describe the number and types of roots for the equation. b. 3x2 2x 5 a. 2x2 5x 3 The discriminant is The discriminant is b2 4ac 52 4(2)(3) or 1. b2 4ac ( 2)2 4(3)(5) or 56. The discriminant is a perfect square, so The discriminant is negative, so the the equation has 2 rational roots. equation has 2 complex roots.
Example
Exercises
For Exercises 1 12, complete parts a c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. 1. p2 12p 4 128; 2. 9x2 6x 1 0 0; 3. 2x2 7x 4 0 81;
two irrational roots; 4 6 4 2
4. x2 4x 4 0 32;
one rational root;
1
2 rational roots;
1
,
5. 5x2
36x
7
0 1156;
6. 4x2
4x
11
0
2 irrational roots; roots; 2 2 2
7. x2 7x 6 0 25;
2 rational roots;
1
160; 2 complex
1 i 10
40x 16 0;
,7
8m 14 8;
8. m2
9. 25x2
2 rational roots; 1, 6
10. 4x2 20x 29 0
2 irrational roots; 4 2
1 rational root; 4
12. 4x2 4x 11 0 192;
64; 11. 6x2 26x 8 0 484; 2 complex roots; 2 rational roots;
338
2 irrational roots;
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
6-5
Skills Practice
The Quadratic Formula and the Discriminant
Complete parts a c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. 1. x2 8x 16 0 2. x2 11x 26 0
0; 1 rational root; 4
3. 3x2 2x 0
225; 2 rational roots;
4. 20x2 7x 3 0
2, 13
3 1
4; 2 rational roots; 0,
5. 5x2 6 0
2
289; 2 rational roots;
6. x2 6 0
,
120; 2 irrational roots;
7. x2 8x 13 0
30 5
24; 2 irrational roots;
8. 5x2 x 1 0
6
21 10
12; 2 irrational roots;
9. x2 2x 17 0
4
3
21; 2 irrational roots; 1
10. x2 49 0
72; 2 irrational roots; 1
11. x2 x 1 0
3 2 i 2 3
196; 2 complex roots;
12. 2x2 3x 2
7i i 4 7
3; 2 complex roots; 1
7; 2 complex roots; 3
Solve each equation by using the method of your choice. Find exact solutions. 13. x2 15. x2 17. x2 19. x2 21. 2x2 23. 8x2 64 x 4x 25 10x 1 4x
8
30 11 0
14. x2
30
0 24x 27
30
0
5, 6
0 2
16. 16x2
9
,
3
15
18. x2 20. 3x2
8x 36 7x 2x
17 0 4 3
0 4
33 3
5i
11 0
2i
0 0
5 2
3
22. 2x2 24. 2x2
7 4 2 1
17
4
25. PARACHUTING Ignoring wind resistance, the distance d(t) in feet that a parachutist falls in t seconds can be estimated using the formula d(t) 16t2. If a parachutist jumps from an airplane and falls for 1100 feet before opening her parachute, how many seconds pass before she opens the parachute? about 8.3 s
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339
Glencoe Algebra 2
Lesson 6-5
1
i
i 5
NAME ______________________________________________ DATE
____________ PERIOD _____
6-5
Practice
(Average)
The Quadratic Formula and the Discriminant
Complete parts a c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. 1. x2 16x 64 0 2. x2 3x 3. 9x2 24x 16 0
0; 1 rational; 8
4. x2 3x 40
9; 2 rational; 0, 3
5. 3x2 9x 2 0 105;
0; 1 rational;
6. 2x2 7x 0
4
169; 2 rational;
7. 5x2 2x 4 0
5, 8 76; i 19 5
2 irrational;
8. 12x2 x 6
9 6
0 289;
105
49; 2 rational; 0,
9. 7x2 6x 2 0
7
20; i 7 5
2 complex; 1
10. 12x2 2x 4
2 rational; 3 ,
11. 6x2 2x 1
2
0 28;
2 complex;
12. x2 3x 6 0
3
0 196;
15;
3 i 15
2 rational;
13. 4x2 3x2
1
6
,
2
0 105;
2 irrational; 1
14. 16x2 8x 1 0
7 6 1
2 complex;
15. 2x2 5x 6
0 73;
2 irrational; 3
105 8
0; 1 rational;
2 irrational; 5
73 4
Solve each equation by using the method of your choice. Find exact solutions. 16. 7x2 18. 3x2 20. 3x2 22. x2 24. 3x2 26. 4x2 28. x2 5x 8x 13x 6x 3 54 4x 4x 0 0, 3 4
5
17. 4x2
9 21
0 4x
3
1
,
0
3
1
19. x2
3, 7
8
,4 6
21. 15x2 23. x2 25. 25x2
22x 14x 20x 1 53 6
2
0 7
,
4
0 3
2i
10 5
3i
17 0
2
1 4i
0 2
27. 8x 29. 4x2
4x2 2 7
3 2
0 3
15 2
i
11
12x
2 2
30. GRAVITATION The height h(t) in feet of an object t seconds after it is propelled straight up from the ground with an initial velocity of 60 feet per second is modeled by the equation h(t) 16t2 60t. At what times will the object be at a height of 56 feet? 1.75 s, 2 s 31. STOPPING DISTANCE The formula d 0.05s2 1.1s estimates the minimum stopping distance d in feet for a car traveling s miles per hour. If a car stops in 200 feet, what is the fastest it could have been traveling when the driver applied the brakes? about 53.2 mi/h
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Glencoe/McGraw-Hill
340
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-5
Reading to Learn Mathematics
The Quadratic Formula and the Discriminant
How is blood pressure related to age? Read the introduction to Lesson 6-5 at the top of page 313 in your textbook. Describe how you would calculate your normal blood pressure using one of the formulas in your textbook.
Pre-Activity
Sample answer: Substitute your age for A in the appropriate formula (for females or males) and evaluate the expression.
Reading the Lesson
1. a. Write the Quadratic Formula. x
b
b2 2a
4ac
5x 7, but do
b. Identify the values of a, b, and c that you would use to solve 2x2 not actually solve the equation. a
2
b
5
c
7
2. Suppose that you are solving four quadratic equations with rational coefficients and have found the value of the discriminant for each equation. In each case, give the number of roots and describe the type of roots that the equation will have.
Value of Discriminant 64 8 21 0 Number of Roots Type of Roots
2 2 2 1
real, rational complex real, irrational real, rational
Helping You Remember
3. How can looking at the Quadratic Formula help you remember the relationships between the value of the discriminant and the number of roots of a quadratic equation and whether the roots are real or complex?
Sample answer: The discriminant is the expression under the radical in the Quadratic Formula. Look at the Quadratic Formula and consider what happens when you take the principal square root of b2 4ac and apply in front of the result. If b2 4ac is positive, its principal square root will be a positive number and applying will give two different real solutions, which may be rational or irrational. If b2 4ac 0, its principal square root is 0, so applying in the Quadratic Formula will only lead to one solution, which will be rational (assuming a, b, and c are integers). If b 2 4ac is negative, since the square roots of negative numbers are not real numbers, you will get two complex roots, corresponding to the and in the symbol.
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341
Glencoe Algebra 2
Lesson 6-5
NAME ______________________________________________ DATE
____________ PERIOD _____
6-5
Enrichment
Sum and Product of Roots
Sometimes you may know the roots of a quadratic equation without knowing the equation itself. Using your knowledge of factoring to solve an equation, you can work backward to find the quadratic equation. The rule for finding the sum and product of roots is as follows:
Sum and Product of Roots If the roots of ax 2 then s1 s2 bx c s2 b and s1 a
0, with a ! 0, are s1 and s2, c . a
A road with an initial gradient, or slope, of 3% can be represented by the formula y ax2 0. 03x c, where y is the elevation and x is the distance along the curve. Suppose the elevation of the road is 1105 feet at points 200 feet and 1000 feet along the curve. You can find the equation of the transition curve. Equations of transition curves are used by civil engineers to design smooth and safe roads. The roots are x 3 ( 8) 5 3( 8) 24 Equation: x2 5x 3 and x
Add the roots. Multiply the roots.
Example
8.
10 –8 –6 –4 –2 O –10 –20
5 – (– –, –301) 2 4
y
24
0
2
4
x
–30
Write a quadratic equation that has the given roots. 1. 6, 9 2. 5, 1 3. 6, 6
x2
3x
54
0
x2
2 2 , 5 7 35x 2
4x
5
0
x2
2
12x
3 5
36
0
4. 4
3
6.
6.
7
x2
8x
13
0
4x
4
0
49x 2
42x
205
0
Find k such that the number given is a root of the equation. 7. 7; 2x2 kx 21 0 8. 2; x2 13x k 0
11
30
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342
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-6
Study Guide and Intervention
Analyzing Graphs of Quadratic Functions
The graph of y a(x h)2 k has the following characteristics: • Vertex: (h, k ) • Axis of symmetry: x h • Opens up if a 0 • Opens down if a 0 • Narrower than the graph of y x 2 if "a" 1 • Wider than the graph of y x 2 if "a" 1
Vertex Form of a Quadratic Function
Example each graph.
Identify the vertex, axis of symmetry, and direction of opening of
a. y 2(x 4)2 11 The vertex is at (h, k) or ( 4, 11), and the axis of symmetry is x up, and is narrower than the graph of y x2. a. y
1 (x 4
4. The graph opens
2)2
10 2. The graph opens
The vertex is at (h, k) or (2, 10), and the axis of symmetry is x down, and is wider than the graph of y x2.
Exercises
Each quadratic function is given in vertex form. Identify the vertex, axis of symmetry, and direction of opening of the graph. 1. y (x 2)2 16 2. y 4(x 3)2 7 3. y
1 (x 2
5)2
3
(2, 16); x
4. y 7(x
2; up
1)2 9
( 3,
5. y
1 (x 5
7); x
4)2
3; up
12
(5, 3); x
6. y 6(x
5; up
6)2 6
( 1,
7. y
2 (x 5
9); x
9)2
1; down
12
(4,
8. y
12); x
8(x 3)2
4; up
2
( 6, 6); x
9. y 3(x 1)2
6; up
2
(9, 12); x
10. y
5 (x 2
9; up
5)2 12
(3,
11. y
2); x
4 (x 3
3; up
22
(1,
12. y
2); x
16(x 4)2
1; down
1
7)2
( 5, 12); x
13. y 3(x 1.2)2
5; down
2.7
(7, 22); x
14. y 0.4(x
7; up
0.6)2 0.2
(4, 1); x
15. y 1.2(x
4; up
0.8)2 6.5
(1.2, 2.7); x
1.2; up
(0.6, 0.2); x down
343
0.6;
( 0.8, 6.5); x up
0.8;
©
Glencoe/McGraw-Hill
Glencoe Algebra 2
Lesson 6-6
Analyze Quadratic Functions
NAME ______________________________________________ DATE
____________ PERIOD _____
6-6
Study Guide and Intervention
(continued)
Analyzing Graphs of Quadratic Functions
Write Quadratic Functions in Vertex Form
A quadratic function is easier to graph when it is in vertex form. You can write a quadratic function of the form y ax2 bx c in vertex from by completing the square.
Example y y y y 2x2 2(x2 2(x2 2(x
Write y
2x2
12x
25 in vertex form. Then graph the function. y 12x 25 6x) 25 6x 9) 25 3)2 7
18 2(x 3)2 7.
The vertex form of the equation is y
O
x
Exercises
Write each quadratic function in vertex form. Then graph the function. 1. y x2 10x 32 2. y x2 6x 3. y x2 8x 6
y
(x y 5)2
7
y
(x
3)2 y O
9 x y
(x
8 4 –4 O –4 –8
4)2 y 10
4
8
x
O
x
–12
4. y
4x2
16x
11
5. y
3x2
12x
5
6. y
5x2
10x
9
y
4(x y 2)2
5
y
3(x y 2)2
7
y
5(x y 1)2
4
O
x
O
x
O
x
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344
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-6
Skills Practice
Analyzing Graphs of Quadratic Functions
Lesson 6-6
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. 1. y (x 2)2 2. y x2 4 3. y x2 6
y (x (2, 0); x
4. y 3(x
2)2
0; 2; up 5)2
5
y (x (0, 4); x
5. y 5x2
4; 0; down
9
0)2
y (x 0)2 6; (0, 6); x 0; up
6. y (x 2)2 18
5)2
y 3(x ( 5, 0); x
7. y x2 2x
0; 5; down
y 5(x 9; (0, 9); x 0; down
8. y x2 6x 2
0)2
y (x (2, 18); x
9. y 3x2
2)2
24x
18; 2; up
y (x (1, 6); x
1)2
6; 1; up
y (x ( 3, 7); x
3)2
7; 3; up
y 3(x (4, 48); x
4)2 48; 4; down
Graph each function. 10. y (x y 3)2
1
11. y
(x
1)2 y 2
12. y y O
(x
4)2
4
x
O
x
O
x
13. y
1 (x 2
2)2 y O
14. y
3x2 y 4
15. y
x2
6x
4 y x
O O
x
x
Write an equation for the parabola with the given vertex that passes through the given point. 16. vertex: (4, 36) point: (0, 20) 17. vertex: (3, 1) point: (2, 0) 18. vertex: ( 2, 2) point: ( 1, 3)
y
©
(x
4)2
36
y
(x
3)2
345
1
y
(x
2)2
2
Glencoe/McGraw-Hill
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-6
Practice
(Average)
Analyzing Graphs of Quadratic Functions
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. 1. y 6(x 2)2 1 2. y 2x2 2 3. y 4x2 8x
y 6(x ( 2, 1); x
4. y x2 10x
2)2
1; 2; down 5; 5; up
y 2(x (0, 2); x
5. y 2x2
0)2 0; up
2;
18
y 4(x 1)2 4; (1, 4); x 1; down
3x2 6x
y (x ( 5, 5); x
7. y 2x2
5)2
16x
20
12x
y x
8. y
2(x 3; up
3x2
3)2;
6. y
( 3, 0);
21
y 3(x (1, 2); x
9. y 2x2
1)2 2; 1; up
29
5
y 2(x ( 4, 0); x
10. y (x 3)2
4)2;
32
18x
4; down
1
y
y 3(x 6; (3, 6); x 3; down
11. y x2 y 3)2
y 2(x 4)2 ( 4, 3); x
2x2 2x
16x
3; 4; up
1
Graph each function. 6x 5 12. y
O O
x
x
Write an equation for the parabola with the given vertex that passes through the given point. 13. vertex: (1, 3) point: ( 2, 15) 14. vertex: ( 3, 0) point: (3, 18) 15. vertex: (10, point: (5, 6) 4)
y y y
1
2(x (x (x
1)2 4)2
3 4
y
1
(x
3)2
y
2
(x
10)2
4
16. Write an equation for a parabola with vertex at (4, 4) and x-intercept 6. 17. Write an equation for a parabola with vertex at ( 3, 1) and y-intercept 2.
3)2
1
18. BASEBALL The height h of a baseball t seconds after being hit is given by h(t) 16t2 80t 3. What is the maximum height that the baseball reaches, and when does this occur? 103 ft; 2.5 s 19. SCULPTURE A modern sculpture in a park contains a parabolic arc that starts at the ground and reaches a maximum height of 10 feet after a horizontal distance of 4 feet. Write a quadratic function in vertex form that describes the shape of the outside of the arc, where y is the height of a point on the arc and x is its horizontal distance from the left-hand 5 starting point of the arc. 2
10 ft
y
(x
4)
10
4 ft
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346
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-6
Reading to Learn Mathematics
Analyzing Graphs of Quadratic Equations
Read the introduction to Lesson 6-6 at the top of page 322 in your textbook. • What does adding a positive number to x2 do to the graph of y x2?
It moves the graph up.
• What does subtracting a positive number to x before squaring do to the graph of y x2? It moves the graph to the right.
Reading the Lesson
1. Complete the following information about the graph of y a. What are the coordinates of the vertex? (h, k) b. What is the equation of the axis of symmetry? x c. In which direction does the graph open if a d. What do you know about the graph if "a" a(x h)2 k.
h
0? up; down
0? If a 1?
It is wider than the graph of y
If "a"
x 2. x 2.
1? It is narrower than the graph of y
2. Match each graph with the description of the constants in the equation in vertex form. a. a c. a i.
O
0, h 0, h y 0, k 0, k
0 iii 0 ii ii. y b. a d. a iii. x O
0, h 0, h y 0, k 0, k
0 iv 0 i iv. y O
x
O
x
x
Helping You Remember
3. When graphing quadratic functions such as y (x 4)2 and y (x 5)2, many students have trouble remembering which represents a translation of the graph of y x2 to the left and which represents a translation to the right. What is an easy way to remember this?
Sample answer: In functions like y (x 4)2, the plus sign puts the graph “ahead” so that the vertex comes “sooner” than the origin and the translation is to the left. In functions like y (x 5)2, the minus puts the graph “behind” so that the vertex comes “later” than the origin and the translation is to the right.
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Glencoe/McGraw-Hill
347
Glencoe Algebra 2
Lesson 6-6
Pre-Activity
How can the graph of y function?
x2 be used to graph any quadratic
NAME ______________________________________________ DATE
____________ PERIOD _____
6-6
Enrichment
Patterns with Differences and Sums of Squares
Some whole numbers can be written as the difference of two squares and some cannot. Formulas can be developed to describe the sets of numbers algebraically. If possible, write each number as the difference of two squares. Look for patterns. 1. 0 02 5. 4 22 9. 8 32 13. 12 42
02 02 12 22
2. 1 12 6. 5 32 10. 9 32 14. 13 72
02 22 02 62
3. 2 cannot 7. 6 cannot 11. 10 cannot 15. 14 cannot
4. 3 22 8. 7 42 12. 11 62 16. 15 42
12 32 52 12
Even numbers can be written as 2n, where n is one of the numbers 0, 1, 2, 3, and so on. Odd numbers can be written 2n 1. Use these expressions for these problems. 17. Show that any odd number can be written as the difference of two squares.
2n
1
(n
1)2
n2
18. Show that the even numbers can be divided into two sets: those that can be written in the form 4n and those that can be written in the form 2 4n.
Find 4n for n 0, 1, 2, and so on. You get {0, 4, 8, 12, …}. For 2 4n, you get {2, 6, 10, 12, …}. Together these sets include all even numbers.
19. Describe the even numbers that cannot be written as the difference of two squares. 2 4n, for n 0, 1, 2, 3, … 20. Show that the other even numbers can be written as the difference of two squares. 4n (n 1)2 (n 1)2 Every whole number can be written as the sum of squares. It is never necessary to use more than four squares. Show that this is true for the whole numbers from 0 through 15 by writing each one as the sum of the least number of squares. 21. 0 02 24. 3 12 27. 6 12 30. 9 32 33. 12 12 36. 15 12
©
22. 1 12
23. 2 12 26. 5 12
12 22 22 12 22 32 32
12 12 12 12
12 22 12 22 32 32
25. 4 22 28. 7 12 31. 10 12 34. 13 22
12 32 32
12
22
29. 8 22 32. 11 12 35. 14 12
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348
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-7
Study Guide and Intervention
Graphing and Solving Quadratic Inequalities
To graph a quadratic inequality in two variables, use
the following steps:
Graph Quadratic Inequalities
1. Graph the related quadratic equation, y ax2 bx c. Use a dashed line for or ; use a solid line for or . 2. Test a point inside the parabola. If it satisfies the inequality, shade the region inside the parabola; otherwise, shade the region outside the parabola.
Example
Graph the inequality y
x2
6x
7. y First graph the equation y x2 6x 7. By completing the square, you get the vertex form of the equation y (x 3)2 2, so the vertex is ( 3, 2). Make a table of values around x 3, and graph. Since the inequality includes , use a dashed line. Test the point ( 3, 0), which is inside the parabola. Since ( 3)2 6( 3) 7 2, and 0 2, ( 3, 0) satisfies the inequality. Therefore, shade the region inside the parabola.
O
x
Exercises
Graph each inequality. 1. y x2 y 8x
17
2. y
x2
6x
4 y 3. y
x2
2x y 2
O
x
O
x
O
x
4. y
x2 y O
4x
6
5. y
2x2
4x y 6. y
2x2 y 4x
2
x
O O
x
x
©
Glencoe/McGraw-Hill
349
Glencoe Algebra 2
Lesson 6-7
NAME ______________________________________________ DATE
____________ PERIOD _____
6-7
Study Guide and Intervention
(continued)
Graphing and Solving Quadratic Inequalities graphically or algebraically.
Solve Quadratic Inequalities
Quadratic inequalities in one variable can be solved
Graphical Method
To solve ax 2 bx c 0: First graph y ax 2 bx c. The solution consists of the x-values for which the graph is below the x-axis. To solve ax 2 bx c 0: First graph y ax 2 bx c. The solution consists the x-values for which the graph is above the x-axis. Find the roots of the related quadratic equation by factoring, completing the square, or using the Quadratic Formula. 2 roots divide the number line into 3 intervals. Test a value in each interval to see which intervals are solutions.
Algebraic Method
If the inequality involves solution set.
or
, the roots of the related equation are included in the
Example
Solve the inequality x2
x x2
6
0.
O
y x
x 6 0. The First find the roots of the related equation equation factors as (x 3)(x 2) 0, so the roots are 3 and 2. The graph opens up with x-intercepts 3 and 2, so it must be on or below the x-axis for 2 x 3. Therefore the solution set is {x" 2 x 3}.
Exercises
Solve each inequality. 1. x2 2x 0 2. x2 16 0 3. 0 6x x2 5
{x 2
4. c2 4
x
0}
{x 4
5. 2m2 m
x
1
4}
{x1
6. y2 8
x
5}
{c 2
7. x2 4x
c
12
2}
0
m
8. x2
1
9x
m
14 0
1
9. x2 7x 10 0
{x 2
10. 2x2 5x
x
3
6}
0
{xx
11. 4x2 23x
7 or x
15 0
2}
12.
{x2
6x2
x
11x
5}
2 0
x 3
13. 2x2 11x
x
12
1
0
xx
14. x2 4x
3
5
or x
0
5
xx
15. 3x2 16x
2 or x
5 0
1
xx
3
or x
4
350
x
1
x
5
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
6-7
Skills Practice
Graphing and Solving Quadratic Inequalities
Graph each inequality. 1. y x2 4x y 4
2. y
x2
4 y 3. y
x2
2x y O
5
x
O O
x
x
Use the graph of its related function to write the solutions of each inequality. 4. x2 6x y 9
0
5.
x2
4x y 32
0
6. x2
x
20 y 0
5 O 6 O O 2 2
x
x
x
3
7. x2 9. x2 11. x2 13. x2 15. x2 17. x2 19. x2
8
x
4
8. x2 10. x2 12. x2
x
5 or x
4
Solve each inequality algebraically. 3x 18x 10 0 2x 36 35 0
{x 2 {xx
7x x 10x 3x 64
x
81
5}
0
{xx {x 6
7x 9x 2x 2x 12x
7 or x x
6 0
5}
9}
0
6} 1}
0
{xx {xx
0 or x
12 25 0 0
7} 3}
16.
{x 6
14. x2
x
18
4 or x
0
{x 6 x2 x
15
3}
0
all reals {xx all reals
©
{x 5
18. 2x2
x
4
3} 1}
3 or x
16x
0}
{xx
20. 9x2
2 or x
9 0
Glencoe/McGraw-Hill
351
Glencoe Algebra 2
Lesson 6-7
NAME ______________________________________________ DATE
____________ PERIOD _____
6-7
Practice x2 (Average)
Graphing and Solving Quadratic Inequalities
Graph each inequality. 1. y 4 y 2. y
x2
6x
6 y 3. y
2x2
4x
2
O
x
O
x
Use the graph of its related function to write the solutions of each inequality. 4. x2
6 O –6 –12 2 4 6 8x O
8x y 0
5.
x2
2x y 3
0
6. x2
O
9x y 14
0 x x
x
0 or x
8
3
x
1
2
x
7
Solve each inequality algebraically. 7. x2 x 20 0 8. x2 10x 16 0 9. x2 4x 5 0
{xx
10. x2 14x
4 or x
49 0
5}
{x2
11. x2 5x
x
14
8}
12. x2 15 8x
all reals
13. x2 5x 7 0
{xx
14. 9x2 36x
2 or x
36 0
7}
{x 5
15. 9x 12x2
x 0 or x
3}
3
0
all reals
16. 4x2 4x 1 0
{xx
17. 5x2 10
2}
27x
xx
18. 9x2
31x
12
xx
1
xx
2
or x
5
x 3
x
4
19. FENCING Vanessa has 180 feet of fencing that she intends to use to build a rectangular play area for her dog. She wants the play area to enclose at least 1800 square feet. What are the possible widths of the play area? 30 ft to 60 ft 20. BUSINESS A bicycle maker sold 300 bicycles last year at a profit of $300 each. The maker wants to increase the profit margin this year, but predicts that each $20 increase in profit will reduce the number of bicycles sold by 10. How many $20 increases in profit can the maker add in and expect to make a total profit of at least $100,000? from 5 to 10
©
Glencoe/McGraw-Hill
352
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-7
Reading to Learn Mathematics
Graphing and Solving Quadratic Inequalities
How can you find the time a trampolinist spends above a certain height? Read the introduction to Lesson 6-7 at the top of page 329 in your textbook. • How far above the ground is the trampoline surface? 3.75 feet • Using the quadratic function given in the introduction, write a quadratic inequality that describes the times at which the trampolinist is more than 20 feet above the ground. 16t 2 42t 3.75 20
Pre-Activity
Reading the Lesson
1. Answer the following questions about how you would graph the inequality y x2 x 6. a. What is the related quadratic equation? y
x2
x
6
b. Should the parabola be solid or dashed? How do you know?
solid; The inequality symbol is 0 for x and 02
.
c. The point (0, 2) is inside the parabola. To use this as a test point, substitute
2
0
for y in the quadratic inequality. 6 true or false? true
d. Is the statement 2
e. Should the region inside or outside the parabola be shaded? inside 2. The graph of y x2 4x is shown at the right. Match each of the following related inequalities with its solution set. a. b. c. d. x2 x2 x2 x2 4x 4x 4x 4x 0 ii 0 iii 0 iv 0 i i. {x"x ii. {x"0 iii. {x"x iv. {x"0 0 or x x 4} 4} 4}
(0, 0) O (4, 0)
y
(2, 4)
x
0 or x x 4}
Helping You Remember
3. A quadratic inequality in two variables may have the form y ax2 bx c, y ax2 bx c, y ax2 bx c, or y ax2 bx c. Describe a way to remember which region to shade by looking at the inequality symbol and without using a test point.
Sample answer: If the symbol is or , shade the region above the parabola. If the symbol is or , shade the region below the parabola.
©
Glencoe/McGraw-Hill
353
Glencoe Algebra 2
Lesson 6-7
NAME ______________________________________________ DATE
____________ PERIOD _____
6-7
Enrichment
Graphing Absolute Value Inequalities
You can solve absolute value inequalities by graphing in much the same manner you graphed quadratic inequalities. Graph the related absolute function for each inequality by using a graphing calculator. For and , identify the x-values, if any, for which the graph lies below the x-axis. For and , identify the x values, if any, for which the graph lies above the x-axis. For each inequality, make a sketch of the related graph and find the solutions rounded to the nearest hundredth. 1. |x 3| 0 2. |x| 6 0 3.
|x
12
4|
8
0
6
x
6
x
4
4. 2|x
6|
2
0
5. |3x
3|
0
6. |x
7|
5
x
7 or x
5
all real numbers
2
x
12
7. |7x
1|
13
8. |x
3.6|
4.2
9. |2x
5|
7
x
1.71 or x
2
0.6
x
7.8
6
x
1
©
Glencoe/McGraw-Hill
354
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Test, Form 1
(x 1)2. C. x 3(x
Write the letter for the correct answer in the blank at the right of each question. 1. Find the y-intercept for f(x) A. 1 B. 1 D. 0 6)2 12? D. x 18 D. 27 1. 2. 3.
2. What is the equation of the axis of symmetry of y A. x 2 B. x 6 C. x 6 3. Find the minimum value of f(x) A. 3 B. 6 4. The graph of f(x) 2x2 A. down; maximum C. up; maximum x2 6x. C.
9
x opens _____ and has a _____ value. B. down; minimum D. up; minimum f (x )
4.
O
x
5.
6. The quadratic function f(x) A. no zeros C. exactly two zeros
x2 has _____. B. exactly one zero D. more than two zeros
6.
For Questions 7 and 8, solve each equation by factoring. 7. x2 3x 10 A. { 5, 2} 8. 2x2 6x 0 A. { 3, 0} 0 B. ( 2, 5) B. {0, 3} C. { 2, 5} C. {0, 6} 2 and 3? B. x2 x D. x2 x D. { 10, 1} D. { 3, 3} 7. 8.
9. Which quadratic equation has roots A. x2 x 6 0 C. x2 6x 1 0 10. To solve x2 first rewrite A. (x 4)2 C. (x 4)2
6 6
0 0
9.
8x 16 25 by using the Square Root Property, you would the equation as _____. 25 B. x2 8x 9 0 5 D. x2 8x 9 10x c a perfect square. C. 10 D. 50
10. 11.
11. Find the value of c that makes x2 A. 100 B. 25
© Glencoe/McGraw-Hill
355
Glencoe Algebra 2
Assessment
5. The related graph of a quadratic equation is shown at the right. Use the graph to determine the solutions of the equation. A. 2, 3 B. 3, 2 C. 0, 6 D. 0, 2
NAME
DATE
PERIOD
6
Chapter 6 Test, Form 1
(continued)
12. The quadratic equation x2 6x 1 is to be solved by completing the square. Which equation would be the first step in that solution? A. x2 6x 1 0 B. x2 6x 36 1 36 C. x(x 6) 1 D. x2 6x 9 1 9 13. Find the exact solutions to x2 A.
3 2 5
12.
3x
13
1
0 by using the Quadratic Formula. C.
3 2 13
B. 3
2
D. 3
2
5
13.
For Questions 14 and 15, use the value of the discriminant to determine the number and type of roots for each equation. 14. x2 3x 7 0 A. 2 complex roots C. 2 real, rational roots 15. x2 4x 4 A. 2 real, rational roots C. 1 real, rational root 16. What is the vertex of y 2(x 3)2 A. ( 3, 6) B. (3, 6) 6? C. ( 3, 6) 3(x D. (3, 6) 6)2 1? D. x 6 16. 17. B. 2 real, irrational roots D. 1 real, rational root B. 2 real, irrational roots D. no real roots
14.
15.
17. What is the equation of the axis of symmetry of y A. x 2 B. x 6 C. x 3
18. Which quadratic function has its vertex at (2, 3) and passes through (1, 0)? A. y 2(x 2)2 3 B. y 3(x 2)2 3 C. y 3(x 2)2 3 D. y 2(x 2)2 3 19. Which quadratic inequality is graphed at the right? A. y (x 1)2 4 B. y (x 1)2 4 C. y (x 1)2 4 D. y (x 1)2 4 20. Solve (x 4)(x 2) 0. A. {x x 2 or x 4} C. {x 2 x 4} B. {x 4 D. {x x y 18.
O
x
19.
x 2} 2 or x
4} B:
20.
Bonus Find the x-intercepts and the y-intercept of the graph of y 2(x 4)2 18.
© Glencoe/McGraw-Hill
356
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Test, Form 2A
Write the letter for the correct answer in the blank at the right of each question. 1. Identify the y-intercept and the axis of symmetry for the graph of f(x) 10x2 40x 42. A. 42; x 4 B. 0; x 4 C. 42; x 2 D. 42; x 2. Identify the quadratic function graphed at the right. A. f(x) x2 2x B. f(x) x2 2x C. f(x) x2 2x D. f(x) (x 2)2 3. Determine whether f(x) 4x2 16x value and find that value. A. minimum; 10 B. minimum; 2 f (x )
2
1.
O
x
2.
6 has a maximum or a minimum C. maximum; 10 D. maximum; 2 3.
4.
x2 4x A. 4, 0 C. between
4 and 4 3;
B. D.
4, 0 2, 4 1;
4.
5. x2 2x 5 A. between 4 and between 1 and 2 C. no real solutions 6. x2 3x A. {6} 7. 3x2 20
B. between 2 and between 3 and 4 D. 1, 6
5.
For Questions 6 and 7, solve each equation by factoring. 18 B. { 6, 3} 7x B. 5, 4
3
C. { 9, 2} 4, 5
3 5
D. { 3, 6} 20, 1
3
6.
A. { 10, 2}
C.
D.
7.
8. Which quadratic equation has roots A. x2 4x 4 0 C. 5x2 9x 2 0
2 and 1 ? B. 5x2 D. 5x2 9x 2 0 11x 2 0 8.
9. To solve 9x2 12x 4 49 by using the Square Root Property, you would first rewrite the equation as _____. A. 9x2 12x 45 0 B. (3x 2)2 49 2 2 C. (3x 2) 7 D. (3x 2) 49 10. Find the value of c that makes x2 A. 81 4
© Glencoe/McGraw-Hill
9.
9x
c a perfect square. C.
81 4
B. 9 2
D. 81
10.
Glencoe Algebra 2
357
Assessment
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
NAME
DATE
PERIOD
6
Chapter 6 Test, Form 2A
(continued)
11. The quadratic equation x2 8x 20 is to be solved by completing the square. Which equation would be a step in that solution? A. (x 4)2 4 B. x 4 2i 2 2 C. x 8x 20 0 D. x 8x 16 20 12. Find the exact solutions to 3x2 A.
5 6 13
11. 12.
5x
1 by using the Quadratic Formula. C. 5
6 37
B.
5
3
13
D. 5
6
13
For Questions 13 and 14, use the value of the discriminant to determine the number and type of roots for each equation. 13. 2x2 7x 9 0 A. 2 real, rational C. 2 complex 14. x2 20 12x 16 A. 1 real, irrational C. no real y
1 (x 2
B. 2 real, irrational D. 1 real, rational B. 2 real, rational D. 1 real, rational
13.
14.
15. Identify the vertex, axis of symmetry, and direction of opening for 8)2 2. 8; up 8; up B. ( 8, 2); x 8; down D. (8, 2); x 8; up 15. A. ( 8, 2); x C. (8, 2); x
16. Which quadratic function has its vertex at ( 2, 7) and opens down? A. y 3(x 2)2 7 B. y (x 2)2 7 C. y 12(x 2)2 7 D. y 2(x 2)2 7 17. Write y A. y (x C. y (x x2 2)2 2)2 4x 5 1 1 in vertex form. B. y D. y (x (x 2)2 2)2 5 3
16.
17.
18. Write an equation for the parabola whose vertex is at ( 8, 4) and passes through ( 6, 2). A. y C. y
3 (x 2 3 (x 2
8)2 6)2 2
4
B. y D. y
1 (x 4 3 (x 2
8)2 8)2
4 4 y O
18. x 19. Which quadratic inequality is graphed at the right? A. y (x 2)(x 3) B. y (x 2)(x 3) C. y (x 2)(x 3) D. y (x 2)(x 3) 20. Solve x2 2x 24. A. {x 4 x 6} C. {x x 6 or x
19.
4}
B. {x 6 D. {x x i 3 . 4
x 4} 4 or x
6} B:
20.
Bonus Write a quadratic equation with roots
© Glencoe/McGraw-Hill
358
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Test, Form 2B
Write the letter for the correct answer in the blank at the right of each question. 1. Identify the y-intercept and the axis of symmetry for the graph of f(x) 3x2 6x 12. A. 2; x 12 B. 12; x 1 C. 2; x 0 D. 12; x 2. Identify the quadratic function graphed at the right. A. f(x) x2 4x B. f(x) x2 4x C. f(x) x2 4x D. f(x) (x 4)2 3. Determine whether f(x) 5x2 10x value and find that value. A. minimum; 1 B. maximum; 11 f (x )
1
1.
O
x
2.
6 has a maximum or a minimum C. maximum; 1 D. minimum; 11 3.
4. x2 4x A. 4, 0 C. 2, 4 5. x2 2x 2 A. between 3 and between 0 and 1 C. no real solutions 6. x2 3x 28 A. { 4, 7} 7. 5x2 A. 4 2, 2
5
B. between D. 0, 4 2;
4 and 4 4.
B. between 1 and 0; between 2 and 3 D. 1, 1
5.
For Questions 6 and 7, solve each equation by factoring. B. { 14, 2}
2 ,2 5
C. { 7, 4}
1 ,4 5 2 ? 3
D. { 2, 14} 4, 1
5
6.
19x B. C. D. 7.
8. Which quadratic equation has roots 7 and A. 2x2 C. 3x2 11x 23x 21 14 0 0
B. 3x2 D. 2x2
19x 11x
14 21
0 0
8.
9. To solve 4x2 28x 49 25 by using the Square Root Property, you would first rewrite the equation as _____. A. (2x 7)2 25 B. (2x 7)2 5 C. (2x 7)2 5 D. 4x2 28x 24 0 10. Find the value of c that makes x2 A. 25 16
© Glencoe/McGraw-Hill
9.
5x
c a perfect square trinomial. C. 25
4
B. 5 4
D. 5
2
10.
Glencoe Algebra 2
359
Assessment
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
NAME
DATE
PERIOD
6
Chapter 6 Test, Form 2B
(continued)
11. The quadratic equation x2 18x 106 is to be solved by completing the square. Which equation would be a step in that solution? A. (x 9)2 25 B. x2 18x 106 0 C. x 9 5i D. x2 18x 81 106 12. Find the exact solutions to 2x2 A.
5 4 17
11. 12.
5x
1 by using the Quadratic Formula. C. 5
4 33
B. 5
4
17
D. 5
2
17
For Questions 13 and 14, use the value of the discriminant to determine the number and type of roots for each equation. 13. 3x2 x 12 0 A. 2 complex roots C. 2 real, rational roots 14. x2 10 3x 3 A. 2 complex roots C. 1 real, rational root B. 1 real, rational root D. 2 real, irrational roots B. 2 real, irrational roots D. 2 real, rational roots
13.
14.
15. Identify the vertex, axis of symmetry, and direction of opening for y 8(x 2)2. A. ( 8, 2); x 8 up B. ( 2, 0); x 2; down C. (2, 0); x 2; down D. ( 2, 8); x 2; down 16. Which quadratic function has its vertex at ( 3, 5) and opens down? A. y (x 3)2 5 B. y (x 3)2 5 C. y (x 3)2 5 D. y (x 3)2 5 17. Write y x2 18x 52 in vertex form. A. y (x 9)2 113 B. y 2 C. y (x 9) 52 D. y (x (x 9)2 9)2 29 29
15.
16.
17.
18. Write an equation for the parabola whose vertex is at ( 5, 7) and passes through ( 3, 1). A. y C. y
1 (x 11 1 (x 2
5)2 5)2
7 7
B. y D. y
2(x
1 (x 2
5)2 5)2
7 7 y O
18.
19. Which quadratic inequality is graphed at the right? A. y (x 3)(x 1) B. y (x 3)(x 1) C. y (x 3)(x 1) D. y (x 3)(x 1) 20. Solve 2x 3 x2. A. {x 1 x 3} C. {x x 1 or x B. {x 3 D. {x x i 2 . 3
x
19.
3}
x 1} 3 or x
1} B:
20.
Bonus Write a quadratic equation with roots
© Glencoe/McGraw-Hill
360
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Test, Form 2C
1.
1. Graph f(x) 5x2 10x, labeling the y-intercept, vertex, and axis of symmetry.
f (x )
O
x
2. Determine whether f(x) 3x2 6x 1 has a maximum or a minimum value and find that value. For Questions 3 and 4, solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 3. x2 6x 8
2.
3. y O
x
4. x2
x
5
0
4. y O
x
5. Solve 5x2
13x
6 by factoring.
5. 6.
6. GEOMETRY The length of a rectangle is 7 inches longer than its width. If the area of the rectangle is 144 square inches, what are its dimensions? 7. Write a quadratic equation with Write the equation in the form ax2 a, b, and c are integers. 6 and 3 as its roots. bx
4
7.
c
0, where
Solve each equation by using the Square Root Property. 8. x2 9. 4x2 6x 20x 9 25 25 7 8. 9.
© Glencoe/McGraw-Hill
361
Glencoe Algebra 2
Assessment
NAME
DATE
PERIOD
6
Chapter 6 Test, Form 2C
(continued)
For Questions 10 and 11, solve each equation by completing the square. 10. x2 11. 2x2 4x 3x 9 2 0 0 3x 2 by using the 10. 11. 12.
12. Find the exact solutions to 5x2 Quadratic Formula.
For Questions 13 and 14, find the value of the discriminant for each quadratic equation. Then describe the number and type of roots for the equation. 13. 9x2 14. 4x2 12x 1 9x 4 0 2
2 (x 3
13. 14. 15.
15. Identify the vertex, axis of symmetry, and direction of opening for y 5)2 7. 1)
16. Write an equation for the parabola with vertex at (2, and y-intercept 5. 17. Write y x2 6x 8 in vertex form.
16. 17. 18.
18. PHYSICS The height h (in feet) of a certain rocket t seconds after it leaves the ground is modeled by h(t) 16t2 48t 15. Write the function in vertex form and find the maximum height reached by the rocket. 19. Graph y x2 6x 9.
19.
y
O
x
20. Solve 2x2
5x
3
0 algebraically.
7 . Write the 3
20. B:
Bonus Write a quadratic equation with roots equation in the form ax2 c are integers. bx c
0, where a, b, and
© Glencoe/McGraw-Hill
362
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Test, Form 2D
1.
1. Graph f(x) x2 4x 3, labeling the y-intercept, vertex, and axis of symmetry.
f (x )
O
x
2. Determine whether f(x) 5x2 20x 3 has a maximum or a minimum value and find that value. For Questions 3 and 4, solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 3. x2 2x 3 0
2.
3. y O
x
4. 2x2
2x
3
0
4. y O
x
5. Solve 3x2
x
4 by factoring.
5. 6.
6. GEOMETRY The length of a rectangle is 10 inches longer than its width. If the area of the rectangle is 144 square inches, what are its dimensions? 7. Write a quadratic equation with Write the equation in the form ax2 a, b, and c are integers.
© Glencoe/McGraw-Hill
4 and 3 as its roots. bx
2
7.
c
0, where
363
Glencoe Algebra 2
Assessment
NAME
DATE
PERIOD
6
Chapter 6 Test, Form 2D
(continued)
Solve each equation by using the Square Root Property. 8. x2 9. 9x2 14x 12x 49 4 16 6 8. 9.
Solve each equation by completing the square. 10. x2 11. 3x2 8x x 14 2 0 0 9x 5 by using the 10. 11. 12.
12. Find the exact solutions to 2x2 Quadratic Formula.
For Questions 13 and 14, find the value of the discriminant for each quadratic equation. Then describe the number and type of roots for the equation. 13. 25x2 14. 2x2 20x 10x 4 9 0 2x 13. 14. 15. 16. 17. 18.
15. Identify the vertex, axis of symmetry, and direction of opening for y (x 6)2 5. 16. Write an equation for the parabola with vertex at ( 4, 2) and y-intercept 2. 17. Write y x2 4x 8 in vertex form.
18. PHYSICS The height h (in feet) of a certain rocket t seconds after it leaves the ground is modeled by h(t) 16t2 64t 12. Write the function in vertex form and find the maximum height reached by the rocket. 19. Graph y x2 4x 4.
19.
y
O
x
20. Solve 2x2
7x
15
0 algebraically.
5 . 4
20. B: 0,
Bonus Write a quadratic equation with roots Write the equation in the form ax2 where a, b, and c are integers.
© Glencoe/McGraw-Hill
bx
c
364
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Test, Form 3
1.
1. Graph f(x) 3 3x2 2x, labeling the y-intercept, vertex, and axis of symmetry. 2. Determine whether f(x) 1
3 x 5 3 2 x 4
f (x )
has a maximum or a minimum value and find that value. 3. BUSINESS Khalid charges $10 for a one-year subscription to his on-line newsletter. Khalid currently has 600 subscribers and he estimates that for each $1 decrease in the subscription price, he would gain 100 new subscribers. What subscription price will maximize Khalid’s income? If he charges this price, how much income should Khalid expect? For Questions 4–6, solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 4. 0.5x2 9 4.5x
O
x
2. 3. 4. y 2 O 2
x
5. 2 x
3
3
1 2 x 3
5. y 2 O 2
x
6. 4x(x
3)
9
6. y O
x
7. Solve 18x2
15
39x by factoring.
2 and 1.75 as its roots. 3
7. 8.
8. Write a quadratic equation with Write the equation in the form ax2 a, b, and c are integers.
bx
c
0, where
9. If the roots of an equation are 5 and 3, what is the equation of the axis of symmetry?
© Glencoe/McGraw-Hill
9.
Glencoe Algebra 2
365
Assessment
NAME
DATE
PERIOD
6
Chapter 6 Test, Form 3
1.44 by using the
(continued)
10. Solve 4x2 2x 0.25 Square Root Property.
10.
For Questions 11 and 12, solve each equation by completing the square. 11. 2x2 12. x2
5 x 2
2 3
0 0.5 3x 1 0 by using the
11. 12. 13. 14.
2.5x
13. Find the exact solutions to 1 x2 4 Quadratic Formula.
14. Find the value of the discriminant for 3x(0.2x 0.4) 1 0.9. Then describe the number and type of roots for the equation. 15. Find all values of k such that x2 complex roots.
3 x 5 1 2 2
kx
1
0 has two
15. 16.
16. Write an equation of the parabola with equation y Then identify the vertex, axis of symmetry, and direction of opening of your function. 17. PHYSICS The height h (in feet) of a certain aircraft t seconds after it leaves the ground is modeled by h(t) 9.1t2 591.5t 20,388.125. Write the function in vertex form and find the maximum height reached by the aircraft. 18. Write an equation for the parabola that has the same vertex as y 19. Graph y
1 2 x 3 83 and x-intercept 1. 2 5 , translated 4 units left and 2 units up. 2
17.
18. y 6x 2x)
(x2
5.25.
19.
O
x
20. Solve x
7 (x 2
1)2
0.
3 2i 5 . 4
20. B:
Bonus Write a quadratic equation with roots Write the equation in the form ax2 where a, b, and c are integers.
© Glencoe/McGraw-Hill
bx
c
0,
366
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Open-Ended Assessment
Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem. 1. Mr. Moseley asked the students in his Algebra class to work in groups to solve (x 3)2 25, stating that each student in the first group to solve the equation correctly would earn five bonus points on the next quiz. Mi-Ling’s group solved the equation using the Square Root Property. Emilia’s group used the Quadratic Formula to find the solutions. In which group would you prefer to be? Explain your reasoning. 2. The next day, Mr. Moseley had his students work in pairs to review for their chapter exam. He asked each student to write a practice problem for his or her partner. Len wrote the following problem for his partner, Jocelyn: Write an equation for the parabola whose vertex is ( 3, 4), that passes through ( 1, 0), and that opens downward. a. Jocelyn had trouble solving Len’s problem. Explain why. b. How would you change Len’s problem? c. Make the change you suggested in part b and complete the problem. 3. a. Write a quadratic function in vertex form whose maximum value is 8. b. Write a quadratic function that transforms the graph of your function from part a so that it is shifted horizontally. Explain the change you made and describe the transformation that results from this change. 4. When asked to write f(x) 2x2 12x 5 in vertex form, Joseph wrote: f(x) 2x2 12x 5 Step 1 f(x) 2(x2 6x) 5 Step 2 f(x) 2(x2 6x 9) 5 9 Step 3 f(x) 2(x 3)2 4 Is Joseph’s answer correct? Explain your reasoning. 5. The graph of y x2 4x 4 is shown. Susan used this graph to solve three quadratic inequalities. Her three solutions are given below. Replace each ! with an inequality symbol ( , , , ) so that each solution is correct. Explain your reasoning for each. a. The solution of x2 4x 4 ! 0 is {x x 2 or x 2}. 2 b. The solution of x 4x 4 ! 0 is . 2 c. The solution of x 4x 4 ! 0 is all real numbers.
© Glencoe/McGraw-Hill y O
x
367
Glencoe Algebra 2
Assessment
NAME
DATE
PERIOD SCORE
6
Chapter 6 Vocabulary Test/Review maximum value minimum value parabola quadratic equation Quadratic Formula quadratic function quadratic inequality quadratic term roots Square Root Property
axis of symmetry completing the square constant term discriminant linear term
vertex vertex form Zero Product Property zeros
Write whether each sentence is true or false. If false, replace the underlined word or words to make a true sentence. 1. The Square Root Property is used when a quadratic equation is solved by factoring. 2. In f(x) 3. 2x2 3x 3x2 4 2x 5, the linear term is 5. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
0 is an example of a quadratic equation.
4. The solutions of a quadratic equation are called its zeros. 5. The quadratic function y vertex form. 2(x 3)2 1 is written in
6. If a parabola opens upward, the y-coordinate of the vertex is the maximum value. 7. In f(x) x2 2x 1, the constant term is x2.
8. It is necessary to identify the values of a, b, and c in order to solve a quadratic equation by completing the square. 9. The highest or lowest point on a parabola is called the vertex. 10. In the Quadratic Formula, the expression b2 called the quadratic term. In your own words— Define each term. 11. parabola 4ac is
12. axis of symmetry
© Glencoe/McGraw-Hill
368
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Quiz
(Lessons 6–1 and 6–2) x2 2x 3. 1.
For Questions 1 and 2, consider f(x)
1. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. 2. Graph the function, labeling the y-intercept, vertex, and axis of symmetry.
f (x )
2.
O
x
3. Determine whether f(x) 2x2 8x 9 has a maximum or a minimum value and find that value. Solve each equation. If exact roots cannot be found, state the consecutive integers between which the roots are located. 4. x2 5. x2 2x 4x 3 7 0
3.
4. 5.
NAME
DATE
PERIOD SCORE
6
Chapter 6 Quiz
(Lessons 6–3 and 6–4)
10 13x 2. x2 4x 45
For Questions 1 and 2, solve each equation by factoring. 1. 3x2 3. STANDARDIZED TEST PRACTICE What is the integer solution of 6x2 9 21x? A. 3 B. 3 C. 1
2
1. 2. 3. 4. 5. 6.
D. 2
Write a quadratic equation with the given roots. Write the equation in the form ax2 bx c 0, where a, b, and c are integers. 4. 6 and 2 5. 2 and
3
4
Solve each equation by using the Square Root Property. 6. x2 8. 25x2 9. x2 8x 16 20x 10x 11 4 36 3 10. x2 4x 29 11 7. x2 2x 1 45
7. 8. 9. 10.
Glencoe Algebra 2
Solve each equation by completing the square.
© Glencoe/McGraw-Hill
369
Assessment
NAME
DATE
PERIOD SCORE
6
Chapter 6 Quiz
(Lessons 6–5 and 6–6)
1. 2. 3.
1. Solve x2 4x 1 by using the Quadratic Formula. Find exact solutions. 2. Find the value of the discriminant for 3x2 6x 11. Then describe the number and type of roots for the equation. 3. Graph y (x 2)2 axis of symmetry. 1. Show and label the vertex and
y
x
O
4. Write y
3x2
12x
6 in vertex form.
4. 5.
5. Write an equation for the parabola whose vertex is at ( 5, 0) and passes through (0, 50).
NAME
DATE
PERIOD SCORE
6
Chapter 6 Quiz
(Lesson 6–7)
1 (x 3
1. Graph y
2)2
3.
1.
y
O
x
2. Use the graph of its related function to write the solutions of x2 6x 5 0. 3. Solve 0 x2 4x 3 by graphing.
2. 3.
y
O
x
4. Solve 4x2
1
4x algebraically.
4.
© Glencoe/McGraw-Hill
370
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Mid-Chapter Test
(Lessons 6–1 through 6–4)
Part I Write the letter for the correct answer in the blank at the right of each question.
1. Which function is A. f(x) x2 2x B. f(x) x2 2x C. f(x) x2 x D. f(x) (x 3)2 graphed? 3 3 3 f (x )
O
x
1. 1)(x B. x 5 D. x
5
2. By the Zero Product Property, if (2x A. x C. x 1 or x
1 or x 2
5)
0, then _____.
1 or x 2
5
5 5 2.
1 or x
3. Write a quadratic equation with 7 and 2 as its roots.
B. y D. y
2x2 2x2
9x 9x
35 35
3.
4. The quadratic equation x2 4x 16 is to be solved by completing the square. Which equation would be a step in that solution? A. (x 2)2 20 B. x2 4x 16 0 C. (x 2)2 20 D. (x 2)2 4 6. If exact roots cannot be found, state the 5. Solve x2 6x consecutive integers between which the roots are located. A. 2, 3 B. between 4 and 3; between 2 and C. 3 D. between 5 and 4; between 2 and
4.
1 1
5.
Part II
6. Solve x2 4x 3 0 by graphing. 6. y O
x
7. Determine whether f(x)
has a maximum or a minimum value and find that value. For Questions 8 and 9, solve each equation by factoring. 8. x2 7x 18 6x 1 9. 4x2 x 8. 9. 10.
Glencoe Algebra 2
1 2 x 2
x
9
7.
10. Solve 9x2
5 by using the Square Root Property.
© Glencoe/McGraw-Hill
371
Assessment
Write the equation in the form ax2 a, b, and c are integers. A. y 5x2 37x 14 C. y 5x2 37x 14
bx
c
0, where
NAME
DATE
PERIOD
6
Chapter 6 Cumulative Review
(Chapters 1–6)
36 4 (5 7)2. (Lesson 1-1) 1. 2. 3.
1. Find the value of 12
2. Find the slope of the line that is parallel to the line with equation 3x 4y 10. (Lesson 2-3) 3. Describe the system 2x 3y 21 and y 5
2 x as 3
consistent and independent, consistent and dependent, or inconsistent. (Lesson 3-1) 4. Find the coordinates of the vertices of the figure formed by the system of inequalities. (Lesson 3-3) x 2 x y 6 y 2 x y 2 5. Find the value of 6. Solve 4 2 1 3 a b 5 12 . (Lesson 4-5) 6 4 11 by using inverse matrices. 13 4.
5. 6. 7.
(Lesson 4-8)
7. Use synthetic division to find (2x4 5x3 x2 10x 4) (x 8. Use a calculator to approximate decimal places. (Lesson 5-5) 9. Solve x 2 1 8. (Lesson 5-8)
3). (Lesson 5-3)
4
983 to three
8. 9. 10.
10. PHYSICS An object is thrown straight up from the top of a 100-foot platform at a velocity of 48 feet per second. The height h(t) of the object t seconds after being thrown is given by h(t) 16t2 48t 100. Find the maximum height reached by the object and the time it takes to achieve this height. (Lesson 6-1) 11. Solve x2 2x 3 by graphing. (Lesson 6-2)
11. y O
x
12. Solve 4x2
4x
24 by factoring. (Lesson 6-3) 0.
12. 13.
13. Find the value of the discriminant for 7x2 5x 1 Then describe the number and type of roots for the equation. (Lesson 6-5) 14. Write y x2 7x 5 in vertex form. (Lesson 6-6)
14.
Glencoe Algebra 2
© Glencoe/McGraw-Hill
372
NAME
DATE
PERIOD
6
Standardized Test Practice
(Chapters 1–6)
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
1. If a b 3 , then 8a equals which of the following? 2 A. 16b B. 12b C. 3b 2
D. 8 b
3
1.
A
B
C
D
2. 20% of 3 yards is how many fifths of 9 feet? E. 1 F. 6 G. 10 3. If u v and t 0, which of the following are true? I. ut vt II. u t v t III. u t v A. I only B. III only C. I and II only D. I, II, and III 4. Which of the following is the greatest? E. 2
3
H. 15 t
2.
E
F
G
H
3.
A
B
C
D
F. 7
9
G. 10
15
H. 8
11
4.
E
F
G
H
5. If 2a 3b represents the perimeter of a rectangle and a 2b represents its width, the length is ______. A. 7b B. b C. 7b
2
D. 14b
5.
A
B
C
D
6. In the figure, what is the area of the shaded region? E. 30 F. 36 G. 54 H. 27
6 3
15
6.
E
F
G
H
7. Mr. Salazár rented a car for d days. The rental agency charged x dollars per day plus c cents per mile for the model he selected. When Mr. Salazár returned the car, he paid a total of T dollars. In terms of d, x, c, and T, how many miles did he drive? A. T (xd c) B. T xd c
C.
T xd c
D. T
c
xd
7.
A
B
C
D
8. If P(3, 2) and Q(7, 10) are the endpoints of the diameter of a circle, what is the area of the circle? E. 2 5 F. 80 G. 4 5 H. 20 9. If (x y)2 A. 120 100 and xy 20, what is the value of x2 y2? B. 140 C. 80 D. 60
8.
E
F
G
H
9.
A
B
C
D
10. The tenth term in the sequence 7, 12, 19, 28, … is ______. E. 124 F. 103 G. 57 H. 147
10.
E
F
G
H
© Glencoe/McGraw-Hill
373
Glencoe Algebra 2
Assessment
NAME
DATE
PERIOD
6
Standardized Test Practice
Part 2: Grid In
(continued)
Instructions: Enter your answer by writing each digit of the answer in a column box and then shading in the appropriate oval that corresponds to that entry.
11. If t2
6t
9, what is the value of t
1 2 ? 2
11.
. 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 . 0 1 2 3 4 5 6 7 8 9
12.
. 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 . 0 1 2 3 4 5 6 7 8 9
12. All four walls of a rectangular room that is 14 feet wide, 20 feet long, and 8 feet high, are to be painted. What is the minimum cost of paint if one gallon covers at most 130 square feet and the paint costs $22 per gallon? 13. The bar graph shows the distribution of votes among the candidates for senior class president. If 220 seniors voted in all, how many students voted for either Theo or Pam? 14. Find the median of x, 2x x
Percent of votes received 50 40 30 20 10 0 Theo Pam Ana Joey Candidates
13.
. 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 . 0 1 2 3 4 5 6 7 8 9
14.
. 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 . 0 1 2 3 4 5 6 7 8 9
1, x
22 if the mean of this set of numbers is 83.
2
13, 45, and
Part 3: Quantitative Comparison
Instructions: Compare the quantities in columns A and B. Shade in A if the quantity in column A is greater; B if the quantity in column B is greater; C if the quantities are equal; or D if the relationship cannot be determined from the information given.
Column A 15. x x˚ y˚ y˚
Column B z˚ 15. z A
B
C
D
16. a c
1
a
c c a
16.
A
B
C
D
17. x © Glencoe/McGraw-Hill
x2 y2
25 16 y 17.
A
B
C
D
374
Glencoe Algebra 2
NAME
DATE
PERIOD
6
Standardized Test Practice
Student Record Sheet
(Use with pages 342–343 of the Student Edition.)
Part 1 Multiple Choice
Select the best answer from the choices given and fill in the corresponding oval. 1 2 3
A B C D
4 5 6
A
B
C
D
7 8
A
B
C
D
9 10
A
B
C
D
A
B
C
D
A
B
C
D
A
B
C
D
A
B
C
D
A
B
C
D
A
B
C
D
Part 2 Short Response/Grid In
Solve the problem and write your answer in the blank. For Questions 14–20, also enter your answer by writing each number or symbol in a box. Then fill in the corresponding oval for that number or symbol. 11 12
.
15
/ . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9
17
19
. 0 1 2 3 4 5 6 7 8 9
. 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
. 0 1 2 3 4 5 6 7 8 9
. 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
. 0 1 2 3 4 5 6 7 8 9
13
14
/ . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9
16
18
20
. 1 2 3 4 5 6 7 8 9
. 0 1 2 3 4 5 6 7 8 9
. 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
. 0 1 2 3 4 5 6 7 8 9
. 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
. 0 1 2 3 4 5 6 7 8 9
. 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
. 0 1 2 3 4 5 6 7 8 9
Part 3 Quantitative Comparison
Select the best answer from the choices given and fill in the corresponding oval. 21 22
A B C D
23 24
A
B
C
D
25 26
A
B
C
D
27 28
A
B
C
D
A
B
C
D
A
B
C
D
A
B
C
D
A
B
C
D
© Glencoe/McGraw-Hill
A1
Glencoe Algebra 2
Answers
1 2 3 4 5 6 7 8 9
( 3) or 3 . The x-coordinate of the vertex is 3 . 2(1) 2 2 Next make a table of values for x near 3 . 2 f (x )
x
x2
3x
5
f(x)
(x, f(x))
Lesson 6-1
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-1
(continued)
Study Guide and Intervention
6-1
Graphing Quadratic Functions
Maximum and Minimum Values ax 2 bx b ; 2a
Study Guide and Intervention
Graphing Quadratic Functions c, where a Maximum or Minimum Value of a Quadratic Function 0
Graph Quadratic Functions
The y-coordinate of the vertex of a quadratic function is the maximum or minimum value of the function.
Quadratic Function
A function defined by an equation of the form f (x)
Glencoe/McGraw-Hill
The graph of f(x) ax 2 bx c, where a 0, opens up and has a minimum when a 0. The graph opens down and has a maximum when a 0. b 2a
Graph of a Quadratic Function
A parabola with these characteristics: y intercept: c; axis of symmetry: x
x-coordinate of vertex:
Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex for the graph of f(x) x2 3x 5. Use this information to graph the function. a. f(x) 3x 2 6x 7 For this function, a 3 and b 6. Since a 0, the graph opens up, and the function has a minimum value. The minimum value is the y-coordinate of the vertex. The x-coordinate of the b 6 vertex is 1.
2a 2(3) 2a
Example
Example Determine whether each function has a maximum or minimum value. Then find the maximum or minimum value of each function.
a
1, b
3, and c
5, so the y-intercept is 5. The equation of the axis of symmetry is
x
b. f(x) 100 2x x 2 For this function, a 1 and b 2. Since a 0, the graph opens down, and the function has a maximum value. The maximum value is the y-coordinate of the vertex. The x-coordinate of the vertex b 2 is 1.
2( 1)
0
02
3(0)
5
5
(0, 5)
1
12
3(1)
5
3
(1, 3)
Answers
3 2
O
3 2 2
3 x 3 2
5
11 4
3 11 , 2 4
Evaluate the function at x 1 to find the minimum value. f(1) 3(1)2 6(1) 7 4, so the minimum value of the function is 4.
Evaluate the function at x 1 to find the maximum value. f( 1) 100 2( 1) ( 1)2 101, so the minimum value of the function is 101.
A2
Exercises
1. f(x) 2x2 x 10
2
22
3(2)
5
3
(2, 3)
3
32
3(3)
5
5
(3, 5)
Exercises min., 9 7
8
4. f(x) 16
Determine whether each function has a maximum or minimum value. Then find the maximum or minimum value of each function. 2. f(x) x2 4x 7 3. f(x) 3x2 3x 1
(Lesson 6-1)
min.,
4x x2 5. f(x) x2
11
7x 11
min., 1
4
6. f(x) x2 6x 4
For Exercises 1–3, complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. 1. f(x) x2 6x 8 2. f(x) x2 2x 2 3. f(x) 2x2 4x 3
8, x x f (x) f (x ) f (x )
12 4 4 –4 O (1, 1) 4 8 8x 8 (–1, 3) 4 –8 –4 –8 –4 O
3,
1 3 2 2 1 f (x) 1 3 3 9 0 2 1 x 1 0 2 3
3
2, x
1,
1
3, x
1, 1
max., 20
7. f(x) x2 5x 2
min.,
8. f(x) 20
5 4
6x x2
max., 5
9. f(x) 4x2 x 3
x
3
2
1
4
f (x)
1
0
3
0
min.,
10. f(x)
12
f (x )
17 4 x2 4x 10
max., 29
11. f(x) x2 10x 5
min., 2 15 16
12. f(x) 6x2 12x 21
8
4
max., 14 x min.,
13. f(x) 25x2 100x 350 14. f(x)
20
0.5x2 0.3x 1.4
max., 27
15. f(x) x2 2 x 4
–8
–4 O (–3, –1) –4
4
x
8
min., 250
313
Glencoe Algebra 2
©
min.,
Glencoe/McGraw-Hill
1.445
314
max.,
7 31
32
Glencoe Algebra 2
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
Lesson 6-1
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-1
(Average)
Skills Practice
6-1
Graphing Quadratic Functions
Practice
Graphing Quadratic Functions
For each quadratic function, find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. 2. f(x) 1 6x 15 x2 3. f(x) x2
Glencoe/McGraw-Hill
1; x
5. f(x) 10x 5 8x 7 8x 15 x2 6. f(x) 1. f(x) 2. f(x) 2x2 x2 x2 4x
1. f(x)
3x2
0; x
0; 0
0; 0
15; x
3; 3
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. 12 3. f(x) 2x2 2x 1
4. f(x)
2x2
11
11; x x f (x) 15 3
16 12 8 4 O 2 8x –6 –4 –2
0; 0
0 1 3 15 2 4 6 8 x 6
5; x
5; 5
7; x
2; 2
15; x
4; 4
12; x
2;
4
2
2 0 2
1; x x f (x) 5 f (x )
0.5; 0.5
1 0 0.5 1 1 0.5 1 f (x )
2 5
f (x) 0 12 16 12 0
(–2, 16)
f (x )
16 12 8 4
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. 8. f(x) 4x 4 6x 2 0 1 0 8 3 4 6
4 6 (4, –1)
7. f(x) 9. f(x) 8
2x2
x2
x2
0; x x f (x) 16 4 f (x )
16 12 8 4 (2, 0) O
0; 0
2 0 0 f (x )
4; x
2 4 16 f (x) 8 4 6 x 0
2; 2
8; x
3; 3
Answers
x
2
1 0
1
2
(0.5, 0.5) O 2x O
x
f (x)
8
2 0
2
8
A3
4. f(x)
f (x )
(0, 0)
O
x
Determine whether each function has a maximum or a minimum value. Then find the maximum or minimum value of each function. x2 2x 8 5. f(x) x2 6x 14 6. v(x) x2 14x 57
min.; x 2 4 6x (3, –1)
(Lesson 6-1)
9
7. f(x) 2x2 4x 6
min.; 5
8. f(x) x2 4x 1
max.;
9. f(x)
8
2 2 x 3
–2
O
8x
24
min.;
8
max.; 3
max.; 0
Determine whether each function has a maximum or a minimum value. Then find the maximum or minimum value of each function. 11. f(x) 2x 8x2 12. f(x) x2
10. f(x)
6x2
10. GRAVITATION From 4 feet above a swimming pool, Susan throws a ball upward with a velocity of 32 feet per second. The height h(t) of the ball t seconds after Susan throws it is given by h(t) 16t2 32t 4. Find the maximum height reached by the ball and the time that this height is reached. 20 ft; 1 s
min.; 0
14. f(x) 4x 1 x2 15. f(x) x2
max.; 0
min.;
1
2x 3
13. f(x)
x2
2x
15
min.; 14
17. f(x) 12x 3 3x2
max.; 3
min.;
18. f(x)
4
2x2 4x 1
11. HEALTH CLUBS Last year, the SportsTime Athletic Club charged $20 to participate in an aerobics class. Seventy people attended the classes. The club wants to increase the class price this year. They expect to lose one customer for each $1 increase in the price. a. What price should the club charge to maximize the income from the aerobics classes?
16. f(x)
2x2
4x
3
$45 min.; 1
b. What is the maximum income the SportsTime Athletic Club can expect to make?
max.;
1
min.;
9
$2025
Glencoe Algebra 2
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Glencoe Algebra 2
315
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Glencoe/McGraw-Hill
Glencoe/McGraw-Hill
316
Glencoe Algebra 2
Answers
Lesson 6-1
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-1
Finding the Axis of Symmetry of a Parabola
As you know, if f(x) bx b b2 2a 4ac
Reading to Learn Mathematics
6-1
Enrichment
Graphing Quadratic Functions ax2 c is a quadratic function, the values of x and f(x) Pre-Activity that make f(x) equal to zero are The average of these two number values is 2x2 5x value when x of the graph of f (x) passes through the point where the maximum or minimum occurs, the axis of symmetry has the equation x b . 2a b . Since the axis of symmetry 2a
O
How can income from a rock concert be maximized?
Read the introduction to Lesson 6-1 at the top of page 286 in your textbook. b b2 2a 4ac
Glencoe/McGraw-Hill
.
b . 2a
• Based on the graph in your textbook, for what ticket price is the income the greatest? $40
• Use the graph to estimate the maximum income. about $72,000 The function f(x) has its maximum or minimum
Reading the Lesson
3, 2x2 is the term, term, and 3 is the 4 x 3x2, a ,b , and
b x = – –– 2a
1. a. For the quadratic function f(x)
5x is the
linear 3 1
0.
quadratic constant term.
f(x) = ax 2 + bx + c
x b b (– –– , f (– –– (( 2a 2a
b. For the quadratic function f(x)
c ax2 bx . c, where a
4 parabola
.
2. Consider the quadratic function f(x)
a. The graph of this function is a
Example
Use x b . 2a
Find the vertex and axis of symmetry for f(x)
5x 2
10x
7.
b. The y-intercept is .
c x
. and the function has a
Answers
c. The axis of symmetry is the line
b 2a x 10 2(5)
1
The x-coordinate of the vertex is 5x2 7 10x 12 b , or x 2a
1. 7.
A4 upward value.
d. If a
0, then the graph opens
minimum downward and the function has a using x 1. f(x)
O (0, –1)
Substitute x 1 in f(x) f( 1) 5( 1)2 10( 1) The vertex is ( 1, 12). The axis of symmetry is x
1.
e. If a value.
(–2, 4)
0, then the graph opens
(Lesson 6-1)
maximum f (x )
Find the vertex and axis of symmetry for the graph of each function b . 2a
3. Refer to the graph at the right as you complete the following sentences.
a. The curve is called a . x parabola vertex
. 1 is 1),
x2
4x
8 (2,
12); x
2
2. g(x)
4x2
8x
3 ( 1, 7); x
1
b. The line x
2 is called the axis of symmetry .
c. The point ( 2, 4) is called the
d. Because the graph contains the point (0, .
3. y
x2
8x
3 (4, 19); x
4
4. f(x)
2x2
6x
5
3 1 , ;x 2 2
3 2
the
y-intercept
Helping You Remember
5. A(x) x2 12x 36 ( 6, 0); x
4. How can you remember the way to use the x2 term of a quadratic function to tell whether the function has a maximum or a minimum value? Sample answer:
6
6. k(x)
2x2
2x
6
1 , 2
5
1 ;x 2
1 2
Remember that the graph of f(x) x 2 (with a 0) is a U-shaped curve that opens up and has a minimum. The graph of g(x) x 2 (with a 0) is just the opposite. It opens down and has a maximum.
Glencoe Algebra 2
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Glencoe Algebra 2
317
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Glencoe/McGraw-Hill
Glencoe/McGraw-Hill
318
Glencoe Algebra 2
A5
2. x2 4x 5 0 5,
Exercises
1
f (x ) x
Solve each equation by graphing. 3. x2 5x 4 0 1, 4
Lesson 6-2
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-2
Solving Quadratic Equations by Graphing
Study Guide and Intervention
6-2
Study Guide and Intervention
(continued)
Solving Quadratic Equations by Graphing
A quadratic equation has the form solution(s) of the equation, or the zero(s) of the related quadratic function ax 2 bx c 0, where a 0.
Solve Quadratic Equations
Estimate Solutions Often, you may not be able to find exact solutions to quadratic equations by graphing. But you can use the graph to estimate solutions. Example Solve x 2 2x 2 0 by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
The equation of the axis of symmetry of the related function is x f (x )
O
Quadratic Equation
Glencoe/McGraw-Hill
6 x2 x f (x) x Roots of a Quadratic Equation
The zeros of a quadratic function are the x-intercepts of its graph. Therefore, finding the x-intercepts is one way of solving the related quadratic equation. 0 by graphing. 6.
1 , and the equation of the 2
1 2 3 2 1 x 1 0 1 2 3
f (x )
Example
Solve x2
x
2 2(1)
1, so the vertex has x-coordinate 1. Make a table of values.
O
Graph the related function f(x)
b The x-coordinate of the vertex is 2a 1 axis of symmetry is x . 2 1 . 2
x
Make a table of values using x-values around
The x-intercepts of the graph are between 2 and 3 and between 0 and 1. So one solution is between 2 and 3, and the other solution is between 0 and 1.
x
1
1 2
0
1
2
Exercises
Solve the equations by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 1. x2 4x 2 0 2. x2 6x 6 0 3. x2 4x 2 0
f(x)
6
6
1 4
6
4
0
Answers
From the table and the graph, we can see that the zeros of the function are 2 and
3.
between 0 and 1; between 3 and 4 f (x ) f (x )
O
between between
2 and 5 and f (x )
1; 4
between between
1 and 0; 4 and 3 f (x )
1. x2
2x
8
0 2,
4
f (x )
O
x
(Lesson 6-2)
O
O
x x
O
x
O
x
4. 5. 4x f (x ) f (x )
x2
2x
4
0
5. 2x2
12x
17
0
6.
1 2 x 2
x
5 2
0
4. 6 0 4x 1
x2 6.
10x
21
0
x2
4x2
0
f (x )
between 3 and 4; between 2 and 1 f (x )
between 2 and 3; between 3 and 4 f (x )
between 2 and between 3 and 4 f (x )
1;
O
x
O O
x
O O
x x
x
O
x
3, 7
319
no real solutions
1 2
Glencoe Algebra 2
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Glencoe Algebra 2
©
Glencoe/McGraw-Hill
Glencoe/McGraw-Hill
320
Glencoe Algebra 2
Answers
A6
4 –6 –4 –2 O
f (x ) f (x )
O
f (x ) x
Lesson 6-2
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-2
(Average)
Skills Practice
6-2
Solving Quadratic Equations By Graphing
Use the related graph of each equation to determine its solutions. 3 f (x ) x
O
Practice
Solving Quadratic Equations By Graphing
2. 6x x2 6x O 9
Use the related graph of each equation to determine its solutions. x2 9 f (x ) f (x )
3x 2 3
1. x2 0 f (x ) f (x )
2x f (x )
3
0
3. 3x2 4x 0
1. 3 x 0 3 0 3x
3x2
2. 3x2
3. x2
2 f (x )
0
Glencoe/McGraw-Hill x f (x ) f (x )
O 3x 2 4x 3 3x 2
f (x )
O
x x x
3
f (x )
x2
3x
2 O
f (x ) x
x2
2x
3
O
x
1, 1 3
4. 6x 5 0 2x2 5. x2 10x
no real solutions
1, 2
3, 1 between 0 and 1; between 4 and 3 f (x )
8 2x 2 6x 12 5
no real solutions 6, 4
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 24 0 6. 2x2 x 6 0
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 5. 2x 4 0 6x 4 0 x2 6. x2 f (x )
between 2 f (x )
O
2 and f (x ) x
1,
4. x2
6x
5
0
Answers
1, 5
no real solutions
between 0 and 1; between 5 and 6 x f (x )
O
x2
2x 4
f (x ) x
x2
10x
O 24
x
f (x )
2x 2
x
6
O
x
Use a quadratic equation to find two real numbers that satisfy each situation, or show that no such numbers exist. 7. Their sum is 1, and their product is f (x ) f (x ) x2 x
6
(Lesson 6-2)
f (x )
x2 f (x )
6x 4
6x
5
x2
6.
8. Their sum is 5, and their product is 8.
x2 x 3, 2
6
0;
f (x )
O
f (x )
x2
5x
x 8
Use a quadratic equation to find two real numbers that satisfy each situation, or show that no such numbers exist. 8. Their sum is 0, and their product is 36.
O
x 2 5x 8 0; no such real numbers exist
7. Their sum is
4, and their product is 0.
x
x2 36
36 24
4x f (x )
0; 0,
4
x2 0; 6, 6
f (x )
x2
4x
f (x )
f (x )
x2
36
For Exercises 9 and 10, use the formula h(t) v0t 16t 2, where h(t) is the height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. 9. BASEBALL Marta throws a baseball with an initial upward velocity of 60 feet per second. Ignoring Marta’s height, how long after she releases the ball will it hit the ground? 3.75 s 10. VOLCANOES A volcanic eruption blasts a boulder upward with an initial velocity of 240 feet per second. How long will it take the boulder to hit the ground if it lands at the same elevation from which it was ejected? 15 s
Glencoe Algebra 2
©
O –12 –6 O 6 12 x
x
12
Glencoe Algebra 2
321
©
Glencoe/McGraw-Hill
Glencoe/McGraw-Hill
322
Glencoe Algebra 2
Lesson 6-2
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-2
Graphing Absolute Value Equations
You can solve absolute value equations in much the same way you solved quadratic equations. Graph the related absolute value function for each equation using a graphing calculator. Then use the ZERO feature in the CALC menu to find its real solutions, if any. Recall that solutions are points where the graph intersects the x-axis. For each equation, make a sketch of the related graph and find the solutions rounded to the nearest hundredth. 1. | x 5| 0 2. | 4x 3| 5 0 3. | x 7|
Reading to Learn Mathematics
6-2
Enrichment
Solving Quadratic Equations by Graphing
Pre-Activity
How does a quadratic function model a free-fall ride?
Glencoe/McGraw-Hill y Read the introduction to Lesson 6-2 at the top of page 294 in your textbook.
Write a quadratic function that describes the height of a ball t seconds after 16t 2 125 it is dropped from a height of 125 feet. h(t)
Reading the Lesson
1. The graph of the quadratic function f(x) x2 x 6 is shown at the right. Use the graph to find the solutions of the quadratic equation x2 x 6 0. 2 and 3
0
5
No solutions
7
O
x
Answers
2. Sketch a graph to illustrate each situation. 4. | x 3| 8
A7
11, 5 y y
O
0
5.
|x
9, 3
3|
6
0
6. | x
2|
3
0
a. A parabola that opens b. A parabola that opens downward and represents a upward and represents a quadratic function with two quadratic function with real zeros, both of which are exactly one real zero. The negative numbers. zero is a positive number.
c. A parabola that opens downward and represents a quadratic function with no real zeros.
1, 5
y
(Lesson 6-2)
O
x
x
O
x
7. | 3x
4|
2
8. | x
12 |
10
9. | x |
3
0
Helping You Remember
2,
2 3
22,
2
3, 3
3. Think of a memory aid that can help you recall what is meant by the zeros of a quadratic function.
Sample answer: The basic facts about a subject are sometimes called the ABCs. In the case of zeros, the ABCs are the XYZs, because the zeros are the x-values that make the y-values equal to zero.
10. Explain how solving absolute value equations algebraically and finding zeros of absolute value functions graphically are related.
Sample answer: values of x when solving algebraically are the x-intercepts (or zeros) of the function when graphed.
Glencoe Algebra 2
323
©
Glencoe/McGraw-Hill
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
324
Glencoe Algebra 2
Answers
0, 7
7. x2
6 x 30 0
0, 9
2
8. 2x2 x 3 0
0, 2
3
9. x2 14x 33 0
Lesson 6-3
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Study Guide and Intervention
Solving Quadratic Equations by Factoring
When you use factoring to solve a quadratic equation,
0, then either a 0 or b 0, or both a and b 0.
6-3
Study Guide and Intervention
q)
(continued)
Solving Quadratic Equations by Factoring
Write Quadratic Equations
(x p)(x To write a quadratic equation with roots p and q, let 0. Then multiply using FOIL.
Solve Equations by Factoring you use the following property.
Zero Product Property
For any real numbers a and b, if ab
a. 3x2
Solve each equation by factoring. 15x b. 4x2 5x 21 3x2 15x Original equation 4x2 5x 21 Original equation 3x2 15x 0 Subtract 15x from both sides. 4x2 5x 21 0 Subtract 21 from both sides. 3x(x 5) 0 Factor the binomial. (4x 7)(x 3) 0 Factor the trinomial. 3x 0 or x 5 0 Zero Product Property 4x 7 0 or x 3 0 Zero Product Property 7 x 0 or x 5 Solve each equation. x or x 3 Solve each equation. 4 The solution set is {0, 5}. 7 The solution set is ,3 .
4
Example
Example Write a quadratic equation with the given roots. Write the equation in the form ax2 bx c 0. a. 3, 5 7 1 b. , 8 3 (x p)(x q) 0 Write the pattern. (x p)(x q) 0 (x 3)[x ( 5)] 0 Replace p with 3, q with 5. 1 7 (x 3)(x 5) 0 Simplify. x x 0 8 3 x2 2x 15 0 Use FOIL. 1 7 x x 0 The equation x2 2x 15 0 has roots
3 and 5.
8 3 (8x 8 24 (8x 7)(3x 24 7) (3x 3 1) 1)
0 24 0 0 13x 7 0 has
Answers
Exercises
Solve each equation by factoring. 1. 6x2 2x 0 2. x2 7x 3. 20x2 25x
24x2
13x
7
The equation 24x2 roots
0, 1
3
7x 4. 6x2
{0, 7}
5. 6x2 27x 0
0,
6. 12x2
5 4
8x 0
7 1 and . 8 3
A8
Glencoe Algebra 2
©
(Lesson 6-3)
Exercises
Write a quadratic equation with the given roots. Write the equation in the form ax2 bx c 0. 1. 3, 4 2. 8, 2 3. 1, 9
{5,
10. 4x2
6}
27x 7 0
3 , 2
11. 3x2
1
29x 10 0
{ 11,
12. 6x2 5x
3}
4 0
x2
4. 5
x 10x
12 25
0 0
x2
5. 10, 7
10x 17x
16 70
0
6.
x2 x2
3 9. 7, 4 4x 2 1 12. 9, 6 6x 2 3 1 15. , 7 5
10x 13x
9 30
0 0
2, 15
1 , 4
13. 12x2
7
8x 1 0
10, 1 3
14. 5x2 28x 12 0 15. 2x2
1 4 , 2 3
250x 5000 0
x2
1 7. ,5 3 3x 2 2 10. 3, 5 5x 2 2 13. , 3 2 3
x2
2 8. 2, 3 3x 2 4 11. , 9 2 9x 5 14. , 4 1 2
0
1 1 , 6 2
16. 2x2 11x 40 0
2 , 5
17. 2x2
6
21x 11 0
{100, 25}
18. 3x2 2x 21 0
14x
5
0
8x
1
4
0
25x
21
0
8,
19. 8x2
5 2
14x 3 0
11, 1
2
20. 6x2 11x 2 0
7 , 3
21. 5x2
3
17x 12 0
17x
6
0
13x
4
0
55x
9
0
3 1 , 2 4
22. 12x2 25x 12 0
2, 1 6
23. 12x2 18x 6 0
3 , 5
24. 7x2
4
36x 5 0
9x 2
7 7 16. , 8 2 16x 2
4
0
8x 2
1 3 17. , 2 4 8x 2
6x
5
0
35x 2
1 1 18. , 8 6
22x
3
0
4 , 3
3 4
1 , 2
1
325
1 ,5 7
Glencoe Algebra 2
42x
49
10x
326
3
0
48x 2
14x
1
0
Glencoe/McGraw-Hill
©
Glencoe/McGraw-Hill
Glencoe Algebra 2
Lesson 6-3
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Skills Practice
Solving Quadratic Equations by Factoring
6-3
Practice
(Average)
Solving Quadratic Equations by Factoring
Solve each equation by factoring. 1. x2 4. x2 4x 6x 4x 12 8 0 {6, 0 {2, 4}
Solve each equation by factoring. 1. x2 64 { 8, 8} 2. x2 100 0 {10,
10}
2}
2. x2 5. x2 8. 7x2
16x 3x
64 2
0 {8} 0 { 2,
3. x2
20x 9x 25
100 14
0 {10} 0 {2, 7}
1} 6. x2
9. x2
3. x2
3x
2
0 {1, 2}
4. x2
4x
3
0 {1, 3}
7.
x2
0 {0, 4}
4 4x 0, 7
11. x2 13. 5x2 15. 2x2 2x 35x 8x
10x {5}
5. x2
2x
3
0 {1,
3}
6. x2
3x
10
0 {5,
2}
10. 10x2 12. x2
9 9x 0, 10
12x 25 2x 24x 8x 36 { 6}
99 { 9, 11} 60 90 0 {3, 4} 0 {9,
7.
x2
6x
5
0 {1, 5}
8.
x2
9x
0 {0, 9}
14.
36x2
5 , 6
1 45 0
5 6 1 , 3
5}
Answers
9.
x2
6x
0 {0, 6}
10. x2
6x
8
0 { 2,
4}
16. 3x2 18. 3x2
1 3}
17. 6x2 19. 15x2 21. 6x2
3 9x 0, 2
19x 5x 6 6 0
0 { 5,
3 , 5 2 3
2 3
11. x2
5x {0,
5}
12. x2
14x
49
0 {7}
20. 3x2
4 2,
2 3
3 , 2
13. x2
6
5x {2, 3}
14. x2
18x
81 { 9}
Write a quadratic equation with the given roots. Write the equation in the form ax2 bx c 0, where a, b, and c are integers. 22. 7, 2 23. 0, 3 24. 5, 8
A9
Glencoe Algebra 2
(Lesson 6-3)
15. x2
4x
21 { 3, 7}
16. 2x2
5x
3
0
1 , 2
0
3
2 ,5 3
x2
25. 7,
9x
8
14 56
0
26.
x2
6,
3x
3
0 18 0
x2
27. 3, 4
3x x
7 2
40 12
0 0
17. 4x2
5x
6
0
3 , 4
2
18. 3x2
13x
10
x2
1 28. 1, 2 2x 2 1 31. , 3
15x
0
x2
1 29. , 2 3 3x 2 1 32. 4, 3 3x 2
9x
x2
30. 0,
Write a quadratic equation with the given roots. Write the equation in the form ax2 bx c 0, where a, b, and c are integers. 19. 1, 4 x 2
3x
3
1
0
7x
2
0
33.
2x 2
2 , 3
7x
4 5
0
5x 7x
4
0 10 3 0 0
20. 6,
9 x2
3x 7x 0 2x
54
0
3x 2 24, 26
8x
3
0
13x
4
0
15x 2
22x
8
0
21.
2,
1 , 3
5 x2
22. 0, 7 x 2
1 3 , 2 4
34. NUMBER THEORY Find two consecutive even positive integers whose product is 624. 35. NUMBER THEORY Find two consecutive odd positive integers whose product is 323.
23.
3 3x 2
10x
24.
8x 2
3
0
17, 19
36. GEOMETRY The length of a rectangle is 2 feet more than its width. Find the dimensions of the rectangle if its area is 63 square feet. 7 ft by 9 ft 37. PHOTOGRAPHY The length and width of a 6-inch by 8-inch photograph are reduced by the same amount to make a new photograph whose area is half that of the original. By how many inches will the dimensions of the photograph have to be reduced? 2 in.
Glencoe Algebra 2
©
25. Find two consecutive integers whose product is 272. 16, 17
©
Glencoe/McGraw-Hill
327
Glencoe/McGraw-Hill
328
Glencoe Algebra 2
Answers
Marla (x 7)(x x2 2x
5) 35
0 0
Rosa (x 7)(x x2 2x
5) 35
0 0
Larry (x 7)(x x2 2x
5) 35
0 0
Lesson 6-3
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Reading to Learn Mathematics
Solving Quadratic Equations by Factoring
How is the Zero Product Property used in geometry? Read the introduction to Lesson 6-3 at the top of page 301 in your textbook. What does the expression x(x 5) mean in this situation?
6-3
Enrichment
Pre-Activity
Euler’s Formula for Prime Numbers
Many mathematicians have searched for a formula that would generate prime numbers. One such formula was proposed by Euler and uses a quadratic polynomial, x2 x 41. Find the values of x2 x 41 for the given values of x. State whether each value of the polynomial is or is not a prime number.
It represents the area of the rectangle, since the area is the product of the width and length.
Reading the Lesson
1. The solution of a quadratic equation by factoring is shown below. Give the reason for each step of the solution. x2 x2 (x x 10x 3)(x 3 x 10x 21 7) 0 0 7 x 0 7 21
Original equation
1. x
0
2. x
1
3. x
2
41, prime
43, prime
47, prime
Answers
Add 21 to each side. Factor the trinomial. Zero Product Property Solve each equation.
.
4. x
3
5. x
4
6. x
5
53, prime
61, prime
71, prime
0 or x 3
A10
Glencoe Algebra 2
©
The solution set is
{3, 7}
7. x
6
8. x
17
9. x
28
(Lesson 6-3)
2. On an algebra quiz, students were asked to write a quadratic equation with 7 and 5 as its roots. The work that three students in the class wrote on their papers is shown below.
83, prime
347, prime
853, prime
10. x
29
11. x
30
12. x
35
Who is correct? Rosa Explain the errors in the other two students’ work.
911, prime
971, prime
1301, prime
Sample answer: Marla used the wrong factors. Larry used the correct factors but multiplied them incorrectly.
13. Does the formula produce all prime numbers greater than 40? Give examples in your answer.
Helping You Remember
3. A good way to remember a concept is to represent it in more than one way. Describe an algebraic way and a graphical way to recognize a quadratic equation that has a double root.
No. Among the primes omitted are 59, 67, 73, 79, 89, 101, 103, 107, 109, and 127.
Sample answer: Algebraic: Write the equation in the standard form ax 2 bx c 0 and examine the trinomial. If it is a perfect square trinomial, the quadratic function has a double root. Graphical: Graph the related quadratic function. If the parabola has exactly one x-intercept, then the equation has a double root.
Glencoe/McGraw-Hill
14. Euler’s formula produces primes for many values of x, but it does not work for all of them. Find the first value of x for which the formula fails. (Hint: Try multiples of ten.)
x
©
40 gives 1681, which equals 412.
330
Glencoe Algebra 2
329
Glencoe Algebra 2
Glencoe/McGraw-Hill
Lesson 6-4
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Study Guide and Intervention
Completing the Square x2 6-4
Study Guide and Intervention
Completing the Square
(continued)
Square Root Property Use the following property to solve a quadratic equation that is in the form “perfect square trinomial constant.”
Square Root Property For any real number x if x 2 n, then x n.
Complete the Square
1. Find b . 2
bx, follow these steps.
To complete the square for a quadratic expression of the form b . 2
!
2. Square
!
3. Add
b 2 to x2 2
bx.
Example
a. x2 x2 x
Solve each equation by using the Square Root Property. 8x 16 25 b. 4x2 20x 25 32 4x2 20x 25 8x 16 25 32 (x 4)2 25 (2x 5)2 32 4 2x 5 25 or x 4 25 32 or 2x 5 x 5 4 9 or x 5 4 1 2x 5 4 2 or 2x 5 1}. x
5 4 2 2 5
Example 1 Find the value of c that makes x2 22x c a perfect square trinomial. Then write the trinomial as the square of a binomial.
32 4 2 Step 1 b 22; Step 2 112 121 Step 3 c 121 The trinomial is x2 22x which can be written as (x 11)2. 121, b 2
Solve 2x2 completing the square. 2x2
2x2
Example 2
8x
8x 2
8x
24
0 by
24
24
0
0 2
Original equation Divide each side by 2. x2 4x 12 is not a perfect square.
11
x2 x2
The solution set is {9,
4x 12 x2 4x 4x 4 (x 2)2
0 12 12 16 4
Add 12 to each side.
Answers
4
Since
4 2 2
4, add 4 to each side.
The solution set is
4 2 . 2
Factor the square. Square Root Property Solve each equation.
x 2 x 6 or x 2 The solution set is {6,
2}.
Exercises
Solve each equation by using the Square Root Property. 1. x2 18x 81 49 2. x2 20x 100 64 3. 4x2 4x 1 16
A11
Glencoe Algebra 2
Exercises
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 1. x2 10x c 2. x2 60x c 3. x2 3x c
(Lesson 6-4)
{2, 16}
{ 2,
18}
3 , 2
5 2
25; (x
4. 36x2 12x 1 18 5. 9x2 12x 4 4 6. 25x2 40x 16 28 4. x2 3.2x
5)2 c 900; (x
5. x2
1 x 2
30)2 c 9 ; x 4
6. x2 2.5x
3 2 2 c 1 6
3 2
0, 4
3
4 5
2 7
2.56; (x
1.6)2
1 ; x 16
1 2 4
1.5625; (x
1.25)2
Solve each equation by completing the square. 7. y2 4y 5 0 8. x2 8x 65 0 9. s2 10s 21 0
7. 4x2
28x
49
64
8. 16x2
24x
9
81
9. 100x2
60x
9
121
15 , 2
1 2
3 , 2
3
{ 0.8, 1.4}
1, 5
10. 2x2 3x 1 0
5, 13
11. 2x2 13x 7 0
3, 7
12. 25x2 40x 9 0
10. 25x2
20x
4
75
11. 36x2
48x
16
12
12. 25x2
30x
9
96
1, 1 2
13. x2 4x 1 0 14. y2
1 ,7 2
12y 4 0
1 , 5
15. t2
9 5
3t 8 0
2 5
5 3
2 3
3
3
4 6 5
2
©
3
6
4 2
332
3 2
41
©
Glencoe/McGraw-Hill
331
Glencoe Algebra 2
Glencoe/McGraw-Hill
Glencoe Algebra 2
Answers
Lesson 6-4
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Skills Practice
Completing the Square
6-4
Practice
(Average)
Completing the Square
Solve each equation by using the Square Root Property. 1 2. x2 4x 4
Solve each equation by using the Square Root Property. 1. x2 8x 16 1 3, 5
1,
3
1. x2
8x
16
1
2. x2
6x
9
1
3. x2
10x
25
16
5,
3. x2 12x 36 25
3
14x 49 9
4,
5. 4x2
2
12x 9 4 6. x2
9, 4
1 2 9. 9x2
1
8x 16 8
1,
11 2
4. 4x2
4x
1
9
1, 2 5 8 15
4. x2
4, 10
5. x2 4x 4 2
1 , 2
9 5 8. x2 2x
5 2
2 2
6x 1 2
2 7
6. x2
2x
1
5 1
7. x2
6x
3
7. x2 6x 9 7 3 8. x2 16x 64 15
5
1
2
1 3
2
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 10. x2 12x c 11. x2 20x c 12. x2 11x c
Answers
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 9. x2 10x c 25; (x
36; (x
13. x2 0.8x
6)2 c 100; (x
14. x2 2.2x
10)2 c 121 ; x 4
15. x2 0.36x
11 2 2 c 5)2 12)2
9 2 2
10. x2
14x
c 49; (x
7)2
5 2 2 1 2 2
0.16; (x
16. x2
5 x 6
0.4)2 c 1.21; (x
17. x2
1 x 4
1.1)2 c 0.0324; (x
18. x2
5 x 3
0.18)2
A12
Glencoe Algebra 2
11. x2
24x
c 144; (x
12. x2
5x
c
25 ; x 4 1 ; x 4
c
(Lesson 6-4)
13. x2
9x
c
81 ; x 4
25 ; x 144
5 2 12
1 ; x 64
1 2 8 2 , 3
0
25 ; x 36
5 2 6 2 3
14. x2
x
c
Solve each equation by completing the square. 19. x2 6x 18 8x 8 9x 3 0 0
4,
2
20. 3x2 23. x2
x 14x
2 19 5
0
1
21. 3x2 24. x2
5x 16x
2 7
0 1, 0 0
Solve each equation by completing the square. 15. x2 13x 36 0 4, 9 16. x2 3x 0 0,
22. x2
3
0 2
6, 3
25. 2x2
7
26. x2 x
30
0
8
27. 2x2 10x
71
5
4
28. x2
22 2
3x 6 0
1
29. 2x2
21 2
5x 6 0
5
30. 7x2
15 2
6x 2 0
17. x2
x
6
0 2,
3 4, 1
3 2
18. x2
4x
13
17 1
13 2
3
19. 2x2
7x
4
0
2 33
20. 3x2
2x
1
0
1 , 3
i 2
15
5
i 4
23
3 7
i
5
21. x2
3x
6
0
22. x2
x
3
0
1
31. GEOMETRY When the dimensions of a cube are reduced by 4 inches on each side, the surface area of the new cube is 864 square inches. What were the dimensions of the original cube? 16 in. by 16 in. by 16 in. 32. INVESTMENTS The amount of money A in an account in which P dollars is invested for 2 years is given by the formula A P(1 r)2, where r is the interest rate compounded annually. If an investment of $800 in the account grows to $882 in two years, at what interest rate was it invested? 5%
Glencoe Algebra 2
©
23. x2
11
i
11
24. x2
2x
4
0 1
i
3
©
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334
Glencoe Algebra 2
Lesson 6-4
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Reading to Learn Mathematics
Completing the Square
How can you find the time it takes an accelerating race car to reach the finish line? Read the introduction to Lesson 6-4 at the top of page 306 in your textbook. Explain what it means to say that the driver accelerates at a constant rate of 8 feet per second square.
6-4
Enrichment
Pre-Activity
The Golden Quadratic Equations
A golden rectangle has the property that its length can be written as a b, where a is the width of the rectangle and a a b
If the driver is traveling at a certain speed at a particular moment, then one second later, the driver is traveling 8 feet per second faster.
divided into a square and a smaller golden rectangle, as shown.
a
b
a . Any golden rectangle can be b a a
Reading the Lesson
1. Give the reason for each step in the following solution of an equation by using the Square Root Property. x2 12x (x x x x 6 x 36 6)2 6 6 6 x 9 9 3 81 81 81
Original equation
The proportion used to define golden rectangles can be used to derive two quadratic equations. These are sometimes called golden quadratic equations. Solve each problem.
a
b
Answers
1. In the proportion for the golden rectangle, let a equal 1. Write the resulting quadratic equation and solve for b.
Factor the perfect square trinomial. Square Root Property 81 9 Rewrite as two equations. Solve each equation.
b2
b
1 b
0
1 2 5
9 or x 15
2. In the proportion, let b equal 1. Write the resulting quadratic equation and solve for a.
A13
Glencoe Algebra 2
©
a2
(Lesson 6-4)
a
1 a
0
1 2 5
2. Explain how to find the constant that must be added to make a binomial into a perfect square trinomial.
Sample answer: Find half of the coefficient of the linear term and square it.
3. a. What is the first step in solving the equation 3x2 6x 5x 5 by completing the square? 12 0 by completing the
3. Describe the difference between the two golden quadratic equations you found in exercises 1 and 2.
The signs of the first-degree terms are opposite.
Divide the equation by 3.
b. What is the first step in solving the equation x2 square? Add 12 to each side. 4. Show that the positive solutions of the two equations in exercises 1 and 2 are reciprocals.
1 2
5
1 2
5
(12)
4
(
5)
2
1 4
5
Helping You Remember
4. How can you use the rules for squaring a binomial to help you remember the procedure for changing a binomial into a perfect square trinomial?
1
5. Use the Pythagorean Theorem to find a radical expression for the diagonal of a golden rectangle when a 1.
One of the rules for squaring a binomial is (x y) 2 x 2 2xy y 2. In completing the square, you are starting with x 2 bx and need to find y 2. This shows you that b 2y, so y b . That is why you must take half of 2
d
10 2
2 5
the coefficient and square it to get the constant that must be added to complete the square.
335
6. Find a radical expression for the diagonal of a golden rectangle when b
1.
d
©
10 2
2 5
Glencoe/McGraw-Hill
Glencoe Algebra 2
Glencoe/McGraw-Hill
336
Glencoe Algebra 2
Answers
3
337
4 2
6 2 2
3
2i
3
Glencoe Algebra 2
Glencoe Algebra 2
14. 4x2 12x 63 0 6x 15. x2 21
©
Lesson 6-5
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-5
(continued)
Study Guide and Intervention
6-5
The Quadratic Formula and the Discriminant
Roots and the Discriminant
Discriminant bx c 0, with a 0, are given by x b b2 2a 4ac
Study Guide and Intervention
The Quadratic Formula and the Discriminant
The expression under the radical sign, b2 the discriminant. 4ac, in the Quadratic Formula is called
The Quadratic Formula can be used to solve any quadratic equation once it is written in the form ax2 bx c 0.
.
Quadratic Formula
Glencoe/McGraw-Hill
Roots of a Quadratic Equation
Discriminant b2 4ac 4ac 4ac 4ac 0 0 1 rational root 2 complex roots 0, but not a perfect square 2 irrational roots 0 and a perfect square 2 rational roots b2
Quadratic Formula
Quadratic Formula
The solutions of ax 2
Example
14 by using the Quadratic Formula. 14 b2 5, and c with 14.
Solve 5x 0.
x2
5x
Type and Number of Roots
Rewrite the equation as
x2
x
Replace a with 1, b with
b b2 b2 2a
4ac
( 5)
( 5)2 2(1)
Simplify.
4(1)( 14)
5
81
5
2 9
2
7 or
2
Answers
The solutions are
2 and 7.
Example Find the value of the discriminant for each equation. Then describe the number and types of roots for the equation. b. 3x2 2x 5 a. 2x2 5x 3 The discriminant is The discriminant is b2 4ac 52 4(2)(3) or 1. b2 4ac ( 2)2 4(3)(5) or 56. The discriminant is a perfect square, so The discriminant is negative, so the the equation has 2 rational roots. equation has 2 complex roots. Exercises
A14
2. x2 10x 24 0 11x 1. p2 5. 14x2 9x x 1 0 15 0 6. 2x2 24 0 3. x2
Exercises
Solve each equation by using the Quadratic Formula.
1. x2
2x
35
0
5,
7
4,
6
3, 8
For Exercises 1 12, complete parts a c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. 2. 9x2 12p 4 128; two irrational roots; 6 4 2 4. x2 4x 4 0 32; 3. 2x2 6x 1 0 0; one rational root; 1 7x 4 0 81;
(Lesson 6-5)
4. 4x2
19x
5
0
3
5. 5x2 36x 7 0 1156;
2 rational roots;
6. 4x2 4x 11 0
1 ,4 2
1 , 4
5
8. 2y2 y 15 0 16x 16 0 9. 3x2
1 , 2
1 7
3,
5 2
7. 3x2
5x
2
160; 2 complex roots;
5
7. x2 7x 6 0 25;
2, 1 3
3
10x 50 0
3r 5 2 25
3
11. r2 0 12. x2
5 , 2
4, 4
2 irrational roots; 2 2 2
2 rational roots; 1 ,7
8. m2 8m 14 8;
1
9. 25x2
i 10 2
40x 16 0; 1 rational root; 4
10. 8x2
6x
9
0
3 3 , 2 4
2 1 , 5 5
5
5 3
0
2 rational roots; 1, 6
10. 4x2 20x 29 0
2 irrational roots; 4 2 64; 11. 6x2 26x 8 0 484; 2 complex roots; 2 rational roots;
5
13. x2
6x
23
0
12. 4x2
3
4x 11 0 192; 2 irrational roots;
Glencoe/McGraw-Hill
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Glencoe Algebra 2
Lesson 6-5
©
NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE______________ PERIOD _____
6-5
(Average)
Complete parts a c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. 1. x2 16x 64 0 3x 2. x2 3. 9x2 24x
Skills Practice
6-5
Practice
Glencoe/McGraw-Hill
The Quadratic Formula and the Discriminant
The Quadratic Formula and the Discriminant
Complete parts a c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. 2. x2 11x 26 0
1. x2
8x
16
0
16
0
0; 1 rational root; 4 0; 1 rational; 8
4. x2 3x 40 5. 3x2 9x 2 4. 20x2 7x 3 0
225; 2 rational roots;
2, 13
9; 2 rational; 0, 3
0 105;
0; 1 rational; 4
3 9 6 105
6. 2x2 7x 0
3. 3x2
2x
0
4; 2 rational roots; 0, 2
3
6. x2 6 7. 5x2 2x 4 0 0
289; 2 rational roots; 169; 2 rational; 76; i 19 5
0 196;
3 1 , 5 4
5, 8
2 irrational;
8. 12x2 x 6
49; 2 rational; 0,
0 289; 9. 7x2 6x 2 0
7 2
5. 5x2
6
0
20; 2 rational; 3 ,
4
11. 6x2 2x 1
120; 2 irrational roots;
8. 5x2 x 10. 12x2 2x 4 1 0
30 5
24; 2 irrational roots;
21 10
13. 4x2 3x2 6
6
2 complex; 1
2 3
0 28;
2 complex;
12. x2 3x 6 0
3 7
i
5
7. x2
8x
13
0
15;
2 3
12; 2 irrational roots;
10. x2 49 0
4
3
21; 2 irrational roots; 1 2 rational; 1 , 2 2 irrational; 3
7
2 irrational; 1
0 105; 14. 16x2 8x 1 0
7 6 105 8
2 complex;
15. 2x2 5x 6
3
i 2
0 73;
15
Answers
9. x2
2x
17
0
A15
3 2
12. 2x2 3x 2
72; 2 irrational roots; 1 i 2 3
196; 2 complex roots; 7; 2 complex roots; 3 i 4
7i
0; 1 rational; 1 4
2 irrational; 5
73 4
11. x2
x
1
0
3; 2 complex roots; 1
Solve each equation by using the method of your choice. Find exact solutions. 16. 7x2 18. 3x2 20. 3x2 22. x2 24. 3x2 26. 4x2 5x 8x 13x 6x 3 54 4x 28. x2 4x 0 0, 3 4
(Lesson 6-5)
Solve each equation by using the method of your choice. Find exact solutions. 14. 30 0 x2
5 7 1 , 3
0 0 3
17. 4x2
9
0
3 2
3
1 ,4 3
19. x2 21. 15x2
21
4x 22x
3, 7
8
13.
x2
64
8
16. 16x2 24x 8x 17 0 4 27 0 18. x2
30
9 , 4 3 4
15. x2
x
30
5, 6 33 3
7 4 1 2 i 5 17
2 , 3
4 5
6 3i
17 15 2 0
23. x2
14x
53
0 7
2i 2
1 2 4i
25. 25x2 27. 8x 1 20x 6 4x2 2 0 2
17. x2 20. 3x2 36 0
4x
11
0 2
15 2i
0 0
10 5 3 2
19. x2
25
0
5i
3
22. 2x2 7x 2x 3 4 24. 2x2
21. 2x2
10x
11
0
5
2
i
11
29. 4x2
12x
7
0 3
2 2
30. GRAVITATION The height h(t) in feet of an object t seconds after it is propelled straight up from the ground with an initial velocity of 60 feet per second is modeled by the equation h(t) 16t2 60t. At what times will the object be at a height of 56 feet? 1.75 s, 2 s 31. STOPPING DISTANCE The formula d 0.05s2 1.1s estimates the minimum stopping distance d in feet for a car traveling s miles per hour. If a car stops in 200 feet, what is the fastest it could have been traveling when the driver applied the brakes? about 53.2 mi/h
Glencoe Algebra 2
©
23. 8x2
1
4x
1
i
4
25. PARACHUTING Ignoring wind resistance, the distance d(t) in feet that a parachutist falls in t seconds can be estimated using the formula d(t) 16t2. If a parachutist jumps from an airplane and falls for 1100 feet before opening her parachute, how many seconds pass before she opens the parachute? about 8.3 s
Glencoe Algebra 2
339
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Glencoe Algebra 2
Answers
Sample answer: The discriminant is the expression under the radical in the Quadratic Formula. Look at the Quadratic Formula and consider what happens when you take the principal square root of b2 4ac and apply in front of the result. If b2 4ac is positive, its principal square root will be a positive number and applying will give two different real solutions, which may be rational or irrational. If b2 4ac 0, its principal square root is 0, so applying in the Quadratic Formula will only lead to one solution, which will be rational (assuming a, b, and c are integers). If b 2 4ac is negative, since the square roots of negative numbers are not real numbers, you will get two complex roots, corresponding to the and in the symbol.
341
Glencoe Algebra 2
Glencoe Algebra 2
©
Lesson 6-5
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-5
Sum and Product of Roots
Reading to Learn Mathematics
6-5
Enrichment
The Quadratic Formula and the Discriminant
Sometimes you may know the roots of a quadratic equation without knowing the equation itself. Using your knowledge of factoring to solve an equation, you can work backward to find the quadratic equation. The rule for finding the sum and product of roots is as follows:
Sum and Product of Roots then s1 s2 b and s1 a
Pre-Activity
How is blood pressure related to age?
Glencoe/McGraw-Hill
If the roots of ax 2 bx s2 c . a
Read the introduction to Lesson 6-5 at the top of page 313 in your textbook.
Describe how you would calculate your normal blood pressure using one of the formulas in your textbook. c 0, with a ! 0, are s1 and s2,
Sample answer: Substitute your age for A in the appropriate formula (for females or males) and evaluate the expression.
Reading the Lesson b 5x The roots are x
Add the roots. Multiply the roots.
1. a. Write the Quadratic Formula. x 7, but do 3 and x 8.
b2 2a 4ac
b. Identify the values of a, b, and c that you would use to solve 2x2 not actually solve the equation.
Example A road with an initial gradient, or slope, of 3% can be represented by the formula y ax2 0. 03x c, where y is the elevation and x is the distance along the curve. Suppose the elevation of the road is 1105 feet at points 200 feet and 1000 feet along the curve. You can find the equation of the transition curve. Equations of transition curves are used by civil engineers to design smooth and safe roads. y 10 –8 –6 –4 –2 O –10 –20
5 – (– –, –301) 2 4
a 3 ( 8) 5 3( 8) 24 Equation: x2 5x 24 0
2 c b
5 7
2. Suppose that you are solving four quadratic equations with rational coefficients and have found the value of the discriminant for each equation. In each case, give the number of roots and describe the type of roots that the equation will have.
Number of Roots Type of Roots
2
4
x
Answers
Value of Discriminant
A16
2 2 2 1
1. 6,
64
real, rational complex real, irrational real, rational
–30
8
21
0
Write a quadratic equation that has the given roots. 9 2. 5, 1 3. 6, 6
(Lesson 6-5)
Helping You Remember
x2
3x
54
0
x2
4x
5
0
x2
2
12x
3 5
36
0
3. How can looking at the Quadratic Formula help you remember the relationships between the value of the discriminant and the number of roots of a quadratic equation and whether the roots are real or complex?
4. 4
3
6.
6.
7
x2
8x
13
0
2 2 , 5 7 35x 2
4x
4
0
49x 2
42x
205
0
Find k such that the number given is a root of the equation. 7. 7; 2x2 kx 21 0 8. 2; x2 13x k 0
11
30
Glencoe/McGraw-Hill
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342
Glencoe Algebra 2
Vertex Form of a Quadratic Function
Lesson 6-6
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-6
Analyzing Graphs of Quadratic Functions
Write Quadratic Functions in Vertex Form
Study Guide and Intervention
6-6
Study Guide and Intervention
(continued)
Analyzing Graphs of Quadratic Functions
h)2
Glencoe/McGraw-Hill
Example
Write y 12x 2x2 2x2 2(x2 2(x2 2(x 18 2(x 3)2 7. 12x 25 6x) 25 6x 9) 25 3)2 7 y y y y The vertex form of the equation is y y Analyze Quadratic Functions
A quadratic function is easier to graph when it is in vertex form. You can write a quadratic function of the form 2 y ax bx c in vertex from by completing the square.
The graph of y a(x k has the following characteristics: • Vertex: (h, k) • Axis of symmetry: x h • Opens up if a 0 • Opens down if a 0 • Narrower than the graph of y x 2 if "a" 1 • Wider than the graph of y x 2 if "a" 1
25 in vertex form. Then graph the function.
Example
each graph. 4. The graph opens
Identify the vertex, axis of symmetry, and direction of opening of
4)2
11 a. y 2(x The vertex is at (h, k) or ( 4, 11), and the axis of symmetry is x up, and is narrower than the graph of y x2.
O
x
a. y 2. The graph opens 1. y x2 10x
1 (x 4
2)2
10
Answers
The vertex is at (h, k) or (2, 10), and the axis of symmetry is x down, and is wider than the graph of y x2.
Exercises
Write each quadratic function in vertex form. Then graph the function. 32 2. y x2 6x 3. y x2 8x 6
A17 y (x y Exercises
5)2
7
y
(x
3)2 y O
9 x y
(x
8 4 –4 O –4 –8 O
4)2 y 10
Each quadratic function is given in vertex form. Identify the vertex, axis of symmetry, and direction of opening of the graph. 2. y 4(x 7 3 3)2 3. y
1 (x 2
1. y
(x
2)2
16
5)2
(Lesson 6-6)
(2, 16); x
5. y 12 6(x 6
1 (x 5
2; up
4)2 6. y 6)2
( 3,
7); x
3; up
(5, 3); x
5; up
4
8
x
4. y
7(x
1)2
9
x
( 1,
8. y 8(x 2 3(x 3)2 9. y 1)2 2
9); x
1; down
(4,
12); x
4; up
( 6, 6); x
6; up
4. y 4x2 16x 11 5. y 3x2 12x 5 6. y
–12
7. y
2 (x 5
9)2
12
5x2
10x
9
(9, 12); x
11. y 22
4 (x 3
9; up
7)2 12. y 16(x 4)2
(3,
2); x
3; up
(1,
2); x
1; down
1
y
4(x y 2)2
5
y
3(x y 2)2
7
y
5(x y 1)2
4
10. y
5 (x 2
5)2
12
( 5, 12); x
14. y 0.4(x 0.2 0.6)2 15. y
5; down
(7, 22); x
7; up
(4, 1); x
1.2(x
4; up
0.8)2 6.5
O
O
x
13. y
3(x
1.2)2
2.7
x
(1.2, 2.7); x
1.2; up
(0.6, 0.2); x down
343
0.6;
( 0.8, 6.5); x up
0.8;
O
x
Glencoe Algebra 2
©
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©
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344
Glencoe Algebra 2
Answers
Lesson 6-6
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
NAME ______________________________________________ DATE
____________ PERIOD _____
6-6
Skills Practice
Analyzing Graphs of Quadratic Functions
6-6
Practice
(Average)
Analyzing Graphs of Quadratic Functions
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. 1. y 6(x 2)2 1 2. y 2x2 2 3. y 4x2 8x
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. 1. y (x 2)2 2. y x2 4 3. y x2 6
y (x 2)2 0; (2, 0); x 2; up
4. y 3(x 5)2
y (x (0, 4); x
5. y 5x2
0)2 4; 0; down
9
y (x 0)2 6; (0, 6); x 0; up
6. y (x 2)2 18
y 6(x 2)2 1; ( 2, 1); x 2; down x2 10x 20
y 2(x (0, 2); x
5. y 2x2
0)2 2; 0; up
18
y 4(x 1)2 4; (1, 4); x 1; down
3x2 6x
4. y
y 3(x ( 5, 0); x
7. y x2 2x
5)2 0; 5; down
5
y 5(x 0)2 9; (0, 9); x 0; down
8. y x2 6x 2
y (x 2)2 18; (2, 18); x 2; up
9. y 3x2 24x
y (x 5)2 ( 5, 5); x
2x2 16x
5; 5; up 4)2; 4; down
1
y
y x
8. y
2(x 3)2; ( 3, 0); 3; up
3x2 18x 21
12x
6. y
y 3(x (1, 2); x
9. y 2x2
1)2 2; 1; up
29
5
7. y
32
y (x 1)2 6; (1, 6); x 1; up
Graph each function. 10. y (x y y (x 3)2 ( 3, 7); x
7; 3; up
y 3(x (4, 48); x
4)2 48; 4; down
y 2(x ( 4, 0); x
10. y (x 3)2
y 3(x 3)2 6; (3, 6); x 3; down x2 y
y 2(x 4)2 ( 4, 3); x
2x2
y
16x
3; 4; up
Answers
Graph each function. 11. y 6x 5 12. y 2x 1
3)2
1
11. y
(x
1)2 y 2
12. y y O
(x
4)2
4
O
x
O
A18
O
x
O
(Lesson 6-6)
x
x
x
O
x
Write an equation for the parabola with the given vertex that passes through the given point. 15. y x2 6x 4 y 13. y
1 (x 2
2)2 y O
14. y
3x2 y 4
13. vertex: (1, 3) point: ( 2, 15)
14. vertex: ( 3, 0) point: (3, 18)
y y
2(x (x
1)2 4)2
3 4
y
1 (x 2
15. vertex: (10, point: (5, 6)
4)
3)2
y
2 (x 5
10)2
4
x
O O
16. Write an equation for a parabola with vertex at (4, 4) and x-intercept 6. x x
17. Write an equation for a parabola with vertex at ( 3,
1 (x 3)2 1 3 18. BASEBALL The height h of a baseball t seconds after being hit is given by
1) and y-intercept 2.
y
h(t) 16t2 80t 3. What is the maximum height that the baseball reaches, and when does this occur? 103 ft; 2.5 s
Write an equation for the parabola with the given vertex that passes through the given point. 16. vertex: (4, 36) point: (0, 20) 17. vertex: (3, 1) point: (2, 0) 18. vertex: ( 2, 2) point: ( 1, 3)
y
©
(x
4)2
36
y
(x
3)2
345
1
y
(x
2)2
2
19. SCULPTURE A modern sculpture in a park contains a parabolic arc that starts at the ground and reaches a maximum height of 10 feet after a horizontal distance of 4 feet. Write a quadratic function in vertex form that describes the shape of the outside of the arc, where y is the height of a point on the arc and x is its horizontal distance from the left-hand 5 starting point of the arc. 2
Glencoe Algebra 2
10 ft
y
8
(x
4)
10
4 ft
Glencoe/McGraw-Hill
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
346
Glencoe Algebra 2
It moves the graph up.
Lesson 6-6
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-6 x2 be used to graph any quadratic
Reading to Learn Mathematics
6-6
Patterns with Differences and Sums of Squares
Some whole numbers can be written as the difference of two squares and some cannot. Formulas can be developed to describe the sets of numbers algebraically. If possible, write each number as the difference of two squares. Look for patterns. 1. 0 02 a(x 9. 8 32 h)2 k. 5. 4 22
Enrichment
Analyzing Graphs of Quadratic Equations
Pre-Activity
How can the graph of y function? x2?
Glencoe/McGraw-Hill
02 02 12 22
14. 13 72 10. 9 32 6. 5 32 2. 1 12
Read the introduction to Lesson 6-6 at the top of page 322 in your textbook.
• What does adding a positive number to x2 do to the graph of y
• What does subtracting a positive number to x before squaring do to the graph of y x2? It moves the graph to the right.
Reading the Lesson
22 02 62
02
3. 2 cannot 7. 6 cannot 11. 10 cannot 15. 14 cannot
4. 3 22 8. 7 42 12. 11 62 16. 15 42
12 32 52 12
1. Complete the following information about the graph of y
a. What are the coordinates of the vertex? (h, k)
b. What is the equation of the axis of symmetry? x
h
0? up; down
13. 12 42
c. In which direction does the graph open if a 0? If a 1?
d. What do you know about the graph if "a"
Answers
It is wider than the graph of y x 2. 2n 1
x 2.
Even numbers can be written as 2n, where n is one of the numbers 0, 1, 2, 3, and so on. Odd numbers can be written 2n 1. Use these expressions for these problems. 17. Show that any odd number can be written as the difference of two squares.
A19
b. a 0, h 0, h y O
If "a"
1? It is narrower than the graph of y
(n
1)2
n2
2. Match each graph with the description of the constants in the equation in vertex form. 0, k 0, k iv. y x x
a. a 0 iv 0 i d. a y 0, h
0, k
0 iii
18. Show that the even numbers can be divided into two sets: those that can be written in the form 4n and those that can be written in the form 2 4n.
c. a iii. x O
0, h
0, k
0 ii
Find 4n for n 0, 1, 2, and so on. You get {0, 4, 8, 12, …}. For 2 4n, you get {2, 6, 10, 12, …}. Together these sets include all even numbers.
19. Describe the even numbers that cannot be written as the difference of two squares. 2 4n, for n 0, 1, 2, 3, … 20. Show that the other even numbers can be written as the difference of two squares. 4n (n 1)2 (n 1)2
(Lesson 6-6)
i.
O
y
ii.
O
x
Helping You Remember
Every whole number can be written as the sum of squares. It is never necessary to use more than four squares. Show that this is true for the whole numbers from 0 through 15 by writing each one as the sum of the least number of squares. 21. 0 02 24. 3 12 27. 6 12 30. 9 32 33. 12 12 36. 15 12 22. 1 12 23. 2 12
3. When graphing quadratic functions such as y (x 4)2 and y (x 5)2, many students have trouble remembering which represents a translation of the graph of y x2 to the left and which represents a translation to the right. What is an easy way to remember this?
12 12 12 12 12
Glencoe Algebra 2
©
12 22 12 22
Glencoe/McGraw-Hill
25. 4 22 28. 7 12 31. 10 12
26. 5 12
22 12 32 32 32
348
Glencoe Algebra 2
12
34. 13 22
22 32
29. 8 22 32. 11 12 35. 14 12
22 12 22 32 32
Sample answer: In functions like y (x 4)2, the plus sign puts the graph “ahead” so that the vertex comes “sooner” than the origin and the translation is to the left. In functions like y (x 5)2, the minus puts the graph “behind” so that the vertex comes “later” than the origin and the translation is to the right.
Glencoe Algebra 2
347
©
Glencoe/McGraw-Hill
Answers
Lesson 6-7
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
NAME ______________________________________________ DATE
____________ PERIOD _____
6-7
Study Guide and Intervention
Graphing and Solving Quadratic Inequalities
To graph a quadratic inequality in two variables, use
6-7
Study Guide and Intervention
(continued)
Graphing and Solving Quadratic Inequalities graphically or algebraically.
Graph Quadratic Inequalities the following steps:
Solve Quadratic Inequalities
Quadratic inequalities in one variable can be solved
1. Graph the related quadratic equation, y ax2 bx c. Use a dashed line for or ; use a solid line for or . 2. Test a point inside the parabola. If it satisfies the inequality, shade the region inside the parabola; otherwise, shade the region outside the parabola.
Graphical Method
To solve ax 2 bx c 0: First graph y ax 2 bx c. The solution consists of the x-values for which the graph is below the x-axis. To solve ax 2 bx c 0: First graph y ax 2 bx c. The solution consists the x-values for which the graph is above the x-axis. Find the roots of the related quadratic equation by factoring, completing the square, or using the Quadratic Formula. 2 roots divide the number line into 3 intervals. Test a value in each interval to see which intervals are solutions.
Example
Graph the inequality y
x2
6x
7. y First graph the equation y x2 6x 7. By completing the square, you get the vertex form of the equation y (x 3)2 2, so the vertex is ( 3, 2). Make a table of values around x 3, and graph. Since the inequality includes , use a dashed line. Test the point ( 3, 0), which is inside the parabola. Since ( 3)2 6( 3) 7 2, and 0 2, ( 3, 0) satisfies the inequality. Therefore, shade the region inside the parabola.
Algebraic Method
If the inequality involves solution set.
or
, the roots of the related equation are included in the
Answers
O
x
Example
Solve the inequality x2
x
6
0.
O
y x
Exercises
Graph each inequality. 1. y x2 y First find the roots of the related equation x2 x 6 0. The equation factors as (x 3)(x 2) 0, so the roots are 3 and 2. The graph opens up with x-intercepts 3 and 2, so it must be on or below the x-axis for 2 x 3. Therefore the solution set is {x" 2 x 3}.
A20
Glencoe Algebra 2
(Lesson 6-7)
8x
17
2. y
x2
6x
4 y 3. y
x2
2x y 2
Exercises
Solve each inequality. 1. x2 2x 0 2. x2 16 0 3. 0 6x x2 5
O
x
{x 2
O
x
0}
{x 4
5. 2m2 m
x
1
4}
{x1
6. y2 8
x
5}
O
x
x
4. c2
4
{c 2
4. y x2 y O
c
12
2}
0
m
8. x2
1 2
9x
m
14 0
1
9. x2 7x 10 0
4x
6
5. y
2x2
4x y 6. y
2x2 y 4x
2
7. x2
4x
{x 2
10. 2x2 5x
x
3
6}
0
{xx
11. 4x2 23x
7 or x
15 0
2}
12.
{x2
6x2
x
11x
5}
2 0
x
O O
x
x 3
13. 2x2 11x
x
12
x
1 2
0
xx
14. x2 4x
3 or x 4
5 0
5
xx
15. 3x2 16x
2 or x
5 0
1 6
xx
Glencoe/McGraw-Hill
3 or x 2
4
350
x 1
3
x
5
©
349
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
Glencoe Algebra 2
Lesson 6-7
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-7
(Average)
Graph each inequality. 2. y 4 y O
Skills Practice
6-7
Practice
Graphing and Solving Quadratic Inequalities x2 3. y 2x y x y y
Graphing and Solving Quadratic Inequalities
5 1. y 4 6x 6 x2 2. y x2 3. y 2x2 4x y Graph each inequality. x2
1. y
x2
4x
4
2
Glencoe/McGraw-Hill
O O
y
x
O
x x
O O
x x
Use the graph of its related function to write the solutions of each inequality. 4. x2 8x y 6 O 5 –6 2 O 2 4 6 8x
Use the graph of its related function to write the solutions of each inequality. 0 5. x2 2x y O
4. x2 5. 4x x y y
6x
9
0
x2 32 0 20 0
6. x2
3
0
6. x2
9x y 14
0 x O
y
x
x
–12
Answers
6 O 2
A21 x O
x
x 8 x 4 x 5 or x 4
0 or x
8
3
x
1
2
x
7
3
Solve each inequality algebraically. 8. 2x 35 0 x2
Solve each inequality algebraically. 7. x2 x 20 0 8. x2 10x 16 0 9. x2 4x 5 0
7.
x2
3x
10
0
(Lesson 6-7)
{x 2
10. 36 x2
x {x 6
12. x2 7x 6 0
5} x x
18 0
{xx 6}
7 or x
5}
{xx
10. x2 14x
4 or x
49 0
5} all reals
13. x2 5x 7 0
{x2
11. x2 5x
x
14
8}
12. x2 15 8x
9.
x2
18x
81
0
{xx {x 6
14. x2 9x
9} 1} 3}
0
{xx
14. 9x2 36x
2 or x
36 0
7} all reals
16. 4x2 4x 1 0
{x 5
15. 9x 12x2
x {xx
17. 5x2 10
3} 2}
27x
11. x2
7x
0
{xx {x 6
16. 2x 15 x2
0 or x x x
4
7}
xx
18. 9x2
0 or x
31x 12 0
13. x2
3 4
x
12
0
{xx {x 5
18. 2x2 2x 12x 9
4 or x 3} 1}
0
3}
xx
15.
x2
10x
25
0
1 2
xx
2 or x 5
5
x 3
x
4 9
19. FENCING Vanessa has 180 feet of fencing that she intends to use to build a rectangular play area for her dog. She wants the play area to enclose at least 1800 square feet. What are the possible widths of the play area? 30 ft to 60 ft
all reals {xx
20. 9x2
17. x2
3x
0
{xx
3 or x
0}
2 or x
19.
x2
64
16x
all reals
351
Glencoe Algebra 2
20. BUSINESS A bicycle maker sold 300 bicycles last year at a profit of $300 each. The maker wants to increase the profit margin this year, but predicts that each $20 increase in profit will reduce the number of bicycles sold by 10. How many $20 increases in profit can the maker add in and expect to make a total profit of at least $100,000? from 5 to 10
©
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
Glencoe/McGraw-Hill
352
Glencoe Algebra 2
Answers
1. Answer the following questions about how you would graph the inequality y x2 x 6.
a. What is the related quadratic equation? y
x2 x 6
b. Should the parabola be solid or dashed? How do you know?
solid; The inequality symbol is 2 for y in the quadratic inequality. 6 true or false? true 4. 2|x 6| 2 0 0
.
Answers
c. The point (0, 2) is inside the parabola. To use this as a test point, substitute
A22 x y
(2, 4)
0
for x and
d. Is the statement 2
02
Lesson 6-7
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-7
Graphing Absolute Value Inequalities
You can solve absolute value inequalities by graphing in much the same manner you graphed quadratic inequalities. Graph the related absolute function for each inequality by using a graphing calculator. For and , identify the x-values, if any, for which the graph lies below the x-axis. For and , identify the x values, if any, for which the graph lies above the x-axis. For each inequality, make a sketch of the related graph and find the solutions rounded to the nearest hundredth. 1. |x 3| 0 2. |x| 6 0 3.
Reading to Learn Mathematics
6-7
Enrichment
Graphing and Solving Quadratic Inequalities
Pre-Activity
How can you find the time a trampolinist spends above a certain height?
Glencoe/McGraw-Hill
|x
12
4| 8 0
Read the introduction to Lesson 6-7 at the top of page 329 in your textbook.
• How far above the ground is the trampoline surface? 3.75 feet
• Using the quadratic function given in the introduction, write a quadratic inequality that describes the times at which the trampolinist is more than 20 feet above the ground. 16t 2 42t 3.75 20
Reading the Lesson x 3 or x 3 6 x 6
x
4
5. |3x
3|
0
6. |x
7|
5
e. Should the region inside or outside the parabola be shaded? inside
7 or x
5
all real numbers
2
x
12
2. The graph of y x2 4x is shown at the right. Match each of the following related inequalities with its solution set. i. {x"x 0 or x x x (Lesson 6-7)
a. 4}
(0, 0) O (4, 0)
x2 ii. {x"0 4} 4} iii. {x"x 0 or x x 4} iv. {x"0
4x
0 ii
b.
x2
4x
0 iii
c.
x2
4x
0 iv
d.
x2
4x
0 i
7. |7x
1|
13
8. |x
3.6|
4.2
9. |2x
5|
7
Helping You Remember
x
1.71 or x
2
0.6
x
7.8
6
x
1
3. A quadratic inequality in two variables may have the form y ax2 bx c, y ax2 bx c, y ax2 bx c, or y ax2 bx c. Describe a way to remember which region to shade by looking at the inequality symbol and without using a test point.
Sample answer: If the symbol is or , shade the region above the parabola. If the symbol is or , shade the region below the parabola.
Glencoe Algebra 2
353
©
Glencoe/McGraw-Hill
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
354
Glencoe Algebra 2
Chapter 6 Assessment Answer Key
Form 1 Page 355
1. 2. 3.
Page 356
12.
Form 2A Page 357
1.
B B C
D
C
13.
D
2.
B
4.
A
14.
A
3.
A
15. 5.
C
4.
B
B
16. 17.
D B
6.
B
5.
A
7. 8.
C B
18.
C
6.
D C
7.
9.
B
19.
B
8.
C
20. 10. 11.
C 1 and 7; 14
9. 10.
A B
B:
D A
(continued on the next page)
© Glencoe/McGraw-Hill
A23
Glencoe Algebra 2
Answers
Chapter 6 Assessment Answer Key
Form 2A (continued) Page 358
11. 12.
Form 2B Page 359
1.
Page 360
11. 12.
B D
B
C B
2. 13.
C
13.
C
3.
B
D
14.
D
14.
A
4. 15.
D
15.
D
5.
B
16.
A
C
16.
C
17.
B
6.
A D
17.
D
7. 18.
A
8.
18.
B
B
19.
19.
C
A
9. 20.
A C
D 16x 2 3 0
10.
20.
A 9x 2 2 0
B: Sample answer:
B: Sample answer:
© Glencoe/McGraw-Hill
A24
Glencoe Algebra 2
Chapter 6 Assessment Answer Key
Form 2C Page 361
1.
f (x )
5x
2
Page 362 f (x )
10x (1, 5) 1
10. 11. x { 2 2, 1
2 3
13}
x
(3, 0) O
12. 2.
i 31 10
maximum; 4
13. 0; 1 real, rational root
3.
2, 4 y 14. 33; 2 real, irrational roots 15. ( 5, 7); x
5; down
O
x
16. 17.
y y
3 (x 2
2)2 3)2
1.5)2
1 1
(x
16(t
between
2 and y 1;
4. between 1 and 2
18. h(t )
51; 51 ft
O
x
19.
y
5. 6.
3, 2
5
O
x
9 in. by 16 in. x x
1 or x 2
2 7. 4x
21x
18
0
20. B:
3
9x 2
7
0
8. 9.
{ 8, 2}
5 2 7
© Glencoe/McGraw-Hill
A25
Glencoe Algebra 2
Answers
Chapter 6 Assessment Answer Key
Form 2D Page 363
1.
f (x ) x2 f (x )
(0, 3) 4x 3 O (2,
Page 364
8. 9.
{3, 11}
2 3 6
x 1)
2
x
10. 11.
{4
2} 1, 2
3 41 4
2.
minimum;
17
9
12.
3.
1, y 3
13. 0; 1 real, rational root 14. 8; 2 complex roots 5); x
1 (x 4
O
x
15. (6, 16. y
6; down 4) 2 2
between 1 and 0; 4. between 1 and 2 y 17.
y
(x
2)2
2) 2
4
16(t 18. h(t) 76; 76 ft x O
19.
y
5. 6.
1, 4
3
O
x
8 in. by 18 in.
20.
x
3 2
x 5
5 0
7.
2x 2
5x
12
B:
16x 2
© Glencoe/McGraw-Hill
A26
Glencoe Algebra 2
Chapter 6 Assessment Answer Key
Form 3 Page 365
1.
f (x ) f (x )
3x 2 2x 3
Page 366
10.
{ 0.35, 0.85}
( x – 1, 8
3 3 1 3
)
O
(0, 3)
x
5
11. 12. 13. 14.
i 39 8
{ 3.5, 1} 6 4 2
2. 3. 4.
minimum; 22 25 $8.00; $6400 3, 6 y 1.2; two real, irrational roots
2 O 2
15. x 2 y k
2
16. between 3 and 2; between 4 and 5 5. y 2 1 3 x 7 ; 5 2 2 7 7 , 1 ;x ; 2 2 2
down
2 O 2
x
6. between 1 and 2 y 18.
y
29 (x 200 y 9)2
29 2
O
x
19.
7.
2 8. 12x
1 5 , 2 3
13x 14 0 20. x x
O
x
7 or x 2
1
9.
x
1
2 B: 16x
24x
29
0
© Glencoe/McGraw-Hill
A27
Glencoe Algebra 2
Answers
h(t) 9.1(t 32.5) 2 17. 30,000; 30,000 ft
Chapter 6 Assessment Answer Key
Page 367, Open-Ended Assessment Scoring Rubric
Score 4 General Description Superior A correct solution that is supported by welldeveloped, accurate explanations Specific Criteria • Shows thorough understanding of the concepts of graphing, analyzing, and finding the maximum and minimum values of quadratic functions; solving quadratic equations; and solving inequalities. • Uses appropriate strategies to solve problems. • Computations are correct. • Written explanations are exemplary. • Goes beyond requirements of some or all problems. • Shows an understanding of the concepts of graphing, analyzing, and finding the maximum and minimum values of quadratic functions; solving quadratic equations; and solving inequalities. • Uses appropriate strategies to solve problems. • Computations are mostly correct. • Written explanations are effective. • Satisfies all requirements of problems. • Shows an understanding of most of the concepts of graphing, analyzing, and finding the maximum and minimum values of quadratic functions; solving quadratic equations; and solving inequalities. • May not use appropriate strategies to solve problems. • Computations are mostly correct. • Written explanations are satisfactory. • Satisfies the requirements of most of the problems. • Final computation is correct. • No written explanations or work is shown to substantiate the final computation. • Satisfies minimal requirements of some of the problems. • Shows little or no understanding of most of the concepts of graphing, analyzing, and finding the maximum and minimum values of quadratic functions; solving quadratic equations; and solving inequalities. • Does not use appropriate strategies to solve problems. • Computations are incorrect. • Written explanations are unsatisfactory. • Does not satisfy requirements of problems. • No answer may be given.
3
Satisfactory A generally correct solution, but may contain minor flaws in reasoning or computation
2
Nearly Satisfactory A partially correct interpretation and/or solution to the problem
1
Nearly Unsatisfactory A correct solution with no supporting evidence or explanation Unsatisfactory An incorrect solution indicating no mathematical understanding of the concept or task, or no solution is given
0
© Glencoe/McGraw-Hill
A28
Glencoe Algebra 2
Chapter 6 Assessment Answer Key
Page 367, Open-Ended Assessment Sample Answers
In addition to the scoring rubric found on page A28, the following sample answers may be used as guidance in evaluating open-ended assessment items. 1. Student responses should indicate that using the Square Root Property, as Mi-Ling’s group did, would take less time than the other two methods since the equation is already set up as a perfect square set equal to a constant. To solve using either of the other two methods, the binomial would need to be expanded and the constant on the right brought to the left side of the equal sign. 2a. Jocelyn had trouble because the problem is impossible. No such parabola exists. 2b. Student responses will vary. One of the three conditions must be omitted or modified. Sample answer: Delete “...and passes through ( 1, 0).” 2c. Answers will vary and depend on the answer for part b. For example, for the sample answer in part b above, a possible equation is: y 2(x 3)2 4. 3a. Answer must be of the form y a(x h)2 8 where h is any real number and a 0. 3b. Answers must be of the form y a[x (h n)]2 8 where h and a represent the same values as in part a. The student choice is for the value of n. The student should indicate that the graph will shift to the left n units if his or her value of n is negative, but will shift the graph to the right n units if the chosen value of n is positive. 4. Students should indicate that Joseph’s answer is not correct. In Step 2, when he completed the square by inserting 9 inside the parentheses, he actually added 2(9) 18 to the right side of the equation, so he must subtract 18 from the constant on the same side, rather than add 9, to keep the statements equivalent. The correct solution is f(x) 2(x 3)2 23. 5a. ; The graph is strictly above the x-axis for all values of x other than 2. 5b. ; The graph is never below the x-axis. 5c. ; The graph is always on or above the x-axis.
© Glencoe/McGraw-Hill
A29
Glencoe Algebra 2
Answers
Chapter 6 Assessment Answer Key
Vocabulary Test/Review Page 368
1. false; Zero Product
Quiz (Lessons 6–1 and 6–2) Page 369
1. 2.
Quiz (Lessons 6–5 and 6–6) Page 370
1. 2. 3.
Property
2. false; constant term 3. false; quadratic
2
5
3; x f (x ) x
1
1;
1
96; 2 complex roots
inequality
4. false; roots 5. true 6. false; minimum
y x
2
O
x f (x) x2
2x 3
( 1,
4)
(0,
3)
O
(2,
x 1)
value
7. false; quadratic
3.
minimum; 1
4. y
term
8. false; (the) 4.
3(x y 2(x
2)2 5)2
6
3,
1
5.
Quadratic Formula
9. true 10. false; discriminant 11. Sample answer:
between 1 and 2; between 6 and 5 5.
Quiz (Lesson 6–7) Page 370 Quiz (Lessons 6–3 and 6–4) Page 369
1. 2. 3.
2 4. x
2 5. 3x
A parabola is a smooth curve that is the graph of a quadratic function.
12. Sample answer:
1.
y
5, 2
3
O
An axis of symmetry is a line along which you can fold a graph and get matching parts on both sides of the line.
x
{ 9, 5} B 4x
10x
2.
{x 1 y x
5}
12
8
0
0
3.
6. 7. 8. 9. 10.
© Glencoe/McGraw-Hill
{ 10, 2} {1
2 5
3 5}
3
O
x
{ 1, 11} {2 i
A30
{x x 4.
1 or x
3}
14}
all reals
Glencoe Algebra 2
Chapter 6 Assessment Answer Key
Mid-Chapter Test Page 371
1.
Cumulative Review Page 372
1. 2.
B
17
3 4
2.
B
3.
inconsistent
( 2, 0), ( 2, 8), 4. (0, 2), (8, 2) A
5. 6. 4.
3.
92 (2, 2x 3 4 x2
8 x 3
3) 2x
C
7.
5.
D
8. 9.
5.599
6.
1, 3 y 10.
136 ft; 1.5 s
O
x
11.
1, 3 y 7. 8. 9. 10.
minimum;
91
2
O
{ 2, 9}
0, 1
4 5 3 1
x
12. 13.
{ 2, 3}
3, 2 complex roots
14.
© Glencoe/McGraw-Hill
y
x
7 2 2
29 4
Glencoe Algebra 2
A31
Answers
51
Chapter 6 Assessment Answer Key
Standardized Test Practice Page 373
1.
A B C D
Page 374
11.
4 9 / 4
. / . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 . 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
12.
1 1 0
. 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 . 0 1 2 3 4 5 6 7 8 9
2.
E
F
G
H
3.
A
B
C
D
13. 4.
E F G H
1 5 4
. 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 . 0 1 2 3 4 5 6 7 8 9
14.
8 0
. 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 . 0 1 2 3 4 5 6 7 8 9
5.
A
B
C
D
6.
E
F
G
H
7.
A
B
C
D
15.
A
B
C
D
8.
E
F
G
H
16.
A
B
C
D
9.
A
B
C
D
17. 10.
E F G H
A
B
C
D
© Glencoe/McGraw-Hill
A32
Glencoe Algebra 2
Consumable Workbooks
Many of the worksheets contained in the Chapter Resource Masters booklets are available as consumable workbooks. Study Guide and Intervention Workbook Skills Practice Workbook Practice Workbook 0-07-828029-X 0-07-828023-0 0-07-828024-9
ANSWERS FOR WORKBOOKS The answers for Chapter 6 of these workbooks can be found in the back of this Chapter Resource Masters booklet.
Glencoe/McGraw-Hill
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher. Send all inquiries to: The McGraw-Hill Companies 8787 Orion Place Columbus, OH 43240-4027 ISBN: 0-07-828009-5 1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02 Algebra 2 Chapter 6 Resource Masters
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii Lesson 6-1
Study Guide and Intervention . . . . . . . . 313–314 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 315 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 Reading to Learn Mathematics . . . . . . . . . . 317 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 318
Lesson 6-6
Study Guide and Intervention . . . . . . . . 343–344 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 345 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 Reading to Learn Mathematics . . . . . . . . . . 347 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 348
Lesson 6-7
Study Guide and Intervention . . . . . . . . 349–350 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 351 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Reading to Learn Mathematics . . . . . . . . . . 353 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 354
Lesson 6-2
Study Guide and Intervention . . . . . . . . 319–320 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 321 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Reading to Learn Mathematics . . . . . . . . . . 323 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 324
Chapter 6 Assessment
Chapter 6 Test, Form 1 . . . . . . . . . . . . 355–356 Chapter 6 Test, Form 2A . . . . . . . . . . . 357–358 Chapter 6 Test, Form 2B . . . . . . . . . . . 359–360 Chapter 6 Test, Form 2C . . . . . . . . . . . 361–362 Chapter 6 Test, Form 2D . . . . . . . . . . . 363–364 Chapter 6 Test, Form 3 . . . . . . . . . . . . 365–366 Chapter 6 Open-Ended Assessment . . . . . . 367 Chapter 6 Vocabulary Test/Review . . . . . . . 368 Chapter 6 Quizzes 1 & 2 . . . . . . . . . . . . . . . 369 Chapter 6 Quizzes 3 & 4 . . . . . . . . . . . . . . . 370 Chapter 6 Mid-Chapter Test . . . . . . . . . . . . 371 Chapter 6 Cumulative Review . . . . . . . . . . . 372 Chapter 6 Standardized Test Practice . . 373–374 Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1 ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A32
Lesson 6-3
Study Guide and Intervention . . . . . . . . 325–326 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 327 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 Reading to Learn Mathematics . . . . . . . . . . 329 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 330
Lesson 6-4
Study Guide and Intervention . . . . . . . . 331–332 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 333 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 Reading to Learn Mathematics . . . . . . . . . . 335 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 336
Lesson 6-5
Study Guide and Intervention . . . . . . . . 337–338 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 339 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 Reading to Learn Mathematics . . . . . . . . . . 341 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 342
©
Glencoe/McGraw-Hill
iii
Glencoe Algebra 2
Teacher’s Guide to Using the Chapter 6 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resources you use most often. The Chapter 6 Resource Masters includes the core materials needed for Chapter 6. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet. All of the materials found in this booklet are included for viewing and printing in the Algebra 2 TeacherWorks CD-ROM. Pages vii–viii include a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar.
Vocabulary Builder
There is one master for each lesson. These problems more closely follow the structure of the Practice and Apply section of the Student Edition exercises. These exercises are of average difficulty.
Practice
WHEN TO USE These provide additional practice options or may be used as homework for second day teaching of the lesson.
WHEN TO USE Give these pages to
students before beginning Lesson 6-1. Encourage them to add these pages to their Algebra 2 Study Notebook. Remind them to add definitions and examples as they complete each lesson.
Reading to Learn Mathematics
Study Guide and Intervention
Each lesson in Algebra 2 addresses two objectives. There is one Study Guide and Intervention master for each objective.
WHEN TO USE Use these masters as
One master is included for each lesson. The first section of each master asks questions about the opening paragraph of the lesson in the Student Edition. Additional questions ask students to interpret the context of and relationships among terms in the lesson. Finally, students are asked to summarize what they have learned using various representation techniques.
reteaching activities for students who need additional reinforcement. These pages can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent. There is one master for each lesson. These provide computational practice at a basic level. used with students who have weaker mathematics backgrounds or need additional reinforcement.
WHEN TO USE This master can be used
as a study tool when presenting the lesson or as an informal reading assessment after presenting the lesson. It is also a helpful tool for ELL (English Language Learner) students. There is one extension master for each lesson. These activities may extend the concepts in the lesson, offer an historical or multicultural look at the concepts, or widen students’ perspectives on the mathematics they are learning. These are not written exclusively for honors students, but are accessible for use with all levels of students.
Skills Practice
Enrichment
WHEN TO USE These masters can be
WHEN TO USE These may be used as
extra credit, short-term projects, or as activities for days when class periods are shortened.
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
iv
Assessment Options
The assessment masters in the Chapter 6 Resource Masters offer a wide range of assessment tools for intermediate and final assessment. The following lists describe each assessment master and its intended use.
Intermediate Assessment
• Four free-response quizzes are included to offer assessment at appropriate intervals in the chapter. • A Mid-Chapter Test provides an option to assess the first half of the chapter. It is composed of both multiple-choice and free-response questions.
Chapter Assessment
CHAPTER TESTS
• Form 1 contains multiple-choice questions and is intended for use with basic level students. • Forms 2A and 2B contain multiple-choice questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. • Forms 2C and 2D are composed of freeresponse questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Grids with axes are provided for questions assessing graphing skills. • Form 3 is an advanced level test with free-response questions. Grids without axes are provided for questions assessing graphing skills. All of the above tests include a freeresponse Bonus question. • The Open-Ended Assessment includes performance assessment tasks that are suitable for all students. A scoring rubric is included for evaluation guidelines. Sample answers are provided for assessment. • A Vocabulary Test, suitable for all students, includes a list of the vocabulary words in the chapter and ten questions assessing students’ knowledge of those terms. This can also be used in conjunction with one of the chapter tests or as a review worksheet.
Continuing Assessment
• The Cumulative Review provides students an opportunity to reinforce and retain skills as they proceed through their study of Algebra 2. It can also be used as a test. This master includes free-response questions. • The Standardized Test Practice offers continuing review of algebra concepts in various formats, which may appear on the standardized tests that they may encounter. This practice includes multiplechoice, grid-in, and quantitativecomparison questions. Bubble-in and grid-in answer sections are provided on the master.
Answers
• Page A1 is an answer sheet for the Standardized Test Practice questions that appear in the Student Edition on pages 342–343. This improves students’ familiarity with the answer formats they may encounter in test taking. • The answers for the lesson-by-lesson masters are provided as reduced pages with answers appearing in red. • Full-size answer keys are provided for the assessment masters in this booklet.
©
Glencoe/McGraw-Hill
v
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6
Reading to Learn Mathematics
Vocabulary Builder
Vocabulary Builder
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 6. As you study the chapter, complete each term’s definition or description. Remember to add the page number where you found the term. Add these pages to your Algebra Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
axis of symmetry
completing the square
constant term
discriminant dihs·KRIH·muh·nuhnt linear term
maximum value
minimum value
parabola puh·RA·buh·luh quadratic equation kwah·DRA·tihk Quadratic Formula
! " # " $
! " # " $
! " " # " " $
(continued on the next page) Glencoe/McGraw-Hill
©
vii
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6
Reading to Learn Mathematics
Vocabulary Builder
Vocabulary Term Found on Page
(continued)
Definition/Description/Example
quadratic function
quadratic inequality
quadratic term
roots
Square Root Property
vertex
vertex form
Zero Product Property
zeros
©
Glencoe/McGraw-Hill
viii
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-1
Study Guide and Intervention
Graphing Quadratic Functions
Graph Quadratic Functions
Quadratic Function Graph of a Quadratic Function A function defined by an equation of the form f (x) b x-coordinate of vertex: 2a
ax 2
bx
c, where a
0 b ; 2a
A parabola with these characteristics: y intercept: c ; axis of symmetry: x
( 3) x or 3 . The x-coordinate of the vertex is 3 . 2(1) 2 2 3 Next make a table of values for x near . 2 x 0 1
3 2
a
1, b
3, and c
5, so the y-intercept is 5. The equation of the axis of symmetry is
x2 02 12
3 2 2
3x 3(0) 3(1)
3 3 2
5 5 5 5 5 5
f(x) 5 3
11 4
(x, f(x)) (0, 5) (1, 3)
3 11 , 2 4
f (x )
2 3
22 32
3(2) 3(3)
3 5
(2, 3) (3, 5)
O
x
Exercises
For Exercises 1–3, complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. 1. f(x) x2 6x 8 2. f(x) x2 2x 2 3. f(x) 2x2 4x 3
8, x x f (x) 3 1
3,
2 0
12 8 4 (x )
3
1 3 4 0
2, x x f (x) 1 3
4 –8 –4 O –4
1,
0 2 f (x )
1
2 2 1 1
3, x x f (x)
1, 1
1 1 f (x )
12
0 3
2 3
3 9
4
8x
8 4 –4 O 4 8
–8
–4
O –4
4
x
–8
x
©
Glencoe/McGraw-Hill
313
Glencoe Algebra 2
Lesson 6-1
Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex for the graph of f(x) x2 3x 5. Use this information to graph the function.
Example
NAME ______________________________________________ DATE
____________ PERIOD _____
6-1
Study Guide and Intervention
Graphing Quadratic Functions
(continued)
Maximum and Minimum Values
Maximum or Minimum Value of a Quadratic Function
The y-coordinate of the vertex of a quadratic function is the maximum or minimum value of the function.
The graph of f(x ) ax 2 bx c, where a 0, opens up and has a minimum when a 0. The graph opens down and has a maximum when a 0.
Determine whether each function has a maximum or minimum value. Then find the maximum or minimum value of each function. a. f(x) 3x 2 6x 7 For this function, a 3 and b 6. Since a 0, the graph opens up, and the function has a minimum value. The minimum value is the y-coordinate of the vertex. The x-coordinate of the b 6 vertex is 1.
2a 2(3)
Example
b. f(x) 100 2x x 2 For this function, a 1 and b 2. Since a 0, the graph opens down, and the function has a maximum value. The maximum value is the y-coordinate of the vertex. The x-coordinate of the vertex b 2 is 1.
2a 2( 1)
Evaluate the function at x 1 to find the minimum value. f(1) 3(1)2 6(1) 7 4, so the minimum value of the function is 4.
Evaluate the function at x 1 to find the maximum value. f( 1) 100 2( 1) ( 1)2 101, so the minimum value of the function is 101.
Exercises
Determine whether each function has a maximum or minimum value. Then find the maximum or minimum value of each function. 1. f(x) 2x2 x 10 2. f(x) x2 4x 7 3. f(x) 3x2 3x 1
min., 9
4. f(x)
7
4x x2
min.,
5. f(x) x2
11
7x 11
min.,
6. f(x)
1 x2 6x 4
16
max., 20
7. f(x) x2 5x 2
min.,
8. f(x) 20
5
6x x2
max., 5
9. f(x) 4x2 x 3
min.,
10. f(x)
17 x2 4x 10
max., 29
11. f(x) x2 10x 5
min., 2 15
12. f(x) 6x2 12x 21
max., 14
13. f(x) 25x2 100x 350
min.,
14. f(x)
20
0.5x2 0.3x 1.4
max., 27
15. f(x) x2 2 x 4
8
min., 250
©
min.,
1.445
314
max.,
7 31
Glencoe Algebra 2
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
6-1
Skills Practice
Graphing Quadratic Functions
For each quadratic function, find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. 1. f(x) 3x2 2. f(x) x2 1 3. f(x) x2 6x 15
0; x
4. f(x)
0; 0
2x2 11
1; x
5. f(x)
0; 0 x2 10x 5
15; x
6. f(x) 2x2
3; 3
8x 7
11; x
0; 0
5; x
5; 5
7; x
2; 2
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. 7. f(x) 2x2 8. f(x) x2 4x 2 0 4 4 6 9. f(x) x2 0 6x 2 0 3 8 4 6 8
0; x x f (x)
0; 0
2 8 1 0 2 0 f (x )
4; x
1 2 2 8 x
2; 2
2 0 4 16
8; x x f (x) 8 f (x )
3; 3
1 0
f (x) 16 4 f (x )
16
O
x
12 8 4 O –2 O 2 4 6x
x
Determine whether each function has a maximum or a minimum value. Then find the maximum or minimum value of each function. 10. f(x) 6x2 11. f(x) 8x2 12. f(x) x2 2x
min.; 0
13. f(x) x2 2x 15
max.; 0
14. f(x) x2 4x 1
min.;
15. f(x) x2
1
2x 3
min.; 14
16. f(x) 2x2 4x 3
max.; 3
17. f(x) 3x2 12x 3
min.;
18. f(x)
4
2x2 4x 1
max.;
1
min.;
9
315
min.;
1
©
Glencoe/McGraw-Hill
Glencoe Algebra 2
Lesson 6-1
NAME ______________________________________________ DATE
____________ PERIOD _____
6-1
Practice
(Average)
Graphing Quadratic Functions
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. 1. f(x) x2 0 8x 15 4 6 8 2. f(x) x2 6 4 4x 12 3. f(x) 2x2 2x 1 2 5
15; x x 4; 4
2 1 3 15
12; x x 2;
2
2 0 2
1; x x f (x) 5
0.5; 0.5
1 0 0.5 1 1 0.5 1
f (x) 15 3
16 12 8 4 O 2 4
f (x) 0 12 16 12 0 f (x )
16 12 8 4
f (x )
6
8x –6 –4 –2
O
2x
Determine whether each function has a maximum or a minimum value. Then find the maximum or minimum value of each function. 4. f(x) x2 2x 8 5. f(x) x2 6x 14 6. v(x) x2 14x 57
min.;
7. f(x)
9
2x2 4x 6
min.; 5
8. f(x) x2 4x 1
max.;
9. f(x)
8
2 2 x 3
8x
24
min.;
8
max.; 3
max.; 0
10. GRAVITATION From 4 feet above a swimming pool, Susan throws a ball upward with a velocity of 32 feet per second. The height h(t) of the ball t seconds after Susan throws it is given by h(t) 16t2 32t 4. Find the maximum height reached by the ball and the time that this height is reached. 20 ft; 1 s 11. HEALTH CLUBS Last year, the SportsTime Athletic Club charged $20 to participate in an aerobics class. Seventy people attended the classes. The club wants to increase the class price this year. They expect to lose one customer for each $1 increase in the price. a. What price should the club charge to maximize the income from the aerobics classes?
$45
b. What is the maximum income the SportsTime Athletic Club can expect to make?
$2025
©
Glencoe/McGraw-Hill
316
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-1
Reading to Learn Mathematics
Graphing Quadratic Functions
How can income from a rock concert be maximized? Read the introduction to Lesson 6-1 at the top of page 286 in your textbook. • Based on the graph in your textbook, for what ticket price is the income the greatest? $40 • Use the graph to estimate the maximum income. about $72,000
Pre-Activity
Reading the Lesson
5x is the
linear 4
.
term, and 3 is the 4 ax2 x 3x2, a
b. For the quadratic function f(x) c 2. Consider the quadratic function f(x) a. The graph of this function is a b. The y-intercept is
3
,b
1
, and
bx
c, where a .
0.
parabola
.
c x
c. The axis of symmetry is the line d. If a 0, then the graph opens
b 2a
. and the function has a
upward
minimum
e. If a
value.
0, then the graph opens
downward
and the function has a
maximum
value.
(–2, 4)
3. Refer to the graph at the right as you complete the following sentences. a. The curve is called a b. The line x
f (x )
parabola
.
O (0, –1)
2 is called the axis of symmetry .
x
c. The point ( 2, 4) is called the
vertex
1), 1 is
.
d. Because the graph contains the point (0, the
y-intercept
.
Helping You Remember
4. How can you remember the way to use the x2 term of a quadratic function to tell whether the function has a maximum or a minimum value? Sample answer:
Remember that the graph of f(x) x 2 (with a 0) is a U-shaped curve that opens up and has a minimum. The graph of g(x) x 2 (with a 0) is just the opposite. It opens down and has a maximum.
317
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
Lesson 6-1
1. a. For the quadratic function f(x)
2x2
5x
3, 2x2 is the
quadratic constant term.
term,
NAME ______________________________________________ DATE
____________ PERIOD _____
6-1
Enrichment
Finding the Axis of Symmetry of a Parabola
As you know, if f(x) ax2 bx c is a quadratic function, the values of x b b2 2a 4ac b . 2a
that make f(x) equal to zero are
and
b
b2 2a f(x) 4ac
.
The average of these two number values is value when x
The function f(x) has its maximum or minimum b . Since the axis of symmetry 2a
O
b x = – –– 2a
f(x) = ax 2 + bx + c
of the graph of f (x) passes through the point where the maximum or minimum occurs, the axis of symmetry has the equation x b . 2a
x b b (– –– , f (– –– (( 2a 2a
Example
Use x x
10 2(5) b . 2a
Find the vertex and axis of symmetry for f(x)
5x 2
10x
7.
1
The x-coordinate of the vertex is 5x2 7 10x 12 7.
1.
Substitute x 1 in f(x) 2 f( 1) 5( 1) 10( 1) The vertex is ( 1, 12). The axis of symmetry is x
b , or x 2a
1.
Find the vertex and axis of symmetry for the graph of each function using x 1. f(x) x2 b . 2a
4x
8
2. g(x)
4x2
8x
3
3. y
x2
8x
3
4. f(x)
2x2
6x
5
5. A(x)
x2
12x
36
6. k(x)
2x2
2x
6
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Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-2
Study Guide and Intervention
Solving Quadratic Equations by Graphing
Solve Quadratic Equations
Quadratic Equation Roots of a Quadratic Equation A quadratic equation has the form ax 2 bx c 0, where a 0. solution(s) of the equation, or the zero(s) of the related quadratic function
The zeros of a quadratic function are the x-intercepts of its graph. Therefore, finding the x-intercepts is one way of solving the related quadratic equation.
Example
Solve x2
x
6 x2
0 by graphing. x 6.
1 , and the equation of the 2 1 . 2
O
Graph the related function f(x)
f (x ) x
b The x-coordinate of the vertex is 2a 1 axis of symmetry is x . 2
Make a table of values using x-values around x f(x) 1 6
1 2
0 6
1 4
2 0
6
1 4
From the table and the graph, we can see that the zeros of the function are 2 and
3.
Exercises
Solve each equation by graphing. 1. x2 2x 8
O
0 2, f (x ) x
4
2. x2
4x
O
5 f (x )
0 5,
1
3. x2
5x
4 f (x )
0 1, 4
x
O
x
4. x2
10x f (x )
21
0
5. x2
4x
6
0 f (x )
6. 4x2
4x
1 f (x )
0
O
x
O
x
O
x
3, 7
©
no real solutions
319
1
Glencoe Algebra 2
Glencoe/McGraw-Hill
Lesson 6-2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-2
Study Guide and Intervention
(continued)
Solving Quadratic Equations by Graphing
Estimate Solutions Example
2 2(1) x f (x) 1 1 0 2
Often, you may not be able to find exact solutions to quadratic equations by graphing. But you can use the graph to estimate solutions.
Solve x 2 2x 2 0 by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. The equation of the axis of symmetry of the related function is x 1, so the vertex has x-coordinate 1. Make a table of values.
1 3 2 2 3 1
O
f (x )
x
The x-intercepts of the graph are between 2 and 3 and between 0 and 1. So one solution is between 2 and 3, and the other solution is between 0 and 1.
Exercises
Solve the equations by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 1. x2 4x 2 0 2. x2 6x 6 0 3. x2 4x 2 0
between 0 and 1; between 3 and 4 f (x )
between between
2 and 5 and f (x )
1; 4
between between
1 and 0; 4 and 3 f (x )
O
O
x
x
O
x
4.
x2
2x
4
0
5. 2x2
12x
17
0
6.
1 2 x 2
x
5 2
0
between 3 and 4; between 2 and 1 f (x )
between 2 and 3; between 3 and 4 f (x )
between 2 and between 3 and 4 f (x )
1;
O O
x
x
O
x
©
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320
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-2
Skills Practice
Solving Quadratic Equations By Graphing
Use the related graph of each equation to determine its solutions. 1. x2 2x 3 f (x )
0
2.
x2 f (x )
6x x2 6x
9
9 O
0 f (x ) x
3. 3x2
4x
3 f (x )
0
O
x
f (x ) f (x ) x2
2x 3
3x 2
4x
3 O
x
3, 1
3
no real solutions
4. x2
6x
5
0
5.
x2
2x
4
0
6. x2
6x
4
0
1, 5 f (x )
no real solutions f (x )
O
between 0 and 1; between 5 and 6 f (x )
x
O
x
O
x
Use a quadratic equation to find two real numbers that satisfy each situation, or show that no such numbers exist. 7. Their sum is 4, and their product is 0. 8. Their sum is 0, and their product is 36.
x2
4x
0; 0, f (x )
4
x2
36
36 24
0;
6, 6
f (x )
O
x
–12 –6
12 O 6 12 x
©
Glencoe/McGraw-Hill
321
Glencoe Algebra 2
Lesson 6-2
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
NAME ______________________________________________ DATE
____________ PERIOD _____
6-2
Practice
(Average)
Solving Quadratic Equations By Graphing
Use the related graph of each equation to determine its solutions. 1. 3x2 f (x )
3
3x 2 3
0 f (x )
2. 3x2
x f (x )
3
0
3. x2
3x
2 f (x )
0
O
x f (x )
O 3x 2
x
3
f (x )
x2
3x
2 O
x
x
1, 1
no real solutions
1, 2
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 4. 2x2 6x 5 0 5. x2 10x 24 0 6. 2x2 x 6 0
between 0 and 1; between 4 and 3
12 8 4 –6 –4 –2 O
6,
4 f (x )
between 2
2 and
1,
f (x )
x
O
x
Use a quadratic equation to find two real numbers that satisfy each situation, or show that no such numbers exist. 7. Their sum is 1, and their product is f (x )
6.
8. Their sum is 5, and their product is 8.
x2 x 3, 2
6
0;
x 2 5x 8 0; no such real numbers exist
O
x
For Exercises 9 and 10, use the formula h(t) v0t 16t 2, where h(t) is the height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. 9. BASEBALL Marta throws a baseball with an initial upward velocity of 60 feet per second. Ignoring Marta’s height, how long after she releases the ball will it hit the ground? 3.75 s 10. VOLCANOES A volcanic eruption blasts a boulder upward with an initial velocity of 240 feet per second. How long will it take the boulder to hit the ground if it lands at the same elevation from which it was ejected? 15 s
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Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-2
Reading to Learn Mathematics
Solving Quadratic Equations by Graphing
How does a quadratic function model a free-fall ride? Read the introduction to Lesson 6-2 at the top of page 294 in your textbook. Write a quadratic function that describes the height of a ball t seconds after 16t 2 125 it is dropped from a height of 125 feet. h(t)
Pre-Activity
Reading the Lesson
1. The graph of the quadratic function f(x) x2 x 6 is shown at the right. Use the graph to find the solutions of the quadratic equation x2 x 6 0. 2 and 3 y O
x
2. Sketch a graph to illustrate each situation. a. A parabola that opens b. A parabola that opens downward and represents a upward and represents a quadratic function with two quadratic function with real zeros, both of which are exactly one real zero. The negative numbers. zero is a positive number. y y
c. A parabola that opens downward and represents a quadratic function with no real zeros. y O
x
O
x
O
x
Helping You Remember
3. Think of a memory aid that can help you recall what is meant by the zeros of a quadratic function.
Sample answer: The basic facts about a subject are sometimes called the ABCs. In the case of zeros, the ABCs are the XYZs, because the zeros are the x-values that make the y-values equal to zero.
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Glencoe Algebra 2
Lesson 6-2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-2
Enrichment
Graphing Absolute Value Equations
You can solve absolute value equations in much the same way you solved quadratic equations. Graph the related absolute value function for each equation using a graphing calculator. Then use the ZERO feature in the CALC menu to find its real solutions, if any. Recall that solutions are points where the graph intersects the x-axis. For each equation, make a sketch of the related graph and find the solutions rounded to the nearest hundredth. 1. | x 5| 0 2. | 4x 3| 5 0 3. | x 7| 0
5
No solutions
7
4. | x
3|
8
0
5.
|x
9, 3
3|
6
0
6. | x
2|
3
0
11, 5
1, 5
7. | 3x
4|
2
2,
2 3
8. | x
12 |
10
9. | x |
3
0
22,
2
3, 3
10. Explain how solving absolute value equations algebraically and finding zeros of absolute value functions graphically are related.
Sample answer: values of x when solving algebraically are the x-intercepts (or zeros) of the function when graphed.
Glencoe/McGraw-Hill
©
324
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Study Guide and Intervention
Solving Quadratic Equations by Factoring
When you use factoring to solve a quadratic equation,
0, then either a 0 or b 0, or both a and b 0.
Solve Equations by Factoring you use the following property.
Zero Product Property
For any real numbers a and b, if ab
Solve each equation by factoring. a. 15x b. 4x2 5x 3x2 15x Original equation 4x2 3x2 15x 0 Subtract 15x from both sides. 4x2 5x 3x(x 5) 0 Factor the binomial. (4x 7)(x 3x 0 or x 5 0 Zero Product Property 4x 7 0 x 0 or x 5 Solve each equation. x 3x2 The solution set is {0, 5}.
Example
21 5x 21 3) or
7 or 4
21 0 0 x
Original equation Subtract 21 from both sides. Factor the trinomial.
3 x
0 3
7 ,3 . 4
Zero Product Property Solve each equation.
The solution set is
Exercises
Solve each equation by factoring. 1. 6x2 2x 0 2. x2 7x 3. 20x2 25x
0,
4. 6x2
1
7x
{0, 7}
5. 6x2 27x 0
0,
6. 12x2
5
8x 0
0,
7. x2
7 x 30 0
0,
8. 2x2
9 x 3 0
0,
9. x2
2
14x 33 0
{5,
10. 4x2
6}
27x 7 0
3
,
1
29x 10 0
{ 11,
12. 6x2 5x
3}
4 0
11. 3x2
1
, ,
7
8x 1 0
10,
14. 5x2
1
12 0 15. 2x2
1 4
,
13. 12x2
28x
250x
5000
0
1 1
16. 2x2 11x 40 0
2
, 11,
6
21x 11 0
{100, 25}
18. 3x2 2x 21 0
17. 2x2
8,
19. 8x2
5
14x 3 0
1
2 0
7
, , ,5
3
17x 12 0
20. 6x2
11x
21. 5x2
3 1
,
2,
25x 12 0 23. 12x2
1
18x 6 0
3
4
36x 5 0
22. 12x2
24. 7x2
4
,
3
1
,
1
325
1
©
Glencoe/McGraw-Hill
Glencoe Algebra 2
Lesson 6-3
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Study Guide and Intervention
q)
(continued)
Solving Quadratic Equations by Factoring
Write Quadratic Equations
(x p)(x To write a quadratic equation with roots p and q, let 0. Then multiply using FOIL.
Example
in the form a. 3, 5 (x p)(x q) (x 3)[x ( 5)] (x 3)(x 5) x2 2x 15 The equation x2 3 and 5.
ax2
Write a quadratic equation with the given roots. Write the equation bx c 0. 0 0 0 0 2x b.
Write the pattern. Replace p with 3, q with Simplify. Use FOIL. 5.
7 1 , 8 3
(x x
7 8
p)(x x x
(3x 3 7)(3x 24
q)
1 3 1 3 1) 1)
0 0 0 0 24 0 0 13x 7 0 has
15
0 has roots
(8x
x
7) 8 24 (8x
7 8
24x2
13x
7
The equation 24x2 roots
7 1 and . 8 3
Exercises
Write a quadratic equation with the given roots. Write the equation in the form ax2 bx c 0. 1. 3, 4 2. 8, 2 3. 1, 9
x2
4. 5
x 10x
12 25
0 0
x2
5. 10, 7
10x 17x
16 70
0
6.
x2 x2
9. 7,
3 4
10x 13x
9 30
0 0
2, 15
x2
7.
1 ,5 3 3x 2 2 5 5x 2 2 , 3 2 3
x2
8. 2,
2 3 3x 2 4 , 9 9x 2 5 , 4 1 2
0
14x
5
0
11.
8x
1
4
0
4x 2
12. 9,
1 6 6x 2 3 1 , 7 5
25x
21
0
10. 3,
17x
6
0
14.
13x
4
0
15.
55x
9
0
13.
9x 2
16.
7 7 , 8 2 16x 2
4
0
17.
8x 2
1 3 , 2 4 8x 2
6x
5
0
18.
35x 2
1 1 , 8 6
22x
3
0
42x
49
10x
326
3
0
48x 2
14x
1
0
©
Glencoe/McGraw-Hill
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Skills Practice
Solving Quadratic Equations by Factoring
Solve each equation by factoring. 1. x2 64 { 8, 8} 2. x2 100 0 {10,
10}
3. x2
3x
2
0 {1, 2}
4. x2
4x
3
0 {1, 3}
5. x2
2x
3
0 {1,
3}
6. x2
3x
10
0 {5,
2}
7. x2
6x
5
0 {1, 5}
8. x2
9x
0 {0, 9}
9.
x2
6x
0 {0, 6}
10. x2
6x
8
0 { 2,
4}
11. x2
5x {0,
5}
12. x2
14x
49
0 {7}
13. x2
6
5x {2, 3}
14. x2
18x
81 { 9}
17. 4x2
5x
6
0
3
,
2
18. 3x2
13x
10
0
2
,5
Write a quadratic equation with the given roots. Write the equation in the form ax2 bx c 0, where a, b, and c are integers. 19. 1, 4 x 2
5x 7x
4
0 10 3 0 0
20. 6,
9 x2
3x 7x 0 2x
54
0
21.
2,
1 , 3
5 x2
22. 0, 7 x 2
1 3 , 2 4
23.
3 3x 2
10x
24.
8x 2
3
0
25. Find two consecutive integers whose product is 272. 16, 17
©
Glencoe/McGraw-Hill
327
Glencoe Algebra 2
Lesson 6-3
15. x2
4x
21 { 3, 7}
16. 2x2
5x
3
0
1
,
3
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Practice
(Average)
Solving Quadratic Equations by Factoring
Solve each equation by factoring. 1. x2 4. x2 7. x2 10. 10x2 12. x2 14. 36x2 16. 3x2 18. 3x2 20. 3x2 4x 6x 4x 12 8 0 {6, 0 {2, 4}
2}
2. x2 5. x2 8. 7x2
16x 3x
64 2
0 {8} 0 { 2,
3. x2
20x 9x 25
100 14
0 {10} 0 {2, 7}
1} 6. x2
9. x2
0 {0, 4} 9x 0,
4x 0,
4
2x 35x 8x
10x {5}
9
11. x2 13. 5x2 15. 2x2
99 { 9, 11} 60 90 0 {3, 4} 0 {9,
12x 25 2x 24x 8x
36 { 6}
5
1
,
0
5 1
5}
,
1 3}
17. 6x2 19. 15x2 21. 6x2
9x 0, 19x 5x 6
3
6 0
45
0 { 5,
3 2
,
2
4 2,
2
3
,
Write a quadratic equation with the given roots. Write the equation in the form ax2 bx c 0, where a, b, and c are integers. 22. 7, 2 23. 0, 3 24. 5, 8
x2
25. 7,
9x
8
14 56
0
26.
x2
6,
3x
3
0 18 0
x2
27. 3, 4
3x x
7 2
40 12
0 0
x2
28. 1,
1 2 2x 2 1 , 3
15x
0
29.
x2
1 ,2 3 3x 2 1 3 3x 2
9x
x2
30. 0,
3x
3
1
0
7x
2
0
33.
2x 2
2 , 3
7x
4 5
0
31.
32. 4,
3x 2 24, 26
8x
3
0
13x
4
0
15x 2
22x
8
0
34. NUMBER THEORY Find two consecutive even positive integers whose product is 624. 35. NUMBER THEORY Find two consecutive odd positive integers whose product is 323.
17, 19
36. GEOMETRY The length of a rectangle is 2 feet more than its width. Find the dimensions of the rectangle if its area is 63 square feet. 7 ft by 9 ft 37. PHOTOGRAPHY The length and width of a 6-inch by 8-inch photograph are reduced by the same amount to make a new photograph whose area is half that of the original. By how many inches will the dimensions of the photograph have to be reduced? 2 in.
©
Glencoe/McGraw-Hill
328
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Reading to Learn Mathematics
Solving Quadratic Equations by Factoring
How is the Zero Product Property used in geometry? Read the introduction to Lesson 6-3 at the top of page 301 in your textbook. What does the expression x(x 5) mean in this situation?
Pre-Activity
It represents the area of the rectangle, since the area is the product of the width and length.
Reading the Lesson
1. The solution of a quadratic equation by factoring is shown below. Give the reason for each step of the solution. x2 x2 (x x 10x 3)(x 3 x 10x 21 7) 0 0 7 x 0 7 21
Original equation
Add 21 to each side. Factor the trinomial. Zero Product Property Solve each equation.
.
0 or x 3
The solution set is
{3, 7}
Marla (x 7)(x x2 2x
5) 35
0 0
Rosa (x 7)(x x2 2x
5) 35
0 0
Larry (x 7)(x x2 2x
5) 35
0 0
Who is correct? Rosa Explain the errors in the other two students’ work.
Sample answer: Marla used the wrong factors. Larry used the correct factors but multiplied them incorrectly.
Helping You Remember
3. A good way to remember a concept is to represent it in more than one way. Describe an algebraic way and a graphical way to recognize a quadratic equation that has a double root.
Sample answer: Algebraic: Write the equation in the standard form ax 2 bx c 0 and examine the trinomial. If it is a perfect square trinomial, the quadratic function has a double root. Graphical: Graph the related quadratic function. If the parabola has exactly one x-intercept, then the equation has a double root.
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329
Glencoe Algebra 2
Lesson 6-3
2. On an algebra quiz, students were asked to write a quadratic equation with 7 and 5 as its roots. The work that three students in the class wrote on their papers is shown below.
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Enrichment
Euler’s Formula for Prime Numbers
Many mathematicians have searched for a formula that would generate prime numbers. One such formula was proposed by Euler and uses a quadratic polynomial, x2 x 41. Find the values of x2 x 41 for the given values of x. State whether each value of the polynomial is or is not a prime number. 1. x 0 2. x 1 3. x 2
4. x
3
5. x
4
6. x
5
7. x
6
8. x
17
9. x
28
10. x
29
11. x
30
12. x
35
13. Does the formula produce all prime numbers greater than 40? Give examples in your answer.
14. Euler’s formula produces primes for many values of x, but it does not work for all of them. Find the first value of x for which the formula fails. (Hint: Try multiples of ten.)
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330
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Study Guide and Intervention
Completing the Square
Square Root Property Use the following property to solve a quadratic equation that is in the form “perfect square trinomial constant.”
Square Root Property For any real number x if x 2 n, then x n.
Example
a. x2 x2 x
Solve each equation by using the Square Root Property. b. 4x2 20x 25 32 8x 16 25 4x2 20x 25 8x 16 25 32 2 2 (x 4) (2x 5) 25 32 4 2x 5 25 or x 4 25 32 or 2x 5 x 5 4 9 or x 5 4 1 2x 5 4 2 or 2x 5 1}. x
5 4 2 2 5
32 4 2
The solution set is {9,
The solution set is
4 2 . 2
Exercises
Solve each equation by using the Square Root Property. 1. x2 18x 81 49 2. x2 20x 100 64 3. 4x2 4x 1 16
{2, 16}
{ 2,
18}
3
,
5
4. 36x2
12x
1
18
5. 9x2
12x
4
4
6. 25x2
40x
16
28
1
3 2
0, 4
4
2 7
7. 4x2
28x
49
64
8. 16x2
24x
9
81
9. 100x2
60x
9
121
15
,
1
3
,
3
{ 0.8, 1.4}
10. 25x2
20x
4
75
11. 36x2
48x
16
12
12. 25x2
30x
9
96
2
5 3
2
3
3
4 6
©
Glencoe/McGraw-Hill
331
Glencoe Algebra 2
Lesson 6-4
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Study Guide and Intervention
Completing the Square
(continued)
Complete the Square x2 1. Find b . 2
bx, follow these steps.
To complete the square for a quadratic expression of the form b . 2
!
2. Square
!
3. Add
b 2 to x2 2
bx.
Find the value of 2 c that makes x 22x c a perfect square trinomial. Then write the trinomial as the square of a binomial. Step 1 b 22; b 2
Example 1
Solve 2x2 completing the square. 2x2
2x2
Example 2
8x
8x 2
8x
24
0 by
24
24
0
0 2
Original equation Divide each side by 2. x2 4x 12 is not a perfect square.
4 2 2
11
x2 x2 121,
Step 2 112 121 Step 3 c 121 The trinomial is x2 22x which can be written as (x 11)2.
4x 12 x2 4x 4x 4 (x 2)2
0 12 12
Add 12 to each side.
4
Since
4, add 4 to each side.
16 x 2 4 x 6 or x 2 The solution set is {6,
Factor the square. Square Root Property Solve each equation.
2}.
Exercises
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 1. x2 10x c 2. x2 60x c 3. x2 3x c
25; (x
4. x2 3.2x
5)2 c 900; (x
5. x2
1 x 2
30)2 c 9
6. x2
; x
2.5x
3 2 c 2.56; (x
1.6)2
1
; x
1 2
1.5625; (x
1.25)2
Solve each equation by completing the square. 7. y2 4y 5 0 8. x2 8x 65 0 9. s2 10s 21 0
1, 5
10. 2x2 3x 1 0
5, 13
11. 2x2 13x 7 0
3, 7
12. 25x2 40x 9 0
1,
13. x2
1
4x 1 0 14. y2
1
,7
12y 4 0
1
15. t2
,
9
3t 8 0
2
©
3
6
4 2
332
3 2
41
Glencoe/McGraw-Hill
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Skills Practice
Completing the Square
Solve each equation by using the Square Root Property. 1. x2 8x 16 1 3, 5 2. x2 4x 4 1
1,
3
3. x2
12x
36
25
1,
11 2
4. 4x2
4x
1
9
1, 2 5 8 15
5. x2
4x
4
2
2 7
6. x2
2x
1
5 1
7. x2
6x
9
7 3
8. x2
16x
64
15
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 9. x2 10x c 25; (x
5)2 12)2
9 2
10. x2
14x
c 49; (x
7)2
5 2
11. x2
24x
c 144; (x
12. x2
5x
c
25
; x
13. x2
9x
c
81
; x
14. x2
x
c
1
; x
1 2
Solve each equation by completing the square. 15. x2 13x 36 0 4, 9 16. x2 3x 0 0,
3
0 2
17. x2
x
6
0 2,
3 4, 1
3 2 33
18. x2
4x
13
17 , 1
13 2
19. 2x2
7x
4
0
20. 3x2
2x
1
0
1
21. x2
3x
6
0
22. x2
x
3
0
1
23. x2
11
i
11
24. x2
2x
4
0 1
i
3
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
333
Lesson 6-4
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Practice
(Average)
Completing the Square
Solve each equation by using the Square Root Property. 1. x2 8x 16 1 2. x2 6x 9 1 3. x2 10x 25 16
5,
4. x2
3
14x 49 9
4,
5. 4x2
2
12x 9 4 6. x2
9, 4
1 2 9. 9x2
1
8x 16 8
4, 10
7. x2 6x 9 5 8. x2
1
,
2x
5
2 2
6x 1 2
3
5
1
2
1 3
2
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 10. x2 12x c 11. x2 20x c 12. x2 11x c
36; (x
13. x2 0.8x
6)2 c 100; (x
14. x2 2.2x
10)2 c 121
15. x2
; x
0.36x
11 2 c 0.16; (x
16. x2
5 x 6
0.4)2 c 1.21; (x
17. x2
1 x 4
1.1)2 c 0.0324; (x
18. x2
5 x 3
0.18)2
c
25
; x
5 2
1
; x
1 2
25
; x
5 2
Solve each equation by completing the square. 19. x2 22. x2 6x 18 8x 8 9x 3 0 0
4,
2 20. 3x2
23. x2
x 14x
2 19 5
0
2
0
,
1
21. 3x2 24. x2
5x 16x
2 7
0 1, 0 0
2
6, 3
25. 2x2
7
26. x2 x
30
0
8
27. 2x2 10x
71
5
4
28. x2
22 2
3x 6 0
1
29. 2x2
21 2
5x 6 0
5
30. 7x2
15 2
6x 2 0
3
i 2
15
5
i 4
23
3 7
i
5
31. GEOMETRY When the dimensions of a cube are reduced by 4 inches on each side, the surface area of the new cube is 864 square inches. What were the dimensions of the original cube? 16 in. by 16 in. by 16 in. 32. INVESTMENTS The amount of money A in an account in which P dollars is invested for 2 years is given by the formula A P(1 r)2, where r is the interest rate compounded annually. If an investment of $800 in the account grows to $882 in two years, at what interest rate was it invested? 5%
©
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334
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Reading to Learn Mathematics
Completing the Square
How can you find the time it takes an accelerating race car to reach the finish line? Read the introduction to Lesson 6-4 at the top of page 306 in your textbook. Explain what it means to say that the driver accelerates at a constant rate of 8 feet per second square.
Pre-Activity
If the driver is traveling at a certain speed at a particular moment, then one second later, the driver is traveling 8 feet per second faster.
Reading the Lesson
1. Give the reason for each step in the following solution of an equation by using the Square Root Property. x2 12x (x x x x 6 x 36 6)2 6 6 6 x 9 9 3 81 81 81
Original equation
Factor the perfect square trinomial. Square Root Property 81 9 Rewrite as two equations. Solve each equation.
9 or x 15
2. Explain how to find the constant that must be added to make a binomial into a perfect square trinomial.
Sample answer: Find half of the coefficient of the linear term and square it.
3. a. What is the first step in solving the equation 3x2 6x 5x 5 by completing the square? 12 0 by completing the
Divide the equation by 3.
b. What is the first step in solving the equation x2 square? Add 12 to each side.
Helping You Remember
4. How can you use the rules for squaring a binomial to help you remember the procedure for changing a binomial into a perfect square trinomial?
One of the rules for squaring a binomial is (x y) 2 x 2 2xy y 2. In completing the square, you are starting with x 2 bx and need to find y 2. This shows you that b 2y, so y b . That is why you must take half of
the coefficient and square it to get the constant that must be added to complete the square.
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Glencoe/McGraw-Hill
335
Glencoe Algebra 2
Lesson 6-4
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Enrichment
The Golden Quadratic Equations
A golden rectangle has the property that its length can be written as a b, where a is the width of the rectangle and a a b
divided into a square and a smaller golden rectangle, as shown.
a
b
a . Any golden rectangle can be b a a
The proportion used to define golden rectangles can be used to derive two quadratic equations. These are sometimes called golden quadratic equations. Solve each problem.
a
b
1. In the proportion for the golden rectangle, let a equal 1. Write the resulting quadratic equation and solve for b.
2. In the proportion, let b equal 1. Write the resulting quadratic equation and solve for a.
3. Describe the difference between the two golden quadratic equations you found in exercises 1 and 2.
4. Show that the positive solutions of the two equations in exercises 1 and 2 are reciprocals.
5. Use the Pythagorean Theorem to find a radical expression for the diagonal of a golden rectangle when a 1.
6. Find a radical expression for the diagonal of a golden rectangle when b
1.
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336
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-5
Study Guide and Intervention
The Quadratic Formula and the Discriminant
Quadratic Formula The Quadratic Formula can be used to solve any quadratic equation once it is written in the form ax2 bx c 0.
Quadratic Formula The solutions of ax 2 bx c 0, with a 0, are given by x b b2 2a 4ac
.
Example x b ( 5) 5 5 2 2 9 81 b2 2a
Solve x2 x2
4ac ( 5)2 2(1)
5x 5x
14 by using the Quadratic Formula. 14 0.
Rewrite the equation as
Quadratic Formula
4(1)( 14)
Replace a with 1, b with
5, and c with
14.
Simplify.
7 or
2 2 and 7.
The solutions are
Exercises
Solve each equation by using the Quadratic Formula. 1. x2 2x 35 0 2. x2 10x 24 0 3. x2 11x 24 0
5,
4. 4x2
7
19x 5 0
4,
5. 14x2
6
9x 1 0
3, 8
6. 2x2 x 15 0
1
,
5
5x 2
1
8. 2y2
, y 1
3,
15 0 9. 3x2
5
7. 3x2
16x
16
0
2, 1
10. 8x2 6x 9 0
5
11. r2
,
3
3r 5 2 25
4, 4
0 12. x2 10x 50 0
13. x2
6x
23
0
14. 4x2
12x
63
0
15. x2
6x
21
0
3
©
4 2
3
6 2
3
2i
3
Glencoe Algebra 2
Glencoe/McGraw-Hill
337
Lesson 6-5
3 3
,
2 1
,
5
5 3
NAME ______________________________________________ DATE
____________ PERIOD _____
6-5
Study Guide and Intervention
(continued)
The Quadratic Formula and the Discriminant
Roots and the Discriminant
Discriminant The expression under the radical sign, b2 the discriminant. 4ac, in the Quadratic Formula is called
Roots of a Quadratic Equation
Discriminant b2 b2 b2 b2 4ac 4ac 4ac 4ac 0 and a perfect square 0, but not a perfect square 0 0 Type and Number of Roots 2 rational roots 2 irrational roots 1 rational root 2 complex roots
Find the value of the discriminant for each equation. Then describe the number and types of roots for the equation. b. 3x2 2x 5 a. 2x2 5x 3 The discriminant is The discriminant is b2 4ac 52 4(2)(3) or 1. b2 4ac ( 2)2 4(3)(5) or 56. The discriminant is a perfect square, so The discriminant is negative, so the the equation has 2 rational roots. equation has 2 complex roots.
Example
Exercises
For Exercises 1 12, complete parts a c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. 1. p2 12p 4 128; 2. 9x2 6x 1 0 0; 3. 2x2 7x 4 0 81;
two irrational roots; 4 6 4 2
4. x2 4x 4 0 32;
one rational root;
1
2 rational roots;
1
,
5. 5x2
36x
7
0 1156;
6. 4x2
4x
11
0
2 irrational roots; roots; 2 2 2
7. x2 7x 6 0 25;
2 rational roots;
1
160; 2 complex
1 i 10
40x 16 0;
,7
8m 14 8;
8. m2
9. 25x2
2 rational roots; 1, 6
10. 4x2 20x 29 0
2 irrational roots; 4 2
1 rational root; 4
12. 4x2 4x 11 0 192;
64; 11. 6x2 26x 8 0 484; 2 complex roots; 2 rational roots;
338
2 irrational roots;
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
6-5
Skills Practice
The Quadratic Formula and the Discriminant
Complete parts a c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. 1. x2 8x 16 0 2. x2 11x 26 0
0; 1 rational root; 4
3. 3x2 2x 0
225; 2 rational roots;
4. 20x2 7x 3 0
2, 13
3 1
4; 2 rational roots; 0,
5. 5x2 6 0
2
289; 2 rational roots;
6. x2 6 0
,
120; 2 irrational roots;
7. x2 8x 13 0
30 5
24; 2 irrational roots;
8. 5x2 x 1 0
6
21 10
12; 2 irrational roots;
9. x2 2x 17 0
4
3
21; 2 irrational roots; 1
10. x2 49 0
72; 2 irrational roots; 1
11. x2 x 1 0
3 2 i 2 3
196; 2 complex roots;
12. 2x2 3x 2
7i i 4 7
3; 2 complex roots; 1
7; 2 complex roots; 3
Solve each equation by using the method of your choice. Find exact solutions. 13. x2 15. x2 17. x2 19. x2 21. 2x2 23. 8x2 64 x 4x 25 10x 1 4x
8
30 11 0
14. x2
30
0 24x 27
30
0
5, 6
0 2
16. 16x2
9
,
3
15
18. x2 20. 3x2
8x 36 7x 2x
17 0 4 3
0 4
33 3
5i
11 0
2i
0 0
5 2
3
22. 2x2 24. 2x2
7 4 2 1
17
4
25. PARACHUTING Ignoring wind resistance, the distance d(t) in feet that a parachutist falls in t seconds can be estimated using the formula d(t) 16t2. If a parachutist jumps from an airplane and falls for 1100 feet before opening her parachute, how many seconds pass before she opens the parachute? about 8.3 s
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Glencoe/McGraw-Hill
339
Glencoe Algebra 2
Lesson 6-5
1
i
i 5
NAME ______________________________________________ DATE
____________ PERIOD _____
6-5
Practice
(Average)
The Quadratic Formula and the Discriminant
Complete parts a c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. 1. x2 16x 64 0 2. x2 3x 3. 9x2 24x 16 0
0; 1 rational; 8
4. x2 3x 40
9; 2 rational; 0, 3
5. 3x2 9x 2 0 105;
0; 1 rational;
6. 2x2 7x 0
4
169; 2 rational;
7. 5x2 2x 4 0
5, 8 76; i 19 5
2 irrational;
8. 12x2 x 6
9 6
0 289;
105
49; 2 rational; 0,
9. 7x2 6x 2 0
7
20; i 7 5
2 complex; 1
10. 12x2 2x 4
2 rational; 3 ,
11. 6x2 2x 1
2
0 28;
2 complex;
12. x2 3x 6 0
3
0 196;
15;
3 i 15
2 rational;
13. 4x2 3x2
1
6
,
2
0 105;
2 irrational; 1
14. 16x2 8x 1 0
7 6 1
2 complex;
15. 2x2 5x 6
0 73;
2 irrational; 3
105 8
0; 1 rational;
2 irrational; 5
73 4
Solve each equation by using the method of your choice. Find exact solutions. 16. 7x2 18. 3x2 20. 3x2 22. x2 24. 3x2 26. 4x2 28. x2 5x 8x 13x 6x 3 54 4x 4x 0 0, 3 4
5
17. 4x2
9 21
0 4x
3
1
,
0
3
1
19. x2
3, 7
8
,4 6
21. 15x2 23. x2 25. 25x2
22x 14x 20x 1 53 6
2
0 7
,
4
0 3
2i
10 5
3i
17 0
2
1 4i
0 2
27. 8x 29. 4x2
4x2 2 7
3 2
0 3
15 2
i
11
12x
2 2
30. GRAVITATION The height h(t) in feet of an object t seconds after it is propelled straight up from the ground with an initial velocity of 60 feet per second is modeled by the equation h(t) 16t2 60t. At what times will the object be at a height of 56 feet? 1.75 s, 2 s 31. STOPPING DISTANCE The formula d 0.05s2 1.1s estimates the minimum stopping distance d in feet for a car traveling s miles per hour. If a car stops in 200 feet, what is the fastest it could have been traveling when the driver applied the brakes? about 53.2 mi/h
©
Glencoe/McGraw-Hill
340
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-5
Reading to Learn Mathematics
The Quadratic Formula and the Discriminant
How is blood pressure related to age? Read the introduction to Lesson 6-5 at the top of page 313 in your textbook. Describe how you would calculate your normal blood pressure using one of the formulas in your textbook.
Pre-Activity
Sample answer: Substitute your age for A in the appropriate formula (for females or males) and evaluate the expression.
Reading the Lesson
1. a. Write the Quadratic Formula. x
b
b2 2a
4ac
5x 7, but do
b. Identify the values of a, b, and c that you would use to solve 2x2 not actually solve the equation. a
2
b
5
c
7
2. Suppose that you are solving four quadratic equations with rational coefficients and have found the value of the discriminant for each equation. In each case, give the number of roots and describe the type of roots that the equation will have.
Value of Discriminant 64 8 21 0 Number of Roots Type of Roots
2 2 2 1
real, rational complex real, irrational real, rational
Helping You Remember
3. How can looking at the Quadratic Formula help you remember the relationships between the value of the discriminant and the number of roots of a quadratic equation and whether the roots are real or complex?
Sample answer: The discriminant is the expression under the radical in the Quadratic Formula. Look at the Quadratic Formula and consider what happens when you take the principal square root of b2 4ac and apply in front of the result. If b2 4ac is positive, its principal square root will be a positive number and applying will give two different real solutions, which may be rational or irrational. If b2 4ac 0, its principal square root is 0, so applying in the Quadratic Formula will only lead to one solution, which will be rational (assuming a, b, and c are integers). If b 2 4ac is negative, since the square roots of negative numbers are not real numbers, you will get two complex roots, corresponding to the and in the symbol.
©
Glencoe/McGraw-Hill
341
Glencoe Algebra 2
Lesson 6-5
NAME ______________________________________________ DATE
____________ PERIOD _____
6-5
Enrichment
Sum and Product of Roots
Sometimes you may know the roots of a quadratic equation without knowing the equation itself. Using your knowledge of factoring to solve an equation, you can work backward to find the quadratic equation. The rule for finding the sum and product of roots is as follows:
Sum and Product of Roots If the roots of ax 2 then s1 s2 bx c s2 b and s1 a
0, with a ! 0, are s1 and s2, c . a
A road with an initial gradient, or slope, of 3% can be represented by the formula y ax2 0. 03x c, where y is the elevation and x is the distance along the curve. Suppose the elevation of the road is 1105 feet at points 200 feet and 1000 feet along the curve. You can find the equation of the transition curve. Equations of transition curves are used by civil engineers to design smooth and safe roads. The roots are x 3 ( 8) 5 3( 8) 24 Equation: x2 5x 3 and x
Add the roots. Multiply the roots.
Example
8.
10 –8 –6 –4 –2 O –10 –20
5 – (– –, –301) 2 4
y
24
0
2
4
x
–30
Write a quadratic equation that has the given roots. 1. 6, 9 2. 5, 1 3. 6, 6
x2
3x
54
0
x2
2 2 , 5 7 35x 2
4x
5
0
x2
2
12x
3 5
36
0
4. 4
3
6.
6.
7
x2
8x
13
0
4x
4
0
49x 2
42x
205
0
Find k such that the number given is a root of the equation. 7. 7; 2x2 kx 21 0 8. 2; x2 13x k 0
11
30
©
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342
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-6
Study Guide and Intervention
Analyzing Graphs of Quadratic Functions
The graph of y a(x h)2 k has the following characteristics: • Vertex: (h, k ) • Axis of symmetry: x h • Opens up if a 0 • Opens down if a 0 • Narrower than the graph of y x 2 if "a" 1 • Wider than the graph of y x 2 if "a" 1
Vertex Form of a Quadratic Function
Example each graph.
Identify the vertex, axis of symmetry, and direction of opening of
a. y 2(x 4)2 11 The vertex is at (h, k) or ( 4, 11), and the axis of symmetry is x up, and is narrower than the graph of y x2. a. y
1 (x 4
4. The graph opens
2)2
10 2. The graph opens
The vertex is at (h, k) or (2, 10), and the axis of symmetry is x down, and is wider than the graph of y x2.
Exercises
Each quadratic function is given in vertex form. Identify the vertex, axis of symmetry, and direction of opening of the graph. 1. y (x 2)2 16 2. y 4(x 3)2 7 3. y
1 (x 2
5)2
3
(2, 16); x
4. y 7(x
2; up
1)2 9
( 3,
5. y
1 (x 5
7); x
4)2
3; up
12
(5, 3); x
6. y 6(x
5; up
6)2 6
( 1,
7. y
2 (x 5
9); x
9)2
1; down
12
(4,
8. y
12); x
8(x 3)2
4; up
2
( 6, 6); x
9. y 3(x 1)2
6; up
2
(9, 12); x
10. y
5 (x 2
9; up
5)2 12
(3,
11. y
2); x
4 (x 3
3; up
22
(1,
12. y
2); x
16(x 4)2
1; down
1
7)2
( 5, 12); x
13. y 3(x 1.2)2
5; down
2.7
(7, 22); x
14. y 0.4(x
7; up
0.6)2 0.2
(4, 1); x
15. y 1.2(x
4; up
0.8)2 6.5
(1.2, 2.7); x
1.2; up
(0.6, 0.2); x down
343
0.6;
( 0.8, 6.5); x up
0.8;
©
Glencoe/McGraw-Hill
Glencoe Algebra 2
Lesson 6-6
Analyze Quadratic Functions
NAME ______________________________________________ DATE
____________ PERIOD _____
6-6
Study Guide and Intervention
(continued)
Analyzing Graphs of Quadratic Functions
Write Quadratic Functions in Vertex Form
A quadratic function is easier to graph when it is in vertex form. You can write a quadratic function of the form y ax2 bx c in vertex from by completing the square.
Example y y y y 2x2 2(x2 2(x2 2(x
Write y
2x2
12x
25 in vertex form. Then graph the function. y 12x 25 6x) 25 6x 9) 25 3)2 7
18 2(x 3)2 7.
The vertex form of the equation is y
O
x
Exercises
Write each quadratic function in vertex form. Then graph the function. 1. y x2 10x 32 2. y x2 6x 3. y x2 8x 6
y
(x y 5)2
7
y
(x
3)2 y O
9 x y
(x
8 4 –4 O –4 –8
4)2 y 10
4
8
x
O
x
–12
4. y
4x2
16x
11
5. y
3x2
12x
5
6. y
5x2
10x
9
y
4(x y 2)2
5
y
3(x y 2)2
7
y
5(x y 1)2
4
O
x
O
x
O
x
©
Glencoe/McGraw-Hill
344
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-6
Skills Practice
Analyzing Graphs of Quadratic Functions
Lesson 6-6
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. 1. y (x 2)2 2. y x2 4 3. y x2 6
y (x (2, 0); x
4. y 3(x
2)2
0; 2; up 5)2
5
y (x (0, 4); x
5. y 5x2
4; 0; down
9
0)2
y (x 0)2 6; (0, 6); x 0; up
6. y (x 2)2 18
5)2
y 3(x ( 5, 0); x
7. y x2 2x
0; 5; down
y 5(x 9; (0, 9); x 0; down
8. y x2 6x 2
0)2
y (x (2, 18); x
9. y 3x2
2)2
24x
18; 2; up
y (x (1, 6); x
1)2
6; 1; up
y (x ( 3, 7); x
3)2
7; 3; up
y 3(x (4, 48); x
4)2 48; 4; down
Graph each function. 10. y (x y 3)2
1
11. y
(x
1)2 y 2
12. y y O
(x
4)2
4
x
O
x
O
x
13. y
1 (x 2
2)2 y O
14. y
3x2 y 4
15. y
x2
6x
4 y x
O O
x
x
Write an equation for the parabola with the given vertex that passes through the given point. 16. vertex: (4, 36) point: (0, 20) 17. vertex: (3, 1) point: (2, 0) 18. vertex: ( 2, 2) point: ( 1, 3)
y
©
(x
4)2
36
y
(x
3)2
345
1
y
(x
2)2
2
Glencoe/McGraw-Hill
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-6
Practice
(Average)
Analyzing Graphs of Quadratic Functions
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. 1. y 6(x 2)2 1 2. y 2x2 2 3. y 4x2 8x
y 6(x ( 2, 1); x
4. y x2 10x
2)2
1; 2; down 5; 5; up
y 2(x (0, 2); x
5. y 2x2
0)2 0; up
2;
18
y 4(x 1)2 4; (1, 4); x 1; down
3x2 6x
y (x ( 5, 5); x
7. y 2x2
5)2
16x
20
12x
y x
8. y
2(x 3; up
3x2
3)2;
6. y
( 3, 0);
21
y 3(x (1, 2); x
9. y 2x2
1)2 2; 1; up
29
5
y 2(x ( 4, 0); x
10. y (x 3)2
4)2;
32
18x
4; down
1
y
y 3(x 6; (3, 6); x 3; down
11. y x2 y 3)2
y 2(x 4)2 ( 4, 3); x
2x2 2x
16x
3; 4; up
1
Graph each function. 6x 5 12. y
O O
x
x
Write an equation for the parabola with the given vertex that passes through the given point. 13. vertex: (1, 3) point: ( 2, 15) 14. vertex: ( 3, 0) point: (3, 18) 15. vertex: (10, point: (5, 6) 4)
y y y
1
2(x (x (x
1)2 4)2
3 4
y
1
(x
3)2
y
2
(x
10)2
4
16. Write an equation for a parabola with vertex at (4, 4) and x-intercept 6. 17. Write an equation for a parabola with vertex at ( 3, 1) and y-intercept 2.
3)2
1
18. BASEBALL The height h of a baseball t seconds after being hit is given by h(t) 16t2 80t 3. What is the maximum height that the baseball reaches, and when does this occur? 103 ft; 2.5 s 19. SCULPTURE A modern sculpture in a park contains a parabolic arc that starts at the ground and reaches a maximum height of 10 feet after a horizontal distance of 4 feet. Write a quadratic function in vertex form that describes the shape of the outside of the arc, where y is the height of a point on the arc and x is its horizontal distance from the left-hand 5 starting point of the arc. 2
10 ft
y
(x
4)
10
4 ft
©
Glencoe/McGraw-Hill
346
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-6
Reading to Learn Mathematics
Analyzing Graphs of Quadratic Equations
Read the introduction to Lesson 6-6 at the top of page 322 in your textbook. • What does adding a positive number to x2 do to the graph of y x2?
It moves the graph up.
• What does subtracting a positive number to x before squaring do to the graph of y x2? It moves the graph to the right.
Reading the Lesson
1. Complete the following information about the graph of y a. What are the coordinates of the vertex? (h, k) b. What is the equation of the axis of symmetry? x c. In which direction does the graph open if a d. What do you know about the graph if "a" a(x h)2 k.
h
0? up; down
0? If a 1?
It is wider than the graph of y
If "a"
x 2. x 2.
1? It is narrower than the graph of y
2. Match each graph with the description of the constants in the equation in vertex form. a. a c. a i.
O
0, h 0, h y 0, k 0, k
0 iii 0 ii ii. y b. a d. a iii. x O
0, h 0, h y 0, k 0, k
0 iv 0 i iv. y O
x
O
x
x
Helping You Remember
3. When graphing quadratic functions such as y (x 4)2 and y (x 5)2, many students have trouble remembering which represents a translation of the graph of y x2 to the left and which represents a translation to the right. What is an easy way to remember this?
Sample answer: In functions like y (x 4)2, the plus sign puts the graph “ahead” so that the vertex comes “sooner” than the origin and the translation is to the left. In functions like y (x 5)2, the minus puts the graph “behind” so that the vertex comes “later” than the origin and the translation is to the right.
©
Glencoe/McGraw-Hill
347
Glencoe Algebra 2
Lesson 6-6
Pre-Activity
How can the graph of y function?
x2 be used to graph any quadratic
NAME ______________________________________________ DATE
____________ PERIOD _____
6-6
Enrichment
Patterns with Differences and Sums of Squares
Some whole numbers can be written as the difference of two squares and some cannot. Formulas can be developed to describe the sets of numbers algebraically. If possible, write each number as the difference of two squares. Look for patterns. 1. 0 02 5. 4 22 9. 8 32 13. 12 42
02 02 12 22
2. 1 12 6. 5 32 10. 9 32 14. 13 72
02 22 02 62
3. 2 cannot 7. 6 cannot 11. 10 cannot 15. 14 cannot
4. 3 22 8. 7 42 12. 11 62 16. 15 42
12 32 52 12
Even numbers can be written as 2n, where n is one of the numbers 0, 1, 2, 3, and so on. Odd numbers can be written 2n 1. Use these expressions for these problems. 17. Show that any odd number can be written as the difference of two squares.
2n
1
(n
1)2
n2
18. Show that the even numbers can be divided into two sets: those that can be written in the form 4n and those that can be written in the form 2 4n.
Find 4n for n 0, 1, 2, and so on. You get {0, 4, 8, 12, …}. For 2 4n, you get {2, 6, 10, 12, …}. Together these sets include all even numbers.
19. Describe the even numbers that cannot be written as the difference of two squares. 2 4n, for n 0, 1, 2, 3, … 20. Show that the other even numbers can be written as the difference of two squares. 4n (n 1)2 (n 1)2 Every whole number can be written as the sum of squares. It is never necessary to use more than four squares. Show that this is true for the whole numbers from 0 through 15 by writing each one as the sum of the least number of squares. 21. 0 02 24. 3 12 27. 6 12 30. 9 32 33. 12 12 36. 15 12
©
22. 1 12
23. 2 12 26. 5 12
12 22 22 12 22 32 32
12 12 12 12
12 22 12 22 32 32
25. 4 22 28. 7 12 31. 10 12 34. 13 22
12 32 32
12
22
29. 8 22 32. 11 12 35. 14 12
Glencoe/McGraw-Hill
348
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-7
Study Guide and Intervention
Graphing and Solving Quadratic Inequalities
To graph a quadratic inequality in two variables, use
the following steps:
Graph Quadratic Inequalities
1. Graph the related quadratic equation, y ax2 bx c. Use a dashed line for or ; use a solid line for or . 2. Test a point inside the parabola. If it satisfies the inequality, shade the region inside the parabola; otherwise, shade the region outside the parabola.
Example
Graph the inequality y
x2
6x
7. y First graph the equation y x2 6x 7. By completing the square, you get the vertex form of the equation y (x 3)2 2, so the vertex is ( 3, 2). Make a table of values around x 3, and graph. Since the inequality includes , use a dashed line. Test the point ( 3, 0), which is inside the parabola. Since ( 3)2 6( 3) 7 2, and 0 2, ( 3, 0) satisfies the inequality. Therefore, shade the region inside the parabola.
O
x
Exercises
Graph each inequality. 1. y x2 y 8x
17
2. y
x2
6x
4 y 3. y
x2
2x y 2
O
x
O
x
O
x
4. y
x2 y O
4x
6
5. y
2x2
4x y 6. y
2x2 y 4x
2
x
O O
x
x
©
Glencoe/McGraw-Hill
349
Glencoe Algebra 2
Lesson 6-7
NAME ______________________________________________ DATE
____________ PERIOD _____
6-7
Study Guide and Intervention
(continued)
Graphing and Solving Quadratic Inequalities graphically or algebraically.
Solve Quadratic Inequalities
Quadratic inequalities in one variable can be solved
Graphical Method
To solve ax 2 bx c 0: First graph y ax 2 bx c. The solution consists of the x-values for which the graph is below the x-axis. To solve ax 2 bx c 0: First graph y ax 2 bx c. The solution consists the x-values for which the graph is above the x-axis. Find the roots of the related quadratic equation by factoring, completing the square, or using the Quadratic Formula. 2 roots divide the number line into 3 intervals. Test a value in each interval to see which intervals are solutions.
Algebraic Method
If the inequality involves solution set.
or
, the roots of the related equation are included in the
Example
Solve the inequality x2
x x2
6
0.
O
y x
x 6 0. The First find the roots of the related equation equation factors as (x 3)(x 2) 0, so the roots are 3 and 2. The graph opens up with x-intercepts 3 and 2, so it must be on or below the x-axis for 2 x 3. Therefore the solution set is {x" 2 x 3}.
Exercises
Solve each inequality. 1. x2 2x 0 2. x2 16 0 3. 0 6x x2 5
{x 2
4. c2 4
x
0}
{x 4
5. 2m2 m
x
1
4}
{x1
6. y2 8
x
5}
{c 2
7. x2 4x
c
12
2}
0
m
8. x2
1
9x
m
14 0
1
9. x2 7x 10 0
{x 2
10. 2x2 5x
x
3
6}
0
{xx
11. 4x2 23x
7 or x
15 0
2}
12.
{x2
6x2
x
11x
5}
2 0
x 3
13. 2x2 11x
x
12
1
0
xx
14. x2 4x
3
5
or x
0
5
xx
15. 3x2 16x
2 or x
5 0
1
xx
3
or x
4
350
x
1
x
5
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
6-7
Skills Practice
Graphing and Solving Quadratic Inequalities
Graph each inequality. 1. y x2 4x y 4
2. y
x2
4 y 3. y
x2
2x y O
5
x
O O
x
x
Use the graph of its related function to write the solutions of each inequality. 4. x2 6x y 9
0
5.
x2
4x y 32
0
6. x2
x
20 y 0
5 O 6 O O 2 2
x
x
x
3
7. x2 9. x2 11. x2 13. x2 15. x2 17. x2 19. x2
8
x
4
8. x2 10. x2 12. x2
x
5 or x
4
Solve each inequality algebraically. 3x 18x 10 0 2x 36 35 0
{x 2 {xx
7x x 10x 3x 64
x
81
5}
0
{xx {x 6
7x 9x 2x 2x 12x
7 or x x
6 0
5}
9}
0
6} 1}
0
{xx {xx
0 or x
12 25 0 0
7} 3}
16.
{x 6
14. x2
x
18
4 or x
0
{x 6 x2 x
15
3}
0
all reals {xx all reals
©
{x 5
18. 2x2
x
4
3} 1}
3 or x
16x
0}
{xx
20. 9x2
2 or x
9 0
Glencoe/McGraw-Hill
351
Glencoe Algebra 2
Lesson 6-7
NAME ______________________________________________ DATE
____________ PERIOD _____
6-7
Practice x2 (Average)
Graphing and Solving Quadratic Inequalities
Graph each inequality. 1. y 4 y 2. y
x2
6x
6 y 3. y
2x2
4x
2
O
x
O
x
Use the graph of its related function to write the solutions of each inequality. 4. x2
6 O –6 –12 2 4 6 8x O
8x y 0
5.
x2
2x y 3
0
6. x2
O
9x y 14
0 x x
x
0 or x
8
3
x
1
2
x
7
Solve each inequality algebraically. 7. x2 x 20 0 8. x2 10x 16 0 9. x2 4x 5 0
{xx
10. x2 14x
4 or x
49 0
5}
{x2
11. x2 5x
x
14
8}
12. x2 15 8x
all reals
13. x2 5x 7 0
{xx
14. 9x2 36x
2 or x
36 0
7}
{x 5
15. 9x 12x2
x 0 or x
3}
3
0
all reals
16. 4x2 4x 1 0
{xx
17. 5x2 10
2}
27x
xx
18. 9x2
31x
12
xx
1
xx
2
or x
5
x 3
x
4
19. FENCING Vanessa has 180 feet of fencing that she intends to use to build a rectangular play area for her dog. She wants the play area to enclose at least 1800 square feet. What are the possible widths of the play area? 30 ft to 60 ft 20. BUSINESS A bicycle maker sold 300 bicycles last year at a profit of $300 each. The maker wants to increase the profit margin this year, but predicts that each $20 increase in profit will reduce the number of bicycles sold by 10. How many $20 increases in profit can the maker add in and expect to make a total profit of at least $100,000? from 5 to 10
©
Glencoe/McGraw-Hill
352
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
6-7
Reading to Learn Mathematics
Graphing and Solving Quadratic Inequalities
How can you find the time a trampolinist spends above a certain height? Read the introduction to Lesson 6-7 at the top of page 329 in your textbook. • How far above the ground is the trampoline surface? 3.75 feet • Using the quadratic function given in the introduction, write a quadratic inequality that describes the times at which the trampolinist is more than 20 feet above the ground. 16t 2 42t 3.75 20
Pre-Activity
Reading the Lesson
1. Answer the following questions about how you would graph the inequality y x2 x 6. a. What is the related quadratic equation? y
x2
x
6
b. Should the parabola be solid or dashed? How do you know?
solid; The inequality symbol is 0 for x and 02
.
c. The point (0, 2) is inside the parabola. To use this as a test point, substitute
2
0
for y in the quadratic inequality. 6 true or false? true
d. Is the statement 2
e. Should the region inside or outside the parabola be shaded? inside 2. The graph of y x2 4x is shown at the right. Match each of the following related inequalities with its solution set. a. b. c. d. x2 x2 x2 x2 4x 4x 4x 4x 0 ii 0 iii 0 iv 0 i i. {x"x ii. {x"0 iii. {x"x iv. {x"0 0 or x x 4} 4} 4}
(0, 0) O (4, 0)
y
(2, 4)
x
0 or x x 4}
Helping You Remember
3. A quadratic inequality in two variables may have the form y ax2 bx c, y ax2 bx c, y ax2 bx c, or y ax2 bx c. Describe a way to remember which region to shade by looking at the inequality symbol and without using a test point.
Sample answer: If the symbol is or , shade the region above the parabola. If the symbol is or , shade the region below the parabola.
©
Glencoe/McGraw-Hill
353
Glencoe Algebra 2
Lesson 6-7
NAME ______________________________________________ DATE
____________ PERIOD _____
6-7
Enrichment
Graphing Absolute Value Inequalities
You can solve absolute value inequalities by graphing in much the same manner you graphed quadratic inequalities. Graph the related absolute function for each inequality by using a graphing calculator. For and , identify the x-values, if any, for which the graph lies below the x-axis. For and , identify the x values, if any, for which the graph lies above the x-axis. For each inequality, make a sketch of the related graph and find the solutions rounded to the nearest hundredth. 1. |x 3| 0 2. |x| 6 0 3.
|x
12
4|
8
0
6
x
6
x
4
4. 2|x
6|
2
0
5. |3x
3|
0
6. |x
7|
5
x
7 or x
5
all real numbers
2
x
12
7. |7x
1|
13
8. |x
3.6|
4.2
9. |2x
5|
7
x
1.71 or x
2
0.6
x
7.8
6
x
1
©
Glencoe/McGraw-Hill
354
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Test, Form 1
(x 1)2. C. x 3(x
Write the letter for the correct answer in the blank at the right of each question. 1. Find the y-intercept for f(x) A. 1 B. 1 D. 0 6)2 12? D. x 18 D. 27 1. 2. 3.
2. What is the equation of the axis of symmetry of y A. x 2 B. x 6 C. x 6 3. Find the minimum value of f(x) A. 3 B. 6 4. The graph of f(x) 2x2 A. down; maximum C. up; maximum x2 6x. C.
9
x opens _____ and has a _____ value. B. down; minimum D. up; minimum f (x )
4.
O
x
5.
6. The quadratic function f(x) A. no zeros C. exactly two zeros
x2 has _____. B. exactly one zero D. more than two zeros
6.
For Questions 7 and 8, solve each equation by factoring. 7. x2 3x 10 A. { 5, 2} 8. 2x2 6x 0 A. { 3, 0} 0 B. ( 2, 5) B. {0, 3} C. { 2, 5} C. {0, 6} 2 and 3? B. x2 x D. x2 x D. { 10, 1} D. { 3, 3} 7. 8.
9. Which quadratic equation has roots A. x2 x 6 0 C. x2 6x 1 0 10. To solve x2 first rewrite A. (x 4)2 C. (x 4)2
6 6
0 0
9.
8x 16 25 by using the Square Root Property, you would the equation as _____. 25 B. x2 8x 9 0 5 D. x2 8x 9 10x c a perfect square. C. 10 D. 50
10. 11.
11. Find the value of c that makes x2 A. 100 B. 25
© Glencoe/McGraw-Hill
355
Glencoe Algebra 2
Assessment
5. The related graph of a quadratic equation is shown at the right. Use the graph to determine the solutions of the equation. A. 2, 3 B. 3, 2 C. 0, 6 D. 0, 2
NAME
DATE
PERIOD
6
Chapter 6 Test, Form 1
(continued)
12. The quadratic equation x2 6x 1 is to be solved by completing the square. Which equation would be the first step in that solution? A. x2 6x 1 0 B. x2 6x 36 1 36 C. x(x 6) 1 D. x2 6x 9 1 9 13. Find the exact solutions to x2 A.
3 2 5
12.
3x
13
1
0 by using the Quadratic Formula. C.
3 2 13
B. 3
2
D. 3
2
5
13.
For Questions 14 and 15, use the value of the discriminant to determine the number and type of roots for each equation. 14. x2 3x 7 0 A. 2 complex roots C. 2 real, rational roots 15. x2 4x 4 A. 2 real, rational roots C. 1 real, rational root 16. What is the vertex of y 2(x 3)2 A. ( 3, 6) B. (3, 6) 6? C. ( 3, 6) 3(x D. (3, 6) 6)2 1? D. x 6 16. 17. B. 2 real, irrational roots D. 1 real, rational root B. 2 real, irrational roots D. no real roots
14.
15.
17. What is the equation of the axis of symmetry of y A. x 2 B. x 6 C. x 3
18. Which quadratic function has its vertex at (2, 3) and passes through (1, 0)? A. y 2(x 2)2 3 B. y 3(x 2)2 3 C. y 3(x 2)2 3 D. y 2(x 2)2 3 19. Which quadratic inequality is graphed at the right? A. y (x 1)2 4 B. y (x 1)2 4 C. y (x 1)2 4 D. y (x 1)2 4 20. Solve (x 4)(x 2) 0. A. {x x 2 or x 4} C. {x 2 x 4} B. {x 4 D. {x x y 18.
O
x
19.
x 2} 2 or x
4} B:
20.
Bonus Find the x-intercepts and the y-intercept of the graph of y 2(x 4)2 18.
© Glencoe/McGraw-Hill
356
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Test, Form 2A
Write the letter for the correct answer in the blank at the right of each question. 1. Identify the y-intercept and the axis of symmetry for the graph of f(x) 10x2 40x 42. A. 42; x 4 B. 0; x 4 C. 42; x 2 D. 42; x 2. Identify the quadratic function graphed at the right. A. f(x) x2 2x B. f(x) x2 2x C. f(x) x2 2x D. f(x) (x 2)2 3. Determine whether f(x) 4x2 16x value and find that value. A. minimum; 10 B. minimum; 2 f (x )
2
1.
O
x
2.
6 has a maximum or a minimum C. maximum; 10 D. maximum; 2 3.
4.
x2 4x A. 4, 0 C. between
4 and 4 3;
B. D.
4, 0 2, 4 1;
4.
5. x2 2x 5 A. between 4 and between 1 and 2 C. no real solutions 6. x2 3x A. {6} 7. 3x2 20
B. between 2 and between 3 and 4 D. 1, 6
5.
For Questions 6 and 7, solve each equation by factoring. 18 B. { 6, 3} 7x B. 5, 4
3
C. { 9, 2} 4, 5
3 5
D. { 3, 6} 20, 1
3
6.
A. { 10, 2}
C.
D.
7.
8. Which quadratic equation has roots A. x2 4x 4 0 C. 5x2 9x 2 0
2 and 1 ? B. 5x2 D. 5x2 9x 2 0 11x 2 0 8.
9. To solve 9x2 12x 4 49 by using the Square Root Property, you would first rewrite the equation as _____. A. 9x2 12x 45 0 B. (3x 2)2 49 2 2 C. (3x 2) 7 D. (3x 2) 49 10. Find the value of c that makes x2 A. 81 4
© Glencoe/McGraw-Hill
9.
9x
c a perfect square. C.
81 4
B. 9 2
D. 81
10.
Glencoe Algebra 2
357
Assessment
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
NAME
DATE
PERIOD
6
Chapter 6 Test, Form 2A
(continued)
11. The quadratic equation x2 8x 20 is to be solved by completing the square. Which equation would be a step in that solution? A. (x 4)2 4 B. x 4 2i 2 2 C. x 8x 20 0 D. x 8x 16 20 12. Find the exact solutions to 3x2 A.
5 6 13
11. 12.
5x
1 by using the Quadratic Formula. C. 5
6 37
B.
5
3
13
D. 5
6
13
For Questions 13 and 14, use the value of the discriminant to determine the number and type of roots for each equation. 13. 2x2 7x 9 0 A. 2 real, rational C. 2 complex 14. x2 20 12x 16 A. 1 real, irrational C. no real y
1 (x 2
B. 2 real, irrational D. 1 real, rational B. 2 real, rational D. 1 real, rational
13.
14.
15. Identify the vertex, axis of symmetry, and direction of opening for 8)2 2. 8; up 8; up B. ( 8, 2); x 8; down D. (8, 2); x 8; up 15. A. ( 8, 2); x C. (8, 2); x
16. Which quadratic function has its vertex at ( 2, 7) and opens down? A. y 3(x 2)2 7 B. y (x 2)2 7 C. y 12(x 2)2 7 D. y 2(x 2)2 7 17. Write y A. y (x C. y (x x2 2)2 2)2 4x 5 1 1 in vertex form. B. y D. y (x (x 2)2 2)2 5 3
16.
17.
18. Write an equation for the parabola whose vertex is at ( 8, 4) and passes through ( 6, 2). A. y C. y
3 (x 2 3 (x 2
8)2 6)2 2
4
B. y D. y
1 (x 4 3 (x 2
8)2 8)2
4 4 y O
18. x 19. Which quadratic inequality is graphed at the right? A. y (x 2)(x 3) B. y (x 2)(x 3) C. y (x 2)(x 3) D. y (x 2)(x 3) 20. Solve x2 2x 24. A. {x 4 x 6} C. {x x 6 or x
19.
4}
B. {x 6 D. {x x i 3 . 4
x 4} 4 or x
6} B:
20.
Bonus Write a quadratic equation with roots
© Glencoe/McGraw-Hill
358
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Test, Form 2B
Write the letter for the correct answer in the blank at the right of each question. 1. Identify the y-intercept and the axis of symmetry for the graph of f(x) 3x2 6x 12. A. 2; x 12 B. 12; x 1 C. 2; x 0 D. 12; x 2. Identify the quadratic function graphed at the right. A. f(x) x2 4x B. f(x) x2 4x C. f(x) x2 4x D. f(x) (x 4)2 3. Determine whether f(x) 5x2 10x value and find that value. A. minimum; 1 B. maximum; 11 f (x )
1
1.
O
x
2.
6 has a maximum or a minimum C. maximum; 1 D. minimum; 11 3.
4. x2 4x A. 4, 0 C. 2, 4 5. x2 2x 2 A. between 3 and between 0 and 1 C. no real solutions 6. x2 3x 28 A. { 4, 7} 7. 5x2 A. 4 2, 2
5
B. between D. 0, 4 2;
4 and 4 4.
B. between 1 and 0; between 2 and 3 D. 1, 1
5.
For Questions 6 and 7, solve each equation by factoring. B. { 14, 2}
2 ,2 5
C. { 7, 4}
1 ,4 5 2 ? 3
D. { 2, 14} 4, 1
5
6.
19x B. C. D. 7.
8. Which quadratic equation has roots 7 and A. 2x2 C. 3x2 11x 23x 21 14 0 0
B. 3x2 D. 2x2
19x 11x
14 21
0 0
8.
9. To solve 4x2 28x 49 25 by using the Square Root Property, you would first rewrite the equation as _____. A. (2x 7)2 25 B. (2x 7)2 5 C. (2x 7)2 5 D. 4x2 28x 24 0 10. Find the value of c that makes x2 A. 25 16
© Glencoe/McGraw-Hill
9.
5x
c a perfect square trinomial. C. 25
4
B. 5 4
D. 5
2
10.
Glencoe Algebra 2
359
Assessment
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
NAME
DATE
PERIOD
6
Chapter 6 Test, Form 2B
(continued)
11. The quadratic equation x2 18x 106 is to be solved by completing the square. Which equation would be a step in that solution? A. (x 9)2 25 B. x2 18x 106 0 C. x 9 5i D. x2 18x 81 106 12. Find the exact solutions to 2x2 A.
5 4 17
11. 12.
5x
1 by using the Quadratic Formula. C. 5
4 33
B. 5
4
17
D. 5
2
17
For Questions 13 and 14, use the value of the discriminant to determine the number and type of roots for each equation. 13. 3x2 x 12 0 A. 2 complex roots C. 2 real, rational roots 14. x2 10 3x 3 A. 2 complex roots C. 1 real, rational root B. 1 real, rational root D. 2 real, irrational roots B. 2 real, irrational roots D. 2 real, rational roots
13.
14.
15. Identify the vertex, axis of symmetry, and direction of opening for y 8(x 2)2. A. ( 8, 2); x 8 up B. ( 2, 0); x 2; down C. (2, 0); x 2; down D. ( 2, 8); x 2; down 16. Which quadratic function has its vertex at ( 3, 5) and opens down? A. y (x 3)2 5 B. y (x 3)2 5 C. y (x 3)2 5 D. y (x 3)2 5 17. Write y x2 18x 52 in vertex form. A. y (x 9)2 113 B. y 2 C. y (x 9) 52 D. y (x (x 9)2 9)2 29 29
15.
16.
17.
18. Write an equation for the parabola whose vertex is at ( 5, 7) and passes through ( 3, 1). A. y C. y
1 (x 11 1 (x 2
5)2 5)2
7 7
B. y D. y
2(x
1 (x 2
5)2 5)2
7 7 y O
18.
19. Which quadratic inequality is graphed at the right? A. y (x 3)(x 1) B. y (x 3)(x 1) C. y (x 3)(x 1) D. y (x 3)(x 1) 20. Solve 2x 3 x2. A. {x 1 x 3} C. {x x 1 or x B. {x 3 D. {x x i 2 . 3
x
19.
3}
x 1} 3 or x
1} B:
20.
Bonus Write a quadratic equation with roots
© Glencoe/McGraw-Hill
360
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Test, Form 2C
1.
1. Graph f(x) 5x2 10x, labeling the y-intercept, vertex, and axis of symmetry.
f (x )
O
x
2. Determine whether f(x) 3x2 6x 1 has a maximum or a minimum value and find that value. For Questions 3 and 4, solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 3. x2 6x 8
2.
3. y O
x
4. x2
x
5
0
4. y O
x
5. Solve 5x2
13x
6 by factoring.
5. 6.
6. GEOMETRY The length of a rectangle is 7 inches longer than its width. If the area of the rectangle is 144 square inches, what are its dimensions? 7. Write a quadratic equation with Write the equation in the form ax2 a, b, and c are integers. 6 and 3 as its roots. bx
4
7.
c
0, where
Solve each equation by using the Square Root Property. 8. x2 9. 4x2 6x 20x 9 25 25 7 8. 9.
© Glencoe/McGraw-Hill
361
Glencoe Algebra 2
Assessment
NAME
DATE
PERIOD
6
Chapter 6 Test, Form 2C
(continued)
For Questions 10 and 11, solve each equation by completing the square. 10. x2 11. 2x2 4x 3x 9 2 0 0 3x 2 by using the 10. 11. 12.
12. Find the exact solutions to 5x2 Quadratic Formula.
For Questions 13 and 14, find the value of the discriminant for each quadratic equation. Then describe the number and type of roots for the equation. 13. 9x2 14. 4x2 12x 1 9x 4 0 2
2 (x 3
13. 14. 15.
15. Identify the vertex, axis of symmetry, and direction of opening for y 5)2 7. 1)
16. Write an equation for the parabola with vertex at (2, and y-intercept 5. 17. Write y x2 6x 8 in vertex form.
16. 17. 18.
18. PHYSICS The height h (in feet) of a certain rocket t seconds after it leaves the ground is modeled by h(t) 16t2 48t 15. Write the function in vertex form and find the maximum height reached by the rocket. 19. Graph y x2 6x 9.
19.
y
O
x
20. Solve 2x2
5x
3
0 algebraically.
7 . Write the 3
20. B:
Bonus Write a quadratic equation with roots equation in the form ax2 c are integers. bx c
0, where a, b, and
© Glencoe/McGraw-Hill
362
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Test, Form 2D
1.
1. Graph f(x) x2 4x 3, labeling the y-intercept, vertex, and axis of symmetry.
f (x )
O
x
2. Determine whether f(x) 5x2 20x 3 has a maximum or a minimum value and find that value. For Questions 3 and 4, solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 3. x2 2x 3 0
2.
3. y O
x
4. 2x2
2x
3
0
4. y O
x
5. Solve 3x2
x
4 by factoring.
5. 6.
6. GEOMETRY The length of a rectangle is 10 inches longer than its width. If the area of the rectangle is 144 square inches, what are its dimensions? 7. Write a quadratic equation with Write the equation in the form ax2 a, b, and c are integers.
© Glencoe/McGraw-Hill
4 and 3 as its roots. bx
2
7.
c
0, where
363
Glencoe Algebra 2
Assessment
NAME
DATE
PERIOD
6
Chapter 6 Test, Form 2D
(continued)
Solve each equation by using the Square Root Property. 8. x2 9. 9x2 14x 12x 49 4 16 6 8. 9.
Solve each equation by completing the square. 10. x2 11. 3x2 8x x 14 2 0 0 9x 5 by using the 10. 11. 12.
12. Find the exact solutions to 2x2 Quadratic Formula.
For Questions 13 and 14, find the value of the discriminant for each quadratic equation. Then describe the number and type of roots for the equation. 13. 25x2 14. 2x2 20x 10x 4 9 0 2x 13. 14. 15. 16. 17. 18.
15. Identify the vertex, axis of symmetry, and direction of opening for y (x 6)2 5. 16. Write an equation for the parabola with vertex at ( 4, 2) and y-intercept 2. 17. Write y x2 4x 8 in vertex form.
18. PHYSICS The height h (in feet) of a certain rocket t seconds after it leaves the ground is modeled by h(t) 16t2 64t 12. Write the function in vertex form and find the maximum height reached by the rocket. 19. Graph y x2 4x 4.
19.
y
O
x
20. Solve 2x2
7x
15
0 algebraically.
5 . 4
20. B: 0,
Bonus Write a quadratic equation with roots Write the equation in the form ax2 where a, b, and c are integers.
© Glencoe/McGraw-Hill
bx
c
364
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Test, Form 3
1.
1. Graph f(x) 3 3x2 2x, labeling the y-intercept, vertex, and axis of symmetry. 2. Determine whether f(x) 1
3 x 5 3 2 x 4
f (x )
has a maximum or a minimum value and find that value. 3. BUSINESS Khalid charges $10 for a one-year subscription to his on-line newsletter. Khalid currently has 600 subscribers and he estimates that for each $1 decrease in the subscription price, he would gain 100 new subscribers. What subscription price will maximize Khalid’s income? If he charges this price, how much income should Khalid expect? For Questions 4–6, solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 4. 0.5x2 9 4.5x
O
x
2. 3. 4. y 2 O 2
x
5. 2 x
3
3
1 2 x 3
5. y 2 O 2
x
6. 4x(x
3)
9
6. y O
x
7. Solve 18x2
15
39x by factoring.
2 and 1.75 as its roots. 3
7. 8.
8. Write a quadratic equation with Write the equation in the form ax2 a, b, and c are integers.
bx
c
0, where
9. If the roots of an equation are 5 and 3, what is the equation of the axis of symmetry?
© Glencoe/McGraw-Hill
9.
Glencoe Algebra 2
365
Assessment
NAME
DATE
PERIOD
6
Chapter 6 Test, Form 3
1.44 by using the
(continued)
10. Solve 4x2 2x 0.25 Square Root Property.
10.
For Questions 11 and 12, solve each equation by completing the square. 11. 2x2 12. x2
5 x 2
2 3
0 0.5 3x 1 0 by using the
11. 12. 13. 14.
2.5x
13. Find the exact solutions to 1 x2 4 Quadratic Formula.
14. Find the value of the discriminant for 3x(0.2x 0.4) 1 0.9. Then describe the number and type of roots for the equation. 15. Find all values of k such that x2 complex roots.
3 x 5 1 2 2
kx
1
0 has two
15. 16.
16. Write an equation of the parabola with equation y Then identify the vertex, axis of symmetry, and direction of opening of your function. 17. PHYSICS The height h (in feet) of a certain aircraft t seconds after it leaves the ground is modeled by h(t) 9.1t2 591.5t 20,388.125. Write the function in vertex form and find the maximum height reached by the aircraft. 18. Write an equation for the parabola that has the same vertex as y 19. Graph y
1 2 x 3 83 and x-intercept 1. 2 5 , translated 4 units left and 2 units up. 2
17.
18. y 6x 2x)
(x2
5.25.
19.
O
x
20. Solve x
7 (x 2
1)2
0.
3 2i 5 . 4
20. B:
Bonus Write a quadratic equation with roots Write the equation in the form ax2 where a, b, and c are integers.
© Glencoe/McGraw-Hill
bx
c
0,
366
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Open-Ended Assessment
Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem. 1. Mr. Moseley asked the students in his Algebra class to work in groups to solve (x 3)2 25, stating that each student in the first group to solve the equation correctly would earn five bonus points on the next quiz. Mi-Ling’s group solved the equation using the Square Root Property. Emilia’s group used the Quadratic Formula to find the solutions. In which group would you prefer to be? Explain your reasoning. 2. The next day, Mr. Moseley had his students work in pairs to review for their chapter exam. He asked each student to write a practice problem for his or her partner. Len wrote the following problem for his partner, Jocelyn: Write an equation for the parabola whose vertex is ( 3, 4), that passes through ( 1, 0), and that opens downward. a. Jocelyn had trouble solving Len’s problem. Explain why. b. How would you change Len’s problem? c. Make the change you suggested in part b and complete the problem. 3. a. Write a quadratic function in vertex form whose maximum value is 8. b. Write a quadratic function that transforms the graph of your function from part a so that it is shifted horizontally. Explain the change you made and describe the transformation that results from this change. 4. When asked to write f(x) 2x2 12x 5 in vertex form, Joseph wrote: f(x) 2x2 12x 5 Step 1 f(x) 2(x2 6x) 5 Step 2 f(x) 2(x2 6x 9) 5 9 Step 3 f(x) 2(x 3)2 4 Is Joseph’s answer correct? Explain your reasoning. 5. The graph of y x2 4x 4 is shown. Susan used this graph to solve three quadratic inequalities. Her three solutions are given below. Replace each ! with an inequality symbol ( , , , ) so that each solution is correct. Explain your reasoning for each. a. The solution of x2 4x 4 ! 0 is {x x 2 or x 2}. 2 b. The solution of x 4x 4 ! 0 is . 2 c. The solution of x 4x 4 ! 0 is all real numbers.
© Glencoe/McGraw-Hill y O
x
367
Glencoe Algebra 2
Assessment
NAME
DATE
PERIOD SCORE
6
Chapter 6 Vocabulary Test/Review maximum value minimum value parabola quadratic equation Quadratic Formula quadratic function quadratic inequality quadratic term roots Square Root Property
axis of symmetry completing the square constant term discriminant linear term
vertex vertex form Zero Product Property zeros
Write whether each sentence is true or false. If false, replace the underlined word or words to make a true sentence. 1. The Square Root Property is used when a quadratic equation is solved by factoring. 2. In f(x) 3. 2x2 3x 3x2 4 2x 5, the linear term is 5. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
0 is an example of a quadratic equation.
4. The solutions of a quadratic equation are called its zeros. 5. The quadratic function y vertex form. 2(x 3)2 1 is written in
6. If a parabola opens upward, the y-coordinate of the vertex is the maximum value. 7. In f(x) x2 2x 1, the constant term is x2.
8. It is necessary to identify the values of a, b, and c in order to solve a quadratic equation by completing the square. 9. The highest or lowest point on a parabola is called the vertex. 10. In the Quadratic Formula, the expression b2 called the quadratic term. In your own words— Define each term. 11. parabola 4ac is
12. axis of symmetry
© Glencoe/McGraw-Hill
368
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Quiz
(Lessons 6–1 and 6–2) x2 2x 3. 1.
For Questions 1 and 2, consider f(x)
1. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. 2. Graph the function, labeling the y-intercept, vertex, and axis of symmetry.
f (x )
2.
O
x
3. Determine whether f(x) 2x2 8x 9 has a maximum or a minimum value and find that value. Solve each equation. If exact roots cannot be found, state the consecutive integers between which the roots are located. 4. x2 5. x2 2x 4x 3 7 0
3.
4. 5.
NAME
DATE
PERIOD SCORE
6
Chapter 6 Quiz
(Lessons 6–3 and 6–4)
10 13x 2. x2 4x 45
For Questions 1 and 2, solve each equation by factoring. 1. 3x2 3. STANDARDIZED TEST PRACTICE What is the integer solution of 6x2 9 21x? A. 3 B. 3 C. 1
2
1. 2. 3. 4. 5. 6.
D. 2
Write a quadratic equation with the given roots. Write the equation in the form ax2 bx c 0, where a, b, and c are integers. 4. 6 and 2 5. 2 and
3
4
Solve each equation by using the Square Root Property. 6. x2 8. 25x2 9. x2 8x 16 20x 10x 11 4 36 3 10. x2 4x 29 11 7. x2 2x 1 45
7. 8. 9. 10.
Glencoe Algebra 2
Solve each equation by completing the square.
© Glencoe/McGraw-Hill
369
Assessment
NAME
DATE
PERIOD SCORE
6
Chapter 6 Quiz
(Lessons 6–5 and 6–6)
1. 2. 3.
1. Solve x2 4x 1 by using the Quadratic Formula. Find exact solutions. 2. Find the value of the discriminant for 3x2 6x 11. Then describe the number and type of roots for the equation. 3. Graph y (x 2)2 axis of symmetry. 1. Show and label the vertex and
y
x
O
4. Write y
3x2
12x
6 in vertex form.
4. 5.
5. Write an equation for the parabola whose vertex is at ( 5, 0) and passes through (0, 50).
NAME
DATE
PERIOD SCORE
6
Chapter 6 Quiz
(Lesson 6–7)
1 (x 3
1. Graph y
2)2
3.
1.
y
O
x
2. Use the graph of its related function to write the solutions of x2 6x 5 0. 3. Solve 0 x2 4x 3 by graphing.
2. 3.
y
O
x
4. Solve 4x2
1
4x algebraically.
4.
© Glencoe/McGraw-Hill
370
Glencoe Algebra 2
NAME
DATE
PERIOD SCORE
6
Chapter 6 Mid-Chapter Test
(Lessons 6–1 through 6–4)
Part I Write the letter for the correct answer in the blank at the right of each question.
1. Which function is A. f(x) x2 2x B. f(x) x2 2x C. f(x) x2 x D. f(x) (x 3)2 graphed? 3 3 3 f (x )
O
x
1. 1)(x B. x 5 D. x
5
2. By the Zero Product Property, if (2x A. x C. x 1 or x
1 or x 2
5)
0, then _____.
1 or x 2
5
5 5 2.
1 or x
3. Write a quadratic equation with 7 and 2 as its roots.
B. y D. y
2x2 2x2
9x 9x
35 35
3.
4. The quadratic equation x2 4x 16 is to be solved by completing the square. Which equation would be a step in that solution? A. (x 2)2 20 B. x2 4x 16 0 C. (x 2)2 20 D. (x 2)2 4 6. If exact roots cannot be found, state the 5. Solve x2 6x consecutive integers between which the roots are located. A. 2, 3 B. between 4 and 3; between 2 and C. 3 D. between 5 and 4; between 2 and
4.
1 1
5.
Part II
6. Solve x2 4x 3 0 by graphing. 6. y O
x
7. Determine whether f(x)
has a maximum or a minimum value and find that value. For Questions 8 and 9, solve each equation by factoring. 8. x2 7x 18 6x 1 9. 4x2 x 8. 9. 10.
Glencoe Algebra 2
1 2 x 2
x
9
7.
10. Solve 9x2
5 by using the Square Root Property.
© Glencoe/McGraw-Hill
371
Assessment
Write the equation in the form ax2 a, b, and c are integers. A. y 5x2 37x 14 C. y 5x2 37x 14
bx
c
0, where
NAME
DATE
PERIOD
6
Chapter 6 Cumulative Review
(Chapters 1–6)
36 4 (5 7)2. (Lesson 1-1) 1. 2. 3.
1. Find the value of 12
2. Find the slope of the line that is parallel to the line with equation 3x 4y 10. (Lesson 2-3) 3. Describe the system 2x 3y 21 and y 5
2 x as 3
consistent and independent, consistent and dependent, or inconsistent. (Lesson 3-1) 4. Find the coordinates of the vertices of the figure formed by the system of inequalities. (Lesson 3-3) x 2 x y 6 y 2 x y 2 5. Find the value of 6. Solve 4 2 1 3 a b 5 12 . (Lesson 4-5) 6 4 11 by using inverse matrices. 13 4.
5. 6. 7.
(Lesson 4-8)
7. Use synthetic division to find (2x4 5x3 x2 10x 4) (x 8. Use a calculator to approximate decimal places. (Lesson 5-5) 9. Solve x 2 1 8. (Lesson 5-8)
3). (Lesson 5-3)
4
983 to three
8. 9. 10.
10. PHYSICS An object is thrown straight up from the top of a 100-foot platform at a velocity of 48 feet per second. The height h(t) of the object t seconds after being thrown is given by h(t) 16t2 48t 100. Find the maximum height reached by the object and the time it takes to achieve this height. (Lesson 6-1) 11. Solve x2 2x 3 by graphing. (Lesson 6-2)
11. y O
x
12. Solve 4x2
4x
24 by factoring. (Lesson 6-3) 0.
12. 13.
13. Find the value of the discriminant for 7x2 5x 1 Then describe the number and type of roots for the equation. (Lesson 6-5) 14. Write y x2 7x 5 in vertex form. (Lesson 6-6)
14.
Glencoe Algebra 2
© Glencoe/McGraw-Hill
372
NAME
DATE
PERIOD
6
Standardized Test Practice
(Chapters 1–6)
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
1. If a b 3 , then 8a equals which of the following? 2 A. 16b B. 12b C. 3b 2
D. 8 b
3
1.
A
B
C
D
2. 20% of 3 yards is how many fifths of 9 feet? E. 1 F. 6 G. 10 3. If u v and t 0, which of the following are true? I. ut vt II. u t v t III. u t v A. I only B. III only C. I and II only D. I, II, and III 4. Which of the following is the greatest? E. 2
3
H. 15 t
2.
E
F
G
H
3.
A
B
C
D
F. 7
9
G. 10
15
H. 8
11
4.
E
F
G
H
5. If 2a 3b represents the perimeter of a rectangle and a 2b represents its width, the length is ______. A. 7b B. b C. 7b
2
D. 14b
5.
A
B
C
D
6. In the figure, what is the area of the shaded region? E. 30 F. 36 G. 54 H. 27
6 3
15
6.
E
F
G
H
7. Mr. Salazár rented a car for d days. The rental agency charged x dollars per day plus c cents per mile for the model he selected. When Mr. Salazár returned the car, he paid a total of T dollars. In terms of d, x, c, and T, how many miles did he drive? A. T (xd c) B. T xd c
C.
T xd c
D. T
c
xd
7.
A
B
C
D
8. If P(3, 2) and Q(7, 10) are the endpoints of the diameter of a circle, what is the area of the circle? E. 2 5 F. 80 G. 4 5 H. 20 9. If (x y)2 A. 120 100 and xy 20, what is the value of x2 y2? B. 140 C. 80 D. 60
8.
E
F
G
H
9.
A
B
C
D
10. The tenth term in the sequence 7, 12, 19, 28, … is ______. E. 124 F. 103 G. 57 H. 147
10.
E
F
G
H
© Glencoe/McGraw-Hill
373
Glencoe Algebra 2
Assessment
NAME
DATE
PERIOD
6
Standardized Test Practice
Part 2: Grid In
(continued)
Instructions: Enter your answer by writing each digit of the answer in a column box and then shading in the appropriate oval that corresponds to that entry.
11. If t2
6t
9, what is the value of t
1 2 ? 2
11.
. 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 . 0 1 2 3 4 5 6 7 8 9
12.
. 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 . 0 1 2 3 4 5 6 7 8 9
12. All four walls of a rectangular room that is 14 feet wide, 20 feet long, and 8 feet high, are to be painted. What is the minimum cost of paint if one gallon covers at most 130 square feet and the paint costs $22 per gallon? 13. The bar graph shows the distribution of votes among the candidates for senior class president. If 220 seniors voted in all, how many students voted for either Theo or Pam? 14. Find the median of x, 2x x
Percent of votes received 50 40 30 20 10 0 Theo Pam Ana Joey Candidates
13.
. 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 . 0 1 2 3 4 5 6 7 8 9
14.
. 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 . 0 1 2 3 4 5 6 7 8 9
1, x
22 if the mean of this set of numbers is 83.
2
13, 45, and
Part 3: Quantitative Comparison
Instructions: Compare the quantities in columns A and B. Shade in A if the quantity in column A is greater; B if the quantity in column B is greater; C if the quantities are equal; or D if the relationship cannot be determined from the information given.
Column A 15. x x˚ y˚ y˚
Column B z˚ 15. z A
B
C
D
16. a c
1
a
c c a
16.
A
B
C
D
17. x © Glencoe/McGraw-Hill
x2 y2
25 16 y 17.
A
B
C
D
374
Glencoe Algebra 2
NAME
DATE
PERIOD
6
Standardized Test Practice
Student Record Sheet
(Use with pages 342–343 of the Student Edition.)
Part 1 Multiple Choice
Select the best answer from the choices given and fill in the corresponding oval. 1 2 3
A B C D
4 5 6
A
B
C
D
7 8
A
B
C
D
9 10
A
B
C
D
A
B
C
D
A
B
C
D
A
B
C
D
A
B
C
D
A
B
C
D
A
B
C
D
Part 2 Short Response/Grid In
Solve the problem and write your answer in the blank. For Questions 14–20, also enter your answer by writing each number or symbol in a box. Then fill in the corresponding oval for that number or symbol. 11 12
.
15
/ . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9
17
19
. 0 1 2 3 4 5 6 7 8 9
. 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
. 0 1 2 3 4 5 6 7 8 9
. 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
. 0 1 2 3 4 5 6 7 8 9
13
14
/ . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9
16
18
20
. 1 2 3 4 5 6 7 8 9
. 0 1 2 3 4 5 6 7 8 9
. 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
. 0 1 2 3 4 5 6 7 8 9
. 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
. 0 1 2 3 4 5 6 7 8 9
. 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
/ . 0 1 2 3 4 5 6 7 8 9
. 0 1 2 3 4 5 6 7 8 9
Part 3 Quantitative Comparison
Select the best answer from the choices given and fill in the corresponding oval. 21 22
A B C D
23 24
A
B
C
D
25 26
A
B
C
D
27 28
A
B
C
D
A
B
C
D
A
B
C
D
A
B
C
D
A
B
C
D
© Glencoe/McGraw-Hill
A1
Glencoe Algebra 2
Answers
1 2 3 4 5 6 7 8 9
( 3) or 3 . The x-coordinate of the vertex is 3 . 2(1) 2 2 Next make a table of values for x near 3 . 2 f (x )
x
x2
3x
5
f(x)
(x, f(x))
Lesson 6-1
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-1
(continued)
Study Guide and Intervention
6-1
Graphing Quadratic Functions
Maximum and Minimum Values ax 2 bx b ; 2a
Study Guide and Intervention
Graphing Quadratic Functions c, where a Maximum or Minimum Value of a Quadratic Function 0
Graph Quadratic Functions
The y-coordinate of the vertex of a quadratic function is the maximum or minimum value of the function.
Quadratic Function
A function defined by an equation of the form f (x)
Glencoe/McGraw-Hill
The graph of f(x) ax 2 bx c, where a 0, opens up and has a minimum when a 0. The graph opens down and has a maximum when a 0. b 2a
Graph of a Quadratic Function
A parabola with these characteristics: y intercept: c; axis of symmetry: x
x-coordinate of vertex:
Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex for the graph of f(x) x2 3x 5. Use this information to graph the function. a. f(x) 3x 2 6x 7 For this function, a 3 and b 6. Since a 0, the graph opens up, and the function has a minimum value. The minimum value is the y-coordinate of the vertex. The x-coordinate of the b 6 vertex is 1.
2a 2(3) 2a
Example
Example Determine whether each function has a maximum or minimum value. Then find the maximum or minimum value of each function.
a
1, b
3, and c
5, so the y-intercept is 5. The equation of the axis of symmetry is
x
b. f(x) 100 2x x 2 For this function, a 1 and b 2. Since a 0, the graph opens down, and the function has a maximum value. The maximum value is the y-coordinate of the vertex. The x-coordinate of the vertex b 2 is 1.
2( 1)
0
02
3(0)
5
5
(0, 5)
1
12
3(1)
5
3
(1, 3)
Answers
3 2
O
3 2 2
3 x 3 2
5
11 4
3 11 , 2 4
Evaluate the function at x 1 to find the minimum value. f(1) 3(1)2 6(1) 7 4, so the minimum value of the function is 4.
Evaluate the function at x 1 to find the maximum value. f( 1) 100 2( 1) ( 1)2 101, so the minimum value of the function is 101.
A2
Exercises
1. f(x) 2x2 x 10
2
22
3(2)
5
3
(2, 3)
3
32
3(3)
5
5
(3, 5)
Exercises min., 9 7
8
4. f(x) 16
Determine whether each function has a maximum or minimum value. Then find the maximum or minimum value of each function. 2. f(x) x2 4x 7 3. f(x) 3x2 3x 1
(Lesson 6-1)
min.,
4x x2 5. f(x) x2
11
7x 11
min., 1
4
6. f(x) x2 6x 4
For Exercises 1–3, complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. 1. f(x) x2 6x 8 2. f(x) x2 2x 2 3. f(x) 2x2 4x 3
8, x x f (x) f (x ) f (x )
12 4 4 –4 O (1, 1) 4 8 8x 8 (–1, 3) 4 –8 –4 –8 –4 O
3,
1 3 2 2 1 f (x) 1 3 3 9 0 2 1 x 1 0 2 3
3
2, x
1,
1
3, x
1, 1
max., 20
7. f(x) x2 5x 2
min.,
8. f(x) 20
5 4
6x x2
max., 5
9. f(x) 4x2 x 3
x
3
2
1
4
f (x)
1
0
3
0
min.,
10. f(x)
12
f (x )
17 4 x2 4x 10
max., 29
11. f(x) x2 10x 5
min., 2 15 16
12. f(x) 6x2 12x 21
8
4
max., 14 x min.,
13. f(x) 25x2 100x 350 14. f(x)
20
0.5x2 0.3x 1.4
max., 27
15. f(x) x2 2 x 4
–8
–4 O (–3, –1) –4
4
x
8
min., 250
313
Glencoe Algebra 2
©
min.,
Glencoe/McGraw-Hill
1.445
314
max.,
7 31
32
Glencoe Algebra 2
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
Lesson 6-1
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-1
(Average)
Skills Practice
6-1
Graphing Quadratic Functions
Practice
Graphing Quadratic Functions
For each quadratic function, find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. 2. f(x) 1 6x 15 x2 3. f(x) x2
Glencoe/McGraw-Hill
1; x
5. f(x) 10x 5 8x 7 8x 15 x2 6. f(x) 1. f(x) 2. f(x) 2x2 x2 x2 4x
1. f(x)
3x2
0; x
0; 0
0; 0
15; x
3; 3
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. 12 3. f(x) 2x2 2x 1
4. f(x)
2x2
11
11; x x f (x) 15 3
16 12 8 4 O 2 8x –6 –4 –2
0; 0
0 1 3 15 2 4 6 8 x 6
5; x
5; 5
7; x
2; 2
15; x
4; 4
12; x
2;
4
2
2 0 2
1; x x f (x) 5 f (x )
0.5; 0.5
1 0 0.5 1 1 0.5 1 f (x )
2 5
f (x) 0 12 16 12 0
(–2, 16)
f (x )
16 12 8 4
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. 8. f(x) 4x 4 6x 2 0 1 0 8 3 4 6
4 6 (4, –1)
7. f(x) 9. f(x) 8
2x2
x2
x2
0; x x f (x) 16 4 f (x )
16 12 8 4 (2, 0) O
0; 0
2 0 0 f (x )
4; x
2 4 16 f (x) 8 4 6 x 0
2; 2
8; x
3; 3
Answers
x
2
1 0
1
2
(0.5, 0.5) O 2x O
x
f (x)
8
2 0
2
8
A3
4. f(x)
f (x )
(0, 0)
O
x
Determine whether each function has a maximum or a minimum value. Then find the maximum or minimum value of each function. x2 2x 8 5. f(x) x2 6x 14 6. v(x) x2 14x 57
min.; x 2 4 6x (3, –1)
(Lesson 6-1)
9
7. f(x) 2x2 4x 6
min.; 5
8. f(x) x2 4x 1
max.;
9. f(x)
8
2 2 x 3
–2
O
8x
24
min.;
8
max.; 3
max.; 0
Determine whether each function has a maximum or a minimum value. Then find the maximum or minimum value of each function. 11. f(x) 2x 8x2 12. f(x) x2
10. f(x)
6x2
10. GRAVITATION From 4 feet above a swimming pool, Susan throws a ball upward with a velocity of 32 feet per second. The height h(t) of the ball t seconds after Susan throws it is given by h(t) 16t2 32t 4. Find the maximum height reached by the ball and the time that this height is reached. 20 ft; 1 s
min.; 0
14. f(x) 4x 1 x2 15. f(x) x2
max.; 0
min.;
1
2x 3
13. f(x)
x2
2x
15
min.; 14
17. f(x) 12x 3 3x2
max.; 3
min.;
18. f(x)
4
2x2 4x 1
11. HEALTH CLUBS Last year, the SportsTime Athletic Club charged $20 to participate in an aerobics class. Seventy people attended the classes. The club wants to increase the class price this year. They expect to lose one customer for each $1 increase in the price. a. What price should the club charge to maximize the income from the aerobics classes?
16. f(x)
2x2
4x
3
$45 min.; 1
b. What is the maximum income the SportsTime Athletic Club can expect to make?
max.;
1
min.;
9
$2025
Glencoe Algebra 2
©
Glencoe Algebra 2
315
©
Glencoe/McGraw-Hill
Glencoe/McGraw-Hill
316
Glencoe Algebra 2
Answers
Lesson 6-1
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-1
Finding the Axis of Symmetry of a Parabola
As you know, if f(x) bx b b2 2a 4ac
Reading to Learn Mathematics
6-1
Enrichment
Graphing Quadratic Functions ax2 c is a quadratic function, the values of x and f(x) Pre-Activity that make f(x) equal to zero are The average of these two number values is 2x2 5x value when x of the graph of f (x) passes through the point where the maximum or minimum occurs, the axis of symmetry has the equation x b . 2a b . Since the axis of symmetry 2a
O
How can income from a rock concert be maximized?
Read the introduction to Lesson 6-1 at the top of page 286 in your textbook. b b2 2a 4ac
Glencoe/McGraw-Hill
.
b . 2a
• Based on the graph in your textbook, for what ticket price is the income the greatest? $40
• Use the graph to estimate the maximum income. about $72,000 The function f(x) has its maximum or minimum
Reading the Lesson
3, 2x2 is the term, term, and 3 is the 4 x 3x2, a ,b , and
b x = – –– 2a
1. a. For the quadratic function f(x)
5x is the
linear 3 1
0.
quadratic constant term.
f(x) = ax 2 + bx + c
x b b (– –– , f (– –– (( 2a 2a
b. For the quadratic function f(x)
c ax2 bx . c, where a
4 parabola
.
2. Consider the quadratic function f(x)
a. The graph of this function is a
Example
Use x b . 2a
Find the vertex and axis of symmetry for f(x)
5x 2
10x
7.
b. The y-intercept is .
c x
. and the function has a
Answers
c. The axis of symmetry is the line
b 2a x 10 2(5)
1
The x-coordinate of the vertex is 5x2 7 10x 12 b , or x 2a
1. 7.
A4 upward value.
d. If a
0, then the graph opens
minimum downward and the function has a using x 1. f(x)
O (0, –1)
Substitute x 1 in f(x) f( 1) 5( 1)2 10( 1) The vertex is ( 1, 12). The axis of symmetry is x
1.
e. If a value.
(–2, 4)
0, then the graph opens
(Lesson 6-1)
maximum f (x )
Find the vertex and axis of symmetry for the graph of each function b . 2a
3. Refer to the graph at the right as you complete the following sentences.
a. The curve is called a . x parabola vertex
. 1 is 1),
x2
4x
8 (2,
12); x
2
2. g(x)
4x2
8x
3 ( 1, 7); x
1
b. The line x
2 is called the axis of symmetry .
c. The point ( 2, 4) is called the
d. Because the graph contains the point (0, .
3. y
x2
8x
3 (4, 19); x
4
4. f(x)
2x2
6x
5
3 1 , ;x 2 2
3 2
the
y-intercept
Helping You Remember
5. A(x) x2 12x 36 ( 6, 0); x
4. How can you remember the way to use the x2 term of a quadratic function to tell whether the function has a maximum or a minimum value? Sample answer:
6
6. k(x)
2x2
2x
6
1 , 2
5
1 ;x 2
1 2
Remember that the graph of f(x) x 2 (with a 0) is a U-shaped curve that opens up and has a minimum. The graph of g(x) x 2 (with a 0) is just the opposite. It opens down and has a maximum.
Glencoe Algebra 2
©
Glencoe Algebra 2
317
©
Glencoe/McGraw-Hill
Glencoe/McGraw-Hill
318
Glencoe Algebra 2
A5
2. x2 4x 5 0 5,
Exercises
1
f (x ) x
Solve each equation by graphing. 3. x2 5x 4 0 1, 4
Lesson 6-2
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-2
Solving Quadratic Equations by Graphing
Study Guide and Intervention
6-2
Study Guide and Intervention
(continued)
Solving Quadratic Equations by Graphing
A quadratic equation has the form solution(s) of the equation, or the zero(s) of the related quadratic function ax 2 bx c 0, where a 0.
Solve Quadratic Equations
Estimate Solutions Often, you may not be able to find exact solutions to quadratic equations by graphing. But you can use the graph to estimate solutions. Example Solve x 2 2x 2 0 by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
The equation of the axis of symmetry of the related function is x f (x )
O
Quadratic Equation
Glencoe/McGraw-Hill
6 x2 x f (x) x Roots of a Quadratic Equation
The zeros of a quadratic function are the x-intercepts of its graph. Therefore, finding the x-intercepts is one way of solving the related quadratic equation. 0 by graphing. 6.
1 , and the equation of the 2
1 2 3 2 1 x 1 0 1 2 3
f (x )
Example
Solve x2
x
2 2(1)
1, so the vertex has x-coordinate 1. Make a table of values.
O
Graph the related function f(x)
b The x-coordinate of the vertex is 2a 1 axis of symmetry is x . 2 1 . 2
x
Make a table of values using x-values around
The x-intercepts of the graph are between 2 and 3 and between 0 and 1. So one solution is between 2 and 3, and the other solution is between 0 and 1.
x
1
1 2
0
1
2
Exercises
Solve the equations by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 1. x2 4x 2 0 2. x2 6x 6 0 3. x2 4x 2 0
f(x)
6
6
1 4
6
4
0
Answers
From the table and the graph, we can see that the zeros of the function are 2 and
3.
between 0 and 1; between 3 and 4 f (x ) f (x )
O
between between
2 and 5 and f (x )
1; 4
between between
1 and 0; 4 and 3 f (x )
1. x2
2x
8
0 2,
4
f (x )
O
x
(Lesson 6-2)
O
O
x x
O
x
O
x
4. 5. 4x f (x ) f (x )
x2
2x
4
0
5. 2x2
12x
17
0
6.
1 2 x 2
x
5 2
0
4. 6 0 4x 1
x2 6.
10x
21
0
x2
4x2
0
f (x )
between 3 and 4; between 2 and 1 f (x )
between 2 and 3; between 3 and 4 f (x )
between 2 and between 3 and 4 f (x )
1;
O
x
O O
x
O O
x x
x
O
x
3, 7
319
no real solutions
1 2
Glencoe Algebra 2
©
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
Glencoe/McGraw-Hill
320
Glencoe Algebra 2
Answers
A6
4 –6 –4 –2 O
f (x ) f (x )
O
f (x ) x
Lesson 6-2
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-2
(Average)
Skills Practice
6-2
Solving Quadratic Equations By Graphing
Use the related graph of each equation to determine its solutions. 3 f (x ) x
O
Practice
Solving Quadratic Equations By Graphing
2. 6x x2 6x O 9
Use the related graph of each equation to determine its solutions. x2 9 f (x ) f (x )
3x 2 3
1. x2 0 f (x ) f (x )
2x f (x )
3
0
3. 3x2 4x 0
1. 3 x 0 3 0 3x
3x2
2. 3x2
3. x2
2 f (x )
0
Glencoe/McGraw-Hill x f (x ) f (x )
O 3x 2 4x 3 3x 2
f (x )
O
x x x
3
f (x )
x2
3x
2 O
f (x ) x
x2
2x
3
O
x
1, 1 3
4. 6x 5 0 2x2 5. x2 10x
no real solutions
1, 2
3, 1 between 0 and 1; between 4 and 3 f (x )
8 2x 2 6x 12 5
no real solutions 6, 4
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 24 0 6. 2x2 x 6 0
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 5. 2x 4 0 6x 4 0 x2 6. x2 f (x )
between 2 f (x )
O
2 and f (x ) x
1,
4. x2
6x
5
0
Answers
1, 5
no real solutions
between 0 and 1; between 5 and 6 x f (x )
O
x2
2x 4
f (x ) x
x2
10x
O 24
x
f (x )
2x 2
x
6
O
x
Use a quadratic equation to find two real numbers that satisfy each situation, or show that no such numbers exist. 7. Their sum is 1, and their product is f (x ) f (x ) x2 x
6
(Lesson 6-2)
f (x )
x2 f (x )
6x 4
6x
5
x2
6.
8. Their sum is 5, and their product is 8.
x2 x 3, 2
6
0;
f (x )
O
f (x )
x2
5x
x 8
Use a quadratic equation to find two real numbers that satisfy each situation, or show that no such numbers exist. 8. Their sum is 0, and their product is 36.
O
x 2 5x 8 0; no such real numbers exist
7. Their sum is
4, and their product is 0.
x
x2 36
36 24
4x f (x )
0; 0,
4
x2 0; 6, 6
f (x )
x2
4x
f (x )
f (x )
x2
36
For Exercises 9 and 10, use the formula h(t) v0t 16t 2, where h(t) is the height of an object in feet, v0 is the object’s initial velocity in feet per second, and t is the time in seconds. 9. BASEBALL Marta throws a baseball with an initial upward velocity of 60 feet per second. Ignoring Marta’s height, how long after she releases the ball will it hit the ground? 3.75 s 10. VOLCANOES A volcanic eruption blasts a boulder upward with an initial velocity of 240 feet per second. How long will it take the boulder to hit the ground if it lands at the same elevation from which it was ejected? 15 s
Glencoe Algebra 2
©
O –12 –6 O 6 12 x
x
12
Glencoe Algebra 2
321
©
Glencoe/McGraw-Hill
Glencoe/McGraw-Hill
322
Glencoe Algebra 2
Lesson 6-2
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-2
Graphing Absolute Value Equations
You can solve absolute value equations in much the same way you solved quadratic equations. Graph the related absolute value function for each equation using a graphing calculator. Then use the ZERO feature in the CALC menu to find its real solutions, if any. Recall that solutions are points where the graph intersects the x-axis. For each equation, make a sketch of the related graph and find the solutions rounded to the nearest hundredth. 1. | x 5| 0 2. | 4x 3| 5 0 3. | x 7|
Reading to Learn Mathematics
6-2
Enrichment
Solving Quadratic Equations by Graphing
Pre-Activity
How does a quadratic function model a free-fall ride?
Glencoe/McGraw-Hill y Read the introduction to Lesson 6-2 at the top of page 294 in your textbook.
Write a quadratic function that describes the height of a ball t seconds after 16t 2 125 it is dropped from a height of 125 feet. h(t)
Reading the Lesson
1. The graph of the quadratic function f(x) x2 x 6 is shown at the right. Use the graph to find the solutions of the quadratic equation x2 x 6 0. 2 and 3
0
5
No solutions
7
O
x
Answers
2. Sketch a graph to illustrate each situation. 4. | x 3| 8
A7
11, 5 y y
O
0
5.
|x
9, 3
3|
6
0
6. | x
2|
3
0
a. A parabola that opens b. A parabola that opens downward and represents a upward and represents a quadratic function with two quadratic function with real zeros, both of which are exactly one real zero. The negative numbers. zero is a positive number.
c. A parabola that opens downward and represents a quadratic function with no real zeros.
1, 5
y
(Lesson 6-2)
O
x
x
O
x
7. | 3x
4|
2
8. | x
12 |
10
9. | x |
3
0
Helping You Remember
2,
2 3
22,
2
3, 3
3. Think of a memory aid that can help you recall what is meant by the zeros of a quadratic function.
Sample answer: The basic facts about a subject are sometimes called the ABCs. In the case of zeros, the ABCs are the XYZs, because the zeros are the x-values that make the y-values equal to zero.
10. Explain how solving absolute value equations algebraically and finding zeros of absolute value functions graphically are related.
Sample answer: values of x when solving algebraically are the x-intercepts (or zeros) of the function when graphed.
Glencoe Algebra 2
323
©
Glencoe/McGraw-Hill
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
324
Glencoe Algebra 2
Answers
0, 7
7. x2
6 x 30 0
0, 9
2
8. 2x2 x 3 0
0, 2
3
9. x2 14x 33 0
Lesson 6-3
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Study Guide and Intervention
Solving Quadratic Equations by Factoring
When you use factoring to solve a quadratic equation,
0, then either a 0 or b 0, or both a and b 0.
6-3
Study Guide and Intervention
q)
(continued)
Solving Quadratic Equations by Factoring
Write Quadratic Equations
(x p)(x To write a quadratic equation with roots p and q, let 0. Then multiply using FOIL.
Solve Equations by Factoring you use the following property.
Zero Product Property
For any real numbers a and b, if ab
a. 3x2
Solve each equation by factoring. 15x b. 4x2 5x 21 3x2 15x Original equation 4x2 5x 21 Original equation 3x2 15x 0 Subtract 15x from both sides. 4x2 5x 21 0 Subtract 21 from both sides. 3x(x 5) 0 Factor the binomial. (4x 7)(x 3) 0 Factor the trinomial. 3x 0 or x 5 0 Zero Product Property 4x 7 0 or x 3 0 Zero Product Property 7 x 0 or x 5 Solve each equation. x or x 3 Solve each equation. 4 The solution set is {0, 5}. 7 The solution set is ,3 .
4
Example
Example Write a quadratic equation with the given roots. Write the equation in the form ax2 bx c 0. a. 3, 5 7 1 b. , 8 3 (x p)(x q) 0 Write the pattern. (x p)(x q) 0 (x 3)[x ( 5)] 0 Replace p with 3, q with 5. 1 7 (x 3)(x 5) 0 Simplify. x x 0 8 3 x2 2x 15 0 Use FOIL. 1 7 x x 0 The equation x2 2x 15 0 has roots
3 and 5.
8 3 (8x 8 24 (8x 7)(3x 24 7) (3x 3 1) 1)
0 24 0 0 13x 7 0 has
Answers
Exercises
Solve each equation by factoring. 1. 6x2 2x 0 2. x2 7x 3. 20x2 25x
24x2
13x
7
The equation 24x2 roots
0, 1
3
7x 4. 6x2
{0, 7}
5. 6x2 27x 0
0,
6. 12x2
5 4
8x 0
7 1 and . 8 3
A8
Glencoe Algebra 2
©
(Lesson 6-3)
Exercises
Write a quadratic equation with the given roots. Write the equation in the form ax2 bx c 0. 1. 3, 4 2. 8, 2 3. 1, 9
{5,
10. 4x2
6}
27x 7 0
3 , 2
11. 3x2
1
29x 10 0
{ 11,
12. 6x2 5x
3}
4 0
x2
4. 5
x 10x
12 25
0 0
x2
5. 10, 7
10x 17x
16 70
0
6.
x2 x2
3 9. 7, 4 4x 2 1 12. 9, 6 6x 2 3 1 15. , 7 5
10x 13x
9 30
0 0
2, 15
1 , 4
13. 12x2
7
8x 1 0
10, 1 3
14. 5x2 28x 12 0 15. 2x2
1 4 , 2 3
250x 5000 0
x2
1 7. ,5 3 3x 2 2 10. 3, 5 5x 2 2 13. , 3 2 3
x2
2 8. 2, 3 3x 2 4 11. , 9 2 9x 5 14. , 4 1 2
0
1 1 , 6 2
16. 2x2 11x 40 0
2 , 5
17. 2x2
6
21x 11 0
{100, 25}
18. 3x2 2x 21 0
14x
5
0
8x
1
4
0
25x
21
0
8,
19. 8x2
5 2
14x 3 0
11, 1
2
20. 6x2 11x 2 0
7 , 3
21. 5x2
3
17x 12 0
17x
6
0
13x
4
0
55x
9
0
3 1 , 2 4
22. 12x2 25x 12 0
2, 1 6
23. 12x2 18x 6 0
3 , 5
24. 7x2
4
36x 5 0
9x 2
7 7 16. , 8 2 16x 2
4
0
8x 2
1 3 17. , 2 4 8x 2
6x
5
0
35x 2
1 1 18. , 8 6
22x
3
0
4 , 3
3 4
1 , 2
1
325
1 ,5 7
Glencoe Algebra 2
42x
49
10x
326
3
0
48x 2
14x
1
0
Glencoe/McGraw-Hill
©
Glencoe/McGraw-Hill
Glencoe Algebra 2
Lesson 6-3
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Skills Practice
Solving Quadratic Equations by Factoring
6-3
Practice
(Average)
Solving Quadratic Equations by Factoring
Solve each equation by factoring. 1. x2 4. x2 4x 6x 4x 12 8 0 {6, 0 {2, 4}
Solve each equation by factoring. 1. x2 64 { 8, 8} 2. x2 100 0 {10,
10}
2}
2. x2 5. x2 8. 7x2
16x 3x
64 2
0 {8} 0 { 2,
3. x2
20x 9x 25
100 14
0 {10} 0 {2, 7}
1} 6. x2
9. x2
3. x2
3x
2
0 {1, 2}
4. x2
4x
3
0 {1, 3}
7.
x2
0 {0, 4}
4 4x 0, 7
11. x2 13. 5x2 15. 2x2 2x 35x 8x
10x {5}
5. x2
2x
3
0 {1,
3}
6. x2
3x
10
0 {5,
2}
10. 10x2 12. x2
9 9x 0, 10
12x 25 2x 24x 8x 36 { 6}
99 { 9, 11} 60 90 0 {3, 4} 0 {9,
7.
x2
6x
5
0 {1, 5}
8.
x2
9x
0 {0, 9}
14.
36x2
5 , 6
1 45 0
5 6 1 , 3
5}
Answers
9.
x2
6x
0 {0, 6}
10. x2
6x
8
0 { 2,
4}
16. 3x2 18. 3x2
1 3}
17. 6x2 19. 15x2 21. 6x2
3 9x 0, 2
19x 5x 6 6 0
0 { 5,
3 , 5 2 3
2 3
11. x2
5x {0,
5}
12. x2
14x
49
0 {7}
20. 3x2
4 2,
2 3
3 , 2
13. x2
6
5x {2, 3}
14. x2
18x
81 { 9}
Write a quadratic equation with the given roots. Write the equation in the form ax2 bx c 0, where a, b, and c are integers. 22. 7, 2 23. 0, 3 24. 5, 8
A9
Glencoe Algebra 2
(Lesson 6-3)
15. x2
4x
21 { 3, 7}
16. 2x2
5x
3
0
1 , 2
0
3
2 ,5 3
x2
25. 7,
9x
8
14 56
0
26.
x2
6,
3x
3
0 18 0
x2
27. 3, 4
3x x
7 2
40 12
0 0
17. 4x2
5x
6
0
3 , 4
2
18. 3x2
13x
10
x2
1 28. 1, 2 2x 2 1 31. , 3
15x
0
x2
1 29. , 2 3 3x 2 1 32. 4, 3 3x 2
9x
x2
30. 0,
Write a quadratic equation with the given roots. Write the equation in the form ax2 bx c 0, where a, b, and c are integers. 19. 1, 4 x 2
3x
3
1
0
7x
2
0
33.
2x 2
2 , 3
7x
4 5
0
5x 7x
4
0 10 3 0 0
20. 6,
9 x2
3x 7x 0 2x
54
0
3x 2 24, 26
8x
3
0
13x
4
0
15x 2
22x
8
0
21.
2,
1 , 3
5 x2
22. 0, 7 x 2
1 3 , 2 4
34. NUMBER THEORY Find two consecutive even positive integers whose product is 624. 35. NUMBER THEORY Find two consecutive odd positive integers whose product is 323.
23.
3 3x 2
10x
24.
8x 2
3
0
17, 19
36. GEOMETRY The length of a rectangle is 2 feet more than its width. Find the dimensions of the rectangle if its area is 63 square feet. 7 ft by 9 ft 37. PHOTOGRAPHY The length and width of a 6-inch by 8-inch photograph are reduced by the same amount to make a new photograph whose area is half that of the original. By how many inches will the dimensions of the photograph have to be reduced? 2 in.
Glencoe Algebra 2
©
25. Find two consecutive integers whose product is 272. 16, 17
©
Glencoe/McGraw-Hill
327
Glencoe/McGraw-Hill
328
Glencoe Algebra 2
Answers
Marla (x 7)(x x2 2x
5) 35
0 0
Rosa (x 7)(x x2 2x
5) 35
0 0
Larry (x 7)(x x2 2x
5) 35
0 0
Lesson 6-3
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
NAME ______________________________________________ DATE
____________ PERIOD _____
6-3
Reading to Learn Mathematics
Solving Quadratic Equations by Factoring
How is the Zero Product Property used in geometry? Read the introduction to Lesson 6-3 at the top of page 301 in your textbook. What does the expression x(x 5) mean in this situation?
6-3
Enrichment
Pre-Activity
Euler’s Formula for Prime Numbers
Many mathematicians have searched for a formula that would generate prime numbers. One such formula was proposed by Euler and uses a quadratic polynomial, x2 x 41. Find the values of x2 x 41 for the given values of x. State whether each value of the polynomial is or is not a prime number.
It represents the area of the rectangle, since the area is the product of the width and length.
Reading the Lesson
1. The solution of a quadratic equation by factoring is shown below. Give the reason for each step of the solution. x2 x2 (x x 10x 3)(x 3 x 10x 21 7) 0 0 7 x 0 7 21
Original equation
1. x
0
2. x
1
3. x
2
41, prime
43, prime
47, prime
Answers
Add 21 to each side. Factor the trinomial. Zero Product Property Solve each equation.
.
4. x
3
5. x
4
6. x
5
53, prime
61, prime
71, prime
0 or x 3
A10
Glencoe Algebra 2
©
The solution set is
{3, 7}
7. x
6
8. x
17
9. x
28
(Lesson 6-3)
2. On an algebra quiz, students were asked to write a quadratic equation with 7 and 5 as its roots. The work that three students in the class wrote on their papers is shown below.
83, prime
347, prime
853, prime
10. x
29
11. x
30
12. x
35
Who is correct? Rosa Explain the errors in the other two students’ work.
911, prime
971, prime
1301, prime
Sample answer: Marla used the wrong factors. Larry used the correct factors but multiplied them incorrectly.
13. Does the formula produce all prime numbers greater than 40? Give examples in your answer.
Helping You Remember
3. A good way to remember a concept is to represent it in more than one way. Describe an algebraic way and a graphical way to recognize a quadratic equation that has a double root.
No. Among the primes omitted are 59, 67, 73, 79, 89, 101, 103, 107, 109, and 127.
Sample answer: Algebraic: Write the equation in the standard form ax 2 bx c 0 and examine the trinomial. If it is a perfect square trinomial, the quadratic function has a double root. Graphical: Graph the related quadratic function. If the parabola has exactly one x-intercept, then the equation has a double root.
Glencoe/McGraw-Hill
14. Euler’s formula produces primes for many values of x, but it does not work for all of them. Find the first value of x for which the formula fails. (Hint: Try multiples of ten.)
x
©
40 gives 1681, which equals 412.
330
Glencoe Algebra 2
329
Glencoe Algebra 2
Glencoe/McGraw-Hill
Lesson 6-4
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Study Guide and Intervention
Completing the Square x2 6-4
Study Guide and Intervention
Completing the Square
(continued)
Square Root Property Use the following property to solve a quadratic equation that is in the form “perfect square trinomial constant.”
Square Root Property For any real number x if x 2 n, then x n.
Complete the Square
1. Find b . 2
bx, follow these steps.
To complete the square for a quadratic expression of the form b . 2
!
2. Square
!
3. Add
b 2 to x2 2
bx.
Example
a. x2 x2 x
Solve each equation by using the Square Root Property. 8x 16 25 b. 4x2 20x 25 32 4x2 20x 25 8x 16 25 32 (x 4)2 25 (2x 5)2 32 4 2x 5 25 or x 4 25 32 or 2x 5 x 5 4 9 or x 5 4 1 2x 5 4 2 or 2x 5 1}. x
5 4 2 2 5
Example 1 Find the value of c that makes x2 22x c a perfect square trinomial. Then write the trinomial as the square of a binomial.
32 4 2 Step 1 b 22; Step 2 112 121 Step 3 c 121 The trinomial is x2 22x which can be written as (x 11)2. 121, b 2
Solve 2x2 completing the square. 2x2
2x2
Example 2
8x
8x 2
8x
24
0 by
24
24
0
0 2
Original equation Divide each side by 2. x2 4x 12 is not a perfect square.
11
x2 x2
The solution set is {9,
4x 12 x2 4x 4x 4 (x 2)2
0 12 12 16 4
Add 12 to each side.
Answers
4
Since
4 2 2
4, add 4 to each side.
The solution set is
4 2 . 2
Factor the square. Square Root Property Solve each equation.
x 2 x 6 or x 2 The solution set is {6,
2}.
Exercises
Solve each equation by using the Square Root Property. 1. x2 18x 81 49 2. x2 20x 100 64 3. 4x2 4x 1 16
A11
Glencoe Algebra 2
Exercises
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 1. x2 10x c 2. x2 60x c 3. x2 3x c
(Lesson 6-4)
{2, 16}
{ 2,
18}
3 , 2
5 2
25; (x
4. 36x2 12x 1 18 5. 9x2 12x 4 4 6. 25x2 40x 16 28 4. x2 3.2x
5)2 c 900; (x
5. x2
1 x 2
30)2 c 9 ; x 4
6. x2 2.5x
3 2 2 c 1 6
3 2
0, 4
3
4 5
2 7
2.56; (x
1.6)2
1 ; x 16
1 2 4
1.5625; (x
1.25)2
Solve each equation by completing the square. 7. y2 4y 5 0 8. x2 8x 65 0 9. s2 10s 21 0
7. 4x2
28x
49
64
8. 16x2
24x
9
81
9. 100x2
60x
9
121
15 , 2
1 2
3 , 2
3
{ 0.8, 1.4}
1, 5
10. 2x2 3x 1 0
5, 13
11. 2x2 13x 7 0
3, 7
12. 25x2 40x 9 0
10. 25x2
20x
4
75
11. 36x2
48x
16
12
12. 25x2
30x
9
96
1, 1 2
13. x2 4x 1 0 14. y2
1 ,7 2
12y 4 0
1 , 5
15. t2
9 5
3t 8 0
2 5
5 3
2 3
3
3
4 6 5
2
©
3
6
4 2
332
3 2
41
©
Glencoe/McGraw-Hill
331
Glencoe Algebra 2
Glencoe/McGraw-Hill
Glencoe Algebra 2
Answers
Lesson 6-4
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Skills Practice
Completing the Square
6-4
Practice
(Average)
Completing the Square
Solve each equation by using the Square Root Property. 1 2. x2 4x 4
Solve each equation by using the Square Root Property. 1. x2 8x 16 1 3, 5
1,
3
1. x2
8x
16
1
2. x2
6x
9
1
3. x2
10x
25
16
5,
3. x2 12x 36 25
3
14x 49 9
4,
5. 4x2
2
12x 9 4 6. x2
9, 4
1 2 9. 9x2
1
8x 16 8
1,
11 2
4. 4x2
4x
1
9
1, 2 5 8 15
4. x2
4, 10
5. x2 4x 4 2
1 , 2
9 5 8. x2 2x
5 2
2 2
6x 1 2
2 7
6. x2
2x
1
5 1
7. x2
6x
3
7. x2 6x 9 7 3 8. x2 16x 64 15
5
1
2
1 3
2
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 10. x2 12x c 11. x2 20x c 12. x2 11x c
Answers
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 9. x2 10x c 25; (x
36; (x
13. x2 0.8x
6)2 c 100; (x
14. x2 2.2x
10)2 c 121 ; x 4
15. x2 0.36x
11 2 2 c 5)2 12)2
9 2 2
10. x2
14x
c 49; (x
7)2
5 2 2 1 2 2
0.16; (x
16. x2
5 x 6
0.4)2 c 1.21; (x
17. x2
1 x 4
1.1)2 c 0.0324; (x
18. x2
5 x 3
0.18)2
A12
Glencoe Algebra 2
11. x2
24x
c 144; (x
12. x2
5x
c
25 ; x 4 1 ; x 4
c
(Lesson 6-4)
13. x2
9x
c
81 ; x 4
25 ; x 144
5 2 12
1 ; x 64
1 2 8 2 , 3
0
25 ; x 36
5 2 6 2 3
14. x2
x
c
Solve each equation by completing the square. 19. x2 6x 18 8x 8 9x 3 0 0
4,
2
20. 3x2 23. x2
x 14x
2 19 5
0
1
21. 3x2 24. x2
5x 16x
2 7
0 1, 0 0
Solve each equation by completing the square. 15. x2 13x 36 0 4, 9 16. x2 3x 0 0,
22. x2
3
0 2
6, 3
25. 2x2
7
26. x2 x
30
0
8
27. 2x2 10x
71
5
4
28. x2
22 2
3x 6 0
1
29. 2x2
21 2
5x 6 0
5
30. 7x2
15 2
6x 2 0
17. x2
x
6
0 2,
3 4, 1
3 2
18. x2
4x
13
17 1
13 2
3
19. 2x2
7x
4
0
2 33
20. 3x2
2x
1
0
1 , 3
i 2
15
5
i 4
23
3 7
i
5
21. x2
3x
6
0
22. x2
x
3
0
1
31. GEOMETRY When the dimensions of a cube are reduced by 4 inches on each side, the surface area of the new cube is 864 square inches. What were the dimensions of the original cube? 16 in. by 16 in. by 16 in. 32. INVESTMENTS The amount of money A in an account in which P dollars is invested for 2 years is given by the formula A P(1 r)2, where r is the interest rate compounded annually. If an investment of $800 in the account grows to $882 in two years, at what interest rate was it invested? 5%
Glencoe Algebra 2
©
23. x2
11
i
11
24. x2
2x
4
0 1
i
3
©
Glencoe/McGraw-Hill
333
Glencoe/McGraw-Hill
334
Glencoe Algebra 2
Lesson 6-4
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
NAME ______________________________________________ DATE
____________ PERIOD _____
6-4
Reading to Learn Mathematics
Completing the Square
How can you find the time it takes an accelerating race car to reach the finish line? Read the introduction to Lesson 6-4 at the top of page 306 in your textbook. Explain what it means to say that the driver accelerates at a constant rate of 8 feet per second square.
6-4
Enrichment
Pre-Activity
The Golden Quadratic Equations
A golden rectangle has the property that its length can be written as a b, where a is the width of the rectangle and a a b
If the driver is traveling at a certain speed at a particular moment, then one second later, the driver is traveling 8 feet per second faster.
divided into a square and a smaller golden rectangle, as shown.
a
b
a . Any golden rectangle can be b a a
Reading the Lesson
1. Give the reason for each step in the following solution of an equation by using the Square Root Property. x2 12x (x x x x 6 x 36 6)2 6 6 6 x 9 9 3 81 81 81
Original equation
The proportion used to define golden rectangles can be used to derive two quadratic equations. These are sometimes called golden quadratic equations. Solve each problem.
a
b
Answers
1. In the proportion for the golden rectangle, let a equal 1. Write the resulting quadratic equation and solve for b.
Factor the perfect square trinomial. Square Root Property 81 9 Rewrite as two equations. Solve each equation.
b2
b
1 b
0
1 2 5
9 or x 15
2. In the proportion, let b equal 1. Write the resulting quadratic equation and solve for a.
A13
Glencoe Algebra 2
©
a2
(Lesson 6-4)
a
1 a
0
1 2 5
2. Explain how to find the constant that must be added to make a binomial into a perfect square trinomial.
Sample answer: Find half of the coefficient of the linear term and square it.
3. a. What is the first step in solving the equation 3x2 6x 5x 5 by completing the square? 12 0 by completing the
3. Describe the difference between the two golden quadratic equations you found in exercises 1 and 2.
The signs of the first-degree terms are opposite.
Divide the equation by 3.
b. What is the first step in solving the equation x2 square? Add 12 to each side. 4. Show that the positive solutions of the two equations in exercises 1 and 2 are reciprocals.
1 2
5
1 2
5
(12)
4
(
5)
2
1 4
5
Helping You Remember
4. How can you use the rules for squaring a binomial to help you remember the procedure for changing a binomial into a perfect square trinomial?
1
5. Use the Pythagorean Theorem to find a radical expression for the diagonal of a golden rectangle when a 1.
One of the rules for squaring a binomial is (x y) 2 x 2 2xy y 2. In completing the square, you are starting with x 2 bx and need to find y 2. This shows you that b 2y, so y b . That is why you must take half of 2
d
10 2
2 5
the coefficient and square it to get the constant that must be added to complete the square.
335
6. Find a radical expression for the diagonal of a golden rectangle when b
1.
d
©
10 2
2 5
Glencoe/McGraw-Hill
Glencoe Algebra 2
Glencoe/McGraw-Hill
336
Glencoe Algebra 2
Answers
3
337
4 2
6 2 2
3
2i
3
Glencoe Algebra 2
Glencoe Algebra 2
14. 4x2 12x 63 0 6x 15. x2 21
©
Lesson 6-5
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-5
(continued)
Study Guide and Intervention
6-5
The Quadratic Formula and the Discriminant
Roots and the Discriminant
Discriminant bx c 0, with a 0, are given by x b b2 2a 4ac
Study Guide and Intervention
The Quadratic Formula and the Discriminant
The expression under the radical sign, b2 the discriminant. 4ac, in the Quadratic Formula is called
The Quadratic Formula can be used to solve any quadratic equation once it is written in the form ax2 bx c 0.
.
Quadratic Formula
Glencoe/McGraw-Hill
Roots of a Quadratic Equation
Discriminant b2 4ac 4ac 4ac 4ac 0 0 1 rational root 2 complex roots 0, but not a perfect square 2 irrational roots 0 and a perfect square 2 rational roots b2
Quadratic Formula
Quadratic Formula
The solutions of ax 2
Example
14 by using the Quadratic Formula. 14 b2 5, and c with 14.
Solve 5x 0.
x2
5x
Type and Number of Roots
Rewrite the equation as
x2
x
Replace a with 1, b with
b b2 b2 2a
4ac
( 5)
( 5)2 2(1)
Simplify.
4(1)( 14)
5
81
5
2 9
2
7 or
2
Answers
The solutions are
2 and 7.
Example Find the value of the discriminant for each equation. Then describe the number and types of roots for the equation. b. 3x2 2x 5 a. 2x2 5x 3 The discriminant is The discriminant is b2 4ac 52 4(2)(3) or 1. b2 4ac ( 2)2 4(3)(5) or 56. The discriminant is a perfect square, so The discriminant is negative, so the the equation has 2 rational roots. equation has 2 complex roots. Exercises
A14
2. x2 10x 24 0 11x 1. p2 5. 14x2 9x x 1 0 15 0 6. 2x2 24 0 3. x2
Exercises
Solve each equation by using the Quadratic Formula.
1. x2
2x
35
0
5,
7
4,
6
3, 8
For Exercises 1 12, complete parts a c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. 2. 9x2 12p 4 128; two irrational roots; 6 4 2 4. x2 4x 4 0 32; 3. 2x2 6x 1 0 0; one rational root; 1 7x 4 0 81;
(Lesson 6-5)
4. 4x2
19x
5
0
3
5. 5x2 36x 7 0 1156;
2 rational roots;
6. 4x2 4x 11 0
1 ,4 2
1 , 4
5
8. 2y2 y 15 0 16x 16 0 9. 3x2
1 , 2
1 7
3,
5 2
7. 3x2
5x
2
160; 2 complex roots;
5
7. x2 7x 6 0 25;
2, 1 3
3
10x 50 0
3r 5 2 25
3
11. r2 0 12. x2
5 , 2
4, 4
2 irrational roots; 2 2 2
2 rational roots; 1 ,7
8. m2 8m 14 8;
1
9. 25x2
i 10 2
40x 16 0; 1 rational root; 4
10. 8x2
6x
9
0
3 3 , 2 4
2 1 , 5 5
5
5 3
0
2 rational roots; 1, 6
10. 4x2 20x 29 0
2 irrational roots; 4 2 64; 11. 6x2 26x 8 0 484; 2 complex roots; 2 rational roots;
5
13. x2
6x
23
0
12. 4x2
3
4x 11 0 192; 2 irrational roots;
Glencoe/McGraw-Hill
©
Glencoe/McGraw-Hill
338
Glencoe Algebra 2
Lesson 6-5
©
NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE______________ PERIOD _____
6-5
(Average)
Complete parts a c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. 1. x2 16x 64 0 3x 2. x2 3. 9x2 24x
Skills Practice
6-5
Practice
Glencoe/McGraw-Hill
The Quadratic Formula and the Discriminant
The Quadratic Formula and the Discriminant
Complete parts a c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. 2. x2 11x 26 0
1. x2
8x
16
0
16
0
0; 1 rational root; 4 0; 1 rational; 8
4. x2 3x 40 5. 3x2 9x 2 4. 20x2 7x 3 0
225; 2 rational roots;
2, 13
9; 2 rational; 0, 3
0 105;
0; 1 rational; 4
3 9 6 105
6. 2x2 7x 0
3. 3x2
2x
0
4; 2 rational roots; 0, 2
3
6. x2 6 7. 5x2 2x 4 0 0
289; 2 rational roots; 169; 2 rational; 76; i 19 5
0 196;
3 1 , 5 4
5, 8
2 irrational;
8. 12x2 x 6
49; 2 rational; 0,
0 289; 9. 7x2 6x 2 0
7 2
5. 5x2
6
0
20; 2 rational; 3 ,
4
11. 6x2 2x 1
120; 2 irrational roots;
8. 5x2 x 10. 12x2 2x 4 1 0
30 5
24; 2 irrational roots;
21 10
13. 4x2 3x2 6
6
2 complex; 1
2 3
0 28;
2 complex;
12. x2 3x 6 0
3 7
i
5
7. x2
8x
13
0
15;
2 3
12; 2 irrational roots;
10. x2 49 0
4
3
21; 2 irrational roots; 1 2 rational; 1 , 2 2 irrational; 3
7
2 irrational; 1
0 105; 14. 16x2 8x 1 0
7 6 105 8
2 complex;
15. 2x2 5x 6
3
i 2
0 73;
15
Answers
9. x2
2x
17
0
A15
3 2
12. 2x2 3x 2
72; 2 irrational roots; 1 i 2 3
196; 2 complex roots; 7; 2 complex roots; 3 i 4
7i
0; 1 rational; 1 4
2 irrational; 5
73 4
11. x2
x
1
0
3; 2 complex roots; 1
Solve each equation by using the method of your choice. Find exact solutions. 16. 7x2 18. 3x2 20. 3x2 22. x2 24. 3x2 26. 4x2 5x 8x 13x 6x 3 54 4x 28. x2 4x 0 0, 3 4
(Lesson 6-5)
Solve each equation by using the method of your choice. Find exact solutions. 14. 30 0 x2
5 7 1 , 3
0 0 3
17. 4x2
9
0
3 2
3
1 ,4 3
19. x2 21. 15x2
21
4x 22x
3, 7
8
13.
x2
64
8
16. 16x2 24x 8x 17 0 4 27 0 18. x2
30
9 , 4 3 4
15. x2
x
30
5, 6 33 3
7 4 1 2 i 5 17
2 , 3
4 5
6 3i
17 15 2 0
23. x2
14x
53
0 7
2i 2
1 2 4i
25. 25x2 27. 8x 1 20x 6 4x2 2 0 2
17. x2 20. 3x2 36 0
4x
11
0 2
15 2i
0 0
10 5 3 2
19. x2
25
0
5i
3
22. 2x2 7x 2x 3 4 24. 2x2
21. 2x2
10x
11
0
5
2
i
11
29. 4x2
12x
7
0 3
2 2
30. GRAVITATION The height h(t) in feet of an object t seconds after it is propelled straight up from the ground with an initial velocity of 60 feet per second is modeled by the equation h(t) 16t2 60t. At what times will the object be at a height of 56 feet? 1.75 s, 2 s 31. STOPPING DISTANCE The formula d 0.05s2 1.1s estimates the minimum stopping distance d in feet for a car traveling s miles per hour. If a car stops in 200 feet, what is the fastest it could have been traveling when the driver applied the brakes? about 53.2 mi/h
Glencoe Algebra 2
©
23. 8x2
1
4x
1
i
4
25. PARACHUTING Ignoring wind resistance, the distance d(t) in feet that a parachutist falls in t seconds can be estimated using the formula d(t) 16t2. If a parachutist jumps from an airplane and falls for 1100 feet before opening her parachute, how many seconds pass before she opens the parachute? about 8.3 s
Glencoe Algebra 2
339
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Glencoe/McGraw-Hill
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Glencoe Algebra 2
Answers
Sample answer: The discriminant is the expression under the radical in the Quadratic Formula. Look at the Quadratic Formula and consider what happens when you take the principal square root of b2 4ac and apply in front of the result. If b2 4ac is positive, its principal square root will be a positive number and applying will give two different real solutions, which may be rational or irrational. If b2 4ac 0, its principal square root is 0, so applying in the Quadratic Formula will only lead to one solution, which will be rational (assuming a, b, and c are integers). If b 2 4ac is negative, since the square roots of negative numbers are not real numbers, you will get two complex roots, corresponding to the and in the symbol.
341
Glencoe Algebra 2
Glencoe Algebra 2
©
Lesson 6-5
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-5
Sum and Product of Roots
Reading to Learn Mathematics
6-5
Enrichment
The Quadratic Formula and the Discriminant
Sometimes you may know the roots of a quadratic equation without knowing the equation itself. Using your knowledge of factoring to solve an equation, you can work backward to find the quadratic equation. The rule for finding the sum and product of roots is as follows:
Sum and Product of Roots then s1 s2 b and s1 a
Pre-Activity
How is blood pressure related to age?
Glencoe/McGraw-Hill
If the roots of ax 2 bx s2 c . a
Read the introduction to Lesson 6-5 at the top of page 313 in your textbook.
Describe how you would calculate your normal blood pressure using one of the formulas in your textbook. c 0, with a ! 0, are s1 and s2,
Sample answer: Substitute your age for A in the appropriate formula (for females or males) and evaluate the expression.
Reading the Lesson b 5x The roots are x
Add the roots. Multiply the roots.
1. a. Write the Quadratic Formula. x 7, but do 3 and x 8.
b2 2a 4ac
b. Identify the values of a, b, and c that you would use to solve 2x2 not actually solve the equation.
Example A road with an initial gradient, or slope, of 3% can be represented by the formula y ax2 0. 03x c, where y is the elevation and x is the distance along the curve. Suppose the elevation of the road is 1105 feet at points 200 feet and 1000 feet along the curve. You can find the equation of the transition curve. Equations of transition curves are used by civil engineers to design smooth and safe roads. y 10 –8 –6 –4 –2 O –10 –20
5 – (– –, –301) 2 4
a 3 ( 8) 5 3( 8) 24 Equation: x2 5x 24 0
2 c b
5 7
2. Suppose that you are solving four quadratic equations with rational coefficients and have found the value of the discriminant for each equation. In each case, give the number of roots and describe the type of roots that the equation will have.
Number of Roots Type of Roots
2
4
x
Answers
Value of Discriminant
A16
2 2 2 1
1. 6,
64
real, rational complex real, irrational real, rational
–30
8
21
0
Write a quadratic equation that has the given roots. 9 2. 5, 1 3. 6, 6
(Lesson 6-5)
Helping You Remember
x2
3x
54
0
x2
4x
5
0
x2
2
12x
3 5
36
0
3. How can looking at the Quadratic Formula help you remember the relationships between the value of the discriminant and the number of roots of a quadratic equation and whether the roots are real or complex?
4. 4
3
6.
6.
7
x2
8x
13
0
2 2 , 5 7 35x 2
4x
4
0
49x 2
42x
205
0
Find k such that the number given is a root of the equation. 7. 7; 2x2 kx 21 0 8. 2; x2 13x k 0
11
30
Glencoe/McGraw-Hill
©
Glencoe/McGraw-Hill
342
Glencoe Algebra 2
Vertex Form of a Quadratic Function
Lesson 6-6
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-6
Analyzing Graphs of Quadratic Functions
Write Quadratic Functions in Vertex Form
Study Guide and Intervention
6-6
Study Guide and Intervention
(continued)
Analyzing Graphs of Quadratic Functions
h)2
Glencoe/McGraw-Hill
Example
Write y 12x 2x2 2x2 2(x2 2(x2 2(x 18 2(x 3)2 7. 12x 25 6x) 25 6x 9) 25 3)2 7 y y y y The vertex form of the equation is y y Analyze Quadratic Functions
A quadratic function is easier to graph when it is in vertex form. You can write a quadratic function of the form 2 y ax bx c in vertex from by completing the square.
The graph of y a(x k has the following characteristics: • Vertex: (h, k) • Axis of symmetry: x h • Opens up if a 0 • Opens down if a 0 • Narrower than the graph of y x 2 if "a" 1 • Wider than the graph of y x 2 if "a" 1
25 in vertex form. Then graph the function.
Example
each graph. 4. The graph opens
Identify the vertex, axis of symmetry, and direction of opening of
4)2
11 a. y 2(x The vertex is at (h, k) or ( 4, 11), and the axis of symmetry is x up, and is narrower than the graph of y x2.
O
x
a. y 2. The graph opens 1. y x2 10x
1 (x 4
2)2
10
Answers
The vertex is at (h, k) or (2, 10), and the axis of symmetry is x down, and is wider than the graph of y x2.
Exercises
Write each quadratic function in vertex form. Then graph the function. 32 2. y x2 6x 3. y x2 8x 6
A17 y (x y Exercises
5)2
7
y
(x
3)2 y O
9 x y
(x
8 4 –4 O –4 –8 O
4)2 y 10
Each quadratic function is given in vertex form. Identify the vertex, axis of symmetry, and direction of opening of the graph. 2. y 4(x 7 3 3)2 3. y
1 (x 2
1. y
(x
2)2
16
5)2
(Lesson 6-6)
(2, 16); x
5. y 12 6(x 6
1 (x 5
2; up
4)2 6. y 6)2
( 3,
7); x
3; up
(5, 3); x
5; up
4
8
x
4. y
7(x
1)2
9
x
( 1,
8. y 8(x 2 3(x 3)2 9. y 1)2 2
9); x
1; down
(4,
12); x
4; up
( 6, 6); x
6; up
4. y 4x2 16x 11 5. y 3x2 12x 5 6. y
–12
7. y
2 (x 5
9)2
12
5x2
10x
9
(9, 12); x
11. y 22
4 (x 3
9; up
7)2 12. y 16(x 4)2
(3,
2); x
3; up
(1,
2); x
1; down
1
y
4(x y 2)2
5
y
3(x y 2)2
7
y
5(x y 1)2
4
10. y
5 (x 2
5)2
12
( 5, 12); x
14. y 0.4(x 0.2 0.6)2 15. y
5; down
(7, 22); x
7; up
(4, 1); x
1.2(x
4; up
0.8)2 6.5
O
O
x
13. y
3(x
1.2)2
2.7
x
(1.2, 2.7); x
1.2; up
(0.6, 0.2); x down
343
0.6;
( 0.8, 6.5); x up
0.8;
O
x
Glencoe Algebra 2
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Glencoe/McGraw-Hill
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©
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Glencoe Algebra 2
Answers
Lesson 6-6
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
NAME ______________________________________________ DATE
____________ PERIOD _____
6-6
Skills Practice
Analyzing Graphs of Quadratic Functions
6-6
Practice
(Average)
Analyzing Graphs of Quadratic Functions
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. 1. y 6(x 2)2 1 2. y 2x2 2 3. y 4x2 8x
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. 1. y (x 2)2 2. y x2 4 3. y x2 6
y (x 2)2 0; (2, 0); x 2; up
4. y 3(x 5)2
y (x (0, 4); x
5. y 5x2
0)2 4; 0; down
9
y (x 0)2 6; (0, 6); x 0; up
6. y (x 2)2 18
y 6(x 2)2 1; ( 2, 1); x 2; down x2 10x 20
y 2(x (0, 2); x
5. y 2x2
0)2 2; 0; up
18
y 4(x 1)2 4; (1, 4); x 1; down
3x2 6x
4. y
y 3(x ( 5, 0); x
7. y x2 2x
5)2 0; 5; down
5
y 5(x 0)2 9; (0, 9); x 0; down
8. y x2 6x 2
y (x 2)2 18; (2, 18); x 2; up
9. y 3x2 24x
y (x 5)2 ( 5, 5); x
2x2 16x
5; 5; up 4)2; 4; down
1
y
y x
8. y
2(x 3)2; ( 3, 0); 3; up
3x2 18x 21
12x
6. y
y 3(x (1, 2); x
9. y 2x2
1)2 2; 1; up
29
5
7. y
32
y (x 1)2 6; (1, 6); x 1; up
Graph each function. 10. y (x y y (x 3)2 ( 3, 7); x
7; 3; up
y 3(x (4, 48); x
4)2 48; 4; down
y 2(x ( 4, 0); x
10. y (x 3)2
y 3(x 3)2 6; (3, 6); x 3; down x2 y
y 2(x 4)2 ( 4, 3); x
2x2
y
16x
3; 4; up
Answers
Graph each function. 11. y 6x 5 12. y 2x 1
3)2
1
11. y
(x
1)2 y 2
12. y y O
(x
4)2
4
O
x
O
A18
O
x
O
(Lesson 6-6)
x
x
x
O
x
Write an equation for the parabola with the given vertex that passes through the given point. 15. y x2 6x 4 y 13. y
1 (x 2
2)2 y O
14. y
3x2 y 4
13. vertex: (1, 3) point: ( 2, 15)
14. vertex: ( 3, 0) point: (3, 18)
y y
2(x (x
1)2 4)2
3 4
y
1 (x 2
15. vertex: (10, point: (5, 6)
4)
3)2
y
2 (x 5
10)2
4
x
O O
16. Write an equation for a parabola with vertex at (4, 4) and x-intercept 6. x x
17. Write an equation for a parabola with vertex at ( 3,
1 (x 3)2 1 3 18. BASEBALL The height h of a baseball t seconds after being hit is given by
1) and y-intercept 2.
y
h(t) 16t2 80t 3. What is the maximum height that the baseball reaches, and when does this occur? 103 ft; 2.5 s
Write an equation for the parabola with the given vertex that passes through the given point. 16. vertex: (4, 36) point: (0, 20) 17. vertex: (3, 1) point: (2, 0) 18. vertex: ( 2, 2) point: ( 1, 3)
y
©
(x
4)2
36
y
(x
3)2
345
1
y
(x
2)2
2
19. SCULPTURE A modern sculpture in a park contains a parabolic arc that starts at the ground and reaches a maximum height of 10 feet after a horizontal distance of 4 feet. Write a quadratic function in vertex form that describes the shape of the outside of the arc, where y is the height of a point on the arc and x is its horizontal distance from the left-hand 5 starting point of the arc. 2
Glencoe Algebra 2
10 ft
y
8
(x
4)
10
4 ft
Glencoe/McGraw-Hill
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
346
Glencoe Algebra 2
It moves the graph up.
Lesson 6-6
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-6 x2 be used to graph any quadratic
Reading to Learn Mathematics
6-6
Patterns with Differences and Sums of Squares
Some whole numbers can be written as the difference of two squares and some cannot. Formulas can be developed to describe the sets of numbers algebraically. If possible, write each number as the difference of two squares. Look for patterns. 1. 0 02 a(x 9. 8 32 h)2 k. 5. 4 22
Enrichment
Analyzing Graphs of Quadratic Equations
Pre-Activity
How can the graph of y function? x2?
Glencoe/McGraw-Hill
02 02 12 22
14. 13 72 10. 9 32 6. 5 32 2. 1 12
Read the introduction to Lesson 6-6 at the top of page 322 in your textbook.
• What does adding a positive number to x2 do to the graph of y
• What does subtracting a positive number to x before squaring do to the graph of y x2? It moves the graph to the right.
Reading the Lesson
22 02 62
02
3. 2 cannot 7. 6 cannot 11. 10 cannot 15. 14 cannot
4. 3 22 8. 7 42 12. 11 62 16. 15 42
12 32 52 12
1. Complete the following information about the graph of y
a. What are the coordinates of the vertex? (h, k)
b. What is the equation of the axis of symmetry? x
h
0? up; down
13. 12 42
c. In which direction does the graph open if a 0? If a 1?
d. What do you know about the graph if "a"
Answers
It is wider than the graph of y x 2. 2n 1
x 2.
Even numbers can be written as 2n, where n is one of the numbers 0, 1, 2, 3, and so on. Odd numbers can be written 2n 1. Use these expressions for these problems. 17. Show that any odd number can be written as the difference of two squares.
A19
b. a 0, h 0, h y O
If "a"
1? It is narrower than the graph of y
(n
1)2
n2
2. Match each graph with the description of the constants in the equation in vertex form. 0, k 0, k iv. y x x
a. a 0 iv 0 i d. a y 0, h
0, k
0 iii
18. Show that the even numbers can be divided into two sets: those that can be written in the form 4n and those that can be written in the form 2 4n.
c. a iii. x O
0, h
0, k
0 ii
Find 4n for n 0, 1, 2, and so on. You get {0, 4, 8, 12, …}. For 2 4n, you get {2, 6, 10, 12, …}. Together these sets include all even numbers.
19. Describe the even numbers that cannot be written as the difference of two squares. 2 4n, for n 0, 1, 2, 3, … 20. Show that the other even numbers can be written as the difference of two squares. 4n (n 1)2 (n 1)2
(Lesson 6-6)
i.
O
y
ii.
O
x
Helping You Remember
Every whole number can be written as the sum of squares. It is never necessary to use more than four squares. Show that this is true for the whole numbers from 0 through 15 by writing each one as the sum of the least number of squares. 21. 0 02 24. 3 12 27. 6 12 30. 9 32 33. 12 12 36. 15 12 22. 1 12 23. 2 12
3. When graphing quadratic functions such as y (x 4)2 and y (x 5)2, many students have trouble remembering which represents a translation of the graph of y x2 to the left and which represents a translation to the right. What is an easy way to remember this?
12 12 12 12 12
Glencoe Algebra 2
©
12 22 12 22
Glencoe/McGraw-Hill
25. 4 22 28. 7 12 31. 10 12
26. 5 12
22 12 32 32 32
348
Glencoe Algebra 2
12
34. 13 22
22 32
29. 8 22 32. 11 12 35. 14 12
22 12 22 32 32
Sample answer: In functions like y (x 4)2, the plus sign puts the graph “ahead” so that the vertex comes “sooner” than the origin and the translation is to the left. In functions like y (x 5)2, the minus puts the graph “behind” so that the vertex comes “later” than the origin and the translation is to the right.
Glencoe Algebra 2
347
©
Glencoe/McGraw-Hill
Answers
Lesson 6-7
©
Glencoe/McGraw-Hill
NAME ______________________________________________ DATE
____________ PERIOD _____
NAME ______________________________________________ DATE
____________ PERIOD _____
6-7
Study Guide and Intervention
Graphing and Solving Quadratic Inequalities
To graph a quadratic inequality in two variables, use
6-7
Study Guide and Intervention
(continued)
Graphing and Solving Quadratic Inequalities graphically or algebraically.
Graph Quadratic Inequalities the following steps:
Solve Quadratic Inequalities
Quadratic inequalities in one variable can be solved
1. Graph the related quadratic equation, y ax2 bx c. Use a dashed line for or ; use a solid line for or . 2. Test a point inside the parabola. If it satisfies the inequality, shade the region inside the parabola; otherwise, shade the region outside the parabola.
Graphical Method
To solve ax 2 bx c 0: First graph y ax 2 bx c. The solution consists of the x-values for which the graph is below the x-axis. To solve ax 2 bx c 0: First graph y ax 2 bx c. The solution consists the x-values for which the graph is above the x-axis. Find the roots of the related quadratic equation by factoring, completing the square, or using the Quadratic Formula. 2 roots divide the number line into 3 intervals. Test a value in each interval to see which intervals are solutions.
Example
Graph the inequality y
x2
6x
7. y First graph the equation y x2 6x 7. By completing the square, you get the vertex form of the equation y (x 3)2 2, so the vertex is ( 3, 2). Make a table of values around x 3, and graph. Since the inequality includes , use a dashed line. Test the point ( 3, 0), which is inside the parabola. Since ( 3)2 6( 3) 7 2, and 0 2, ( 3, 0) satisfies the inequality. Therefore, shade the region inside the parabola.
Algebraic Method
If the inequality involves solution set.
or
, the roots of the related equation are included in the
Answers
O
x
Example
Solve the inequality x2
x
6
0.
O
y x
Exercises
Graph each inequality. 1. y x2 y First find the roots of the related equation x2 x 6 0. The equation factors as (x 3)(x 2) 0, so the roots are 3 and 2. The graph opens up with x-intercepts 3 and 2, so it must be on or below the x-axis for 2 x 3. Therefore the solution set is {x" 2 x 3}.
A20
Glencoe Algebra 2
(Lesson 6-7)
8x
17
2. y
x2
6x
4 y 3. y
x2
2x y 2
Exercises
Solve each inequality. 1. x2 2x 0 2. x2 16 0 3. 0 6x x2 5
O
x
{x 2
O
x
0}
{x 4
5. 2m2 m
x
1
4}
{x1
6. y2 8
x
5}
O
x
x
4. c2
4
{c 2
4. y x2 y O
c
12
2}
0
m
8. x2
1 2
9x
m
14 0
1
9. x2 7x 10 0
4x
6
5. y
2x2
4x y 6. y
2x2 y 4x
2
7. x2
4x
{x 2
10. 2x2 5x
x
3
6}
0
{xx
11. 4x2 23x
7 or x
15 0
2}
12.
{x2
6x2
x
11x
5}
2 0
x
O O
x
x 3
13. 2x2 11x
x
12
x
1 2
0
xx
14. x2 4x
3 or x 4
5 0
5
xx
15. 3x2 16x
2 or x
5 0
1 6
xx
Glencoe/McGraw-Hill
3 or x 2
4
350
x 1
3
x
5
©
349
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
Glencoe Algebra 2
Lesson 6-7
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-7
(Average)
Graph each inequality. 2. y 4 y O
Skills Practice
6-7
Practice
Graphing and Solving Quadratic Inequalities x2 3. y 2x y x y y
Graphing and Solving Quadratic Inequalities
5 1. y 4 6x 6 x2 2. y x2 3. y 2x2 4x y Graph each inequality. x2
1. y
x2
4x
4
2
Glencoe/McGraw-Hill
O O
y
x
O
x x
O O
x x
Use the graph of its related function to write the solutions of each inequality. 4. x2 8x y 6 O 5 –6 2 O 2 4 6 8x
Use the graph of its related function to write the solutions of each inequality. 0 5. x2 2x y O
4. x2 5. 4x x y y
6x
9
0
x2 32 0 20 0
6. x2
3
0
6. x2
9x y 14
0 x O
y
x
x
–12
Answers
6 O 2
A21 x O
x
x 8 x 4 x 5 or x 4
0 or x
8
3
x
1
2
x
7
3
Solve each inequality algebraically. 8. 2x 35 0 x2
Solve each inequality algebraically. 7. x2 x 20 0 8. x2 10x 16 0 9. x2 4x 5 0
7.
x2
3x
10
0
(Lesson 6-7)
{x 2
10. 36 x2
x {x 6
12. x2 7x 6 0
5} x x
18 0
{xx 6}
7 or x
5}
{xx
10. x2 14x
4 or x
49 0
5} all reals
13. x2 5x 7 0
{x2
11. x2 5x
x
14
8}
12. x2 15 8x
9.
x2
18x
81
0
{xx {x 6
14. x2 9x
9} 1} 3}
0
{xx
14. 9x2 36x
2 or x
36 0
7} all reals
16. 4x2 4x 1 0
{x 5
15. 9x 12x2
x {xx
17. 5x2 10
3} 2}
27x
11. x2
7x
0
{xx {x 6
16. 2x 15 x2
0 or x x x
4
7}
xx
18. 9x2
0 or x
31x 12 0
13. x2
3 4
x
12
0
{xx {x 5
18. 2x2 2x 12x 9
4 or x 3} 1}
0
3}
xx
15.
x2
10x
25
0
1 2
xx
2 or x 5
5
x 3
x
4 9
19. FENCING Vanessa has 180 feet of fencing that she intends to use to build a rectangular play area for her dog. She wants the play area to enclose at least 1800 square feet. What are the possible widths of the play area? 30 ft to 60 ft
all reals {xx
20. 9x2
17. x2
3x
0
{xx
3 or x
0}
2 or x
19.
x2
64
16x
all reals
351
Glencoe Algebra 2
20. BUSINESS A bicycle maker sold 300 bicycles last year at a profit of $300 each. The maker wants to increase the profit margin this year, but predicts that each $20 increase in profit will reduce the number of bicycles sold by 10. How many $20 increases in profit can the maker add in and expect to make a total profit of at least $100,000? from 5 to 10
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Glencoe Algebra 2
©
Glencoe/McGraw-Hill
Glencoe/McGraw-Hill
352
Glencoe Algebra 2
Answers
1. Answer the following questions about how you would graph the inequality y x2 x 6.
a. What is the related quadratic equation? y
x2 x 6
b. Should the parabola be solid or dashed? How do you know?
solid; The inequality symbol is 2 for y in the quadratic inequality. 6 true or false? true 4. 2|x 6| 2 0 0
.
Answers
c. The point (0, 2) is inside the parabola. To use this as a test point, substitute
A22 x y
(2, 4)
0
for x and
d. Is the statement 2
02
Lesson 6-7
©
____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____
NAME ______________________________________________ DATE
6-7
Graphing Absolute Value Inequalities
You can solve absolute value inequalities by graphing in much the same manner you graphed quadratic inequalities. Graph the related absolute function for each inequality by using a graphing calculator. For and , identify the x-values, if any, for which the graph lies below the x-axis. For and , identify the x values, if any, for which the graph lies above the x-axis. For each inequality, make a sketch of the related graph and find the solutions rounded to the nearest hundredth. 1. |x 3| 0 2. |x| 6 0 3.
Reading to Learn Mathematics
6-7
Enrichment
Graphing and Solving Quadratic Inequalities
Pre-Activity
How can you find the time a trampolinist spends above a certain height?
Glencoe/McGraw-Hill
|x
12
4| 8 0
Read the introduction to Lesson 6-7 at the top of page 329 in your textbook.
• How far above the ground is the trampoline surface? 3.75 feet
• Using the quadratic function given in the introduction, write a quadratic inequality that describes the times at which the trampolinist is more than 20 feet above the ground. 16t 2 42t 3.75 20
Reading the Lesson x 3 or x 3 6 x 6
x
4
5. |3x
3|
0
6. |x
7|
5
e. Should the region inside or outside the parabola be shaded? inside
7 or x
5
all real numbers
2
x
12
2. The graph of y x2 4x is shown at the right. Match each of the following related inequalities with its solution set. i. {x"x 0 or x x x (Lesson 6-7)
a. 4}
(0, 0) O (4, 0)
x2 ii. {x"0 4} 4} iii. {x"x 0 or x x 4} iv. {x"0
4x
0 ii
b.
x2
4x
0 iii
c.
x2
4x
0 iv
d.
x2
4x
0 i
7. |7x
1|
13
8. |x
3.6|
4.2
9. |2x
5|
7
Helping You Remember
x
1.71 or x
2
0.6
x
7.8
6
x
1
3. A quadratic inequality in two variables may have the form y ax2 bx c, y ax2 bx c, y ax2 bx c, or y ax2 bx c. Describe a way to remember which region to shade by looking at the inequality symbol and without using a test point.
Sample answer: If the symbol is or , shade the region above the parabola. If the symbol is or , shade the region below the parabola.
Glencoe Algebra 2
353
©
Glencoe/McGraw-Hill
Glencoe Algebra 2
©
Glencoe/McGraw-Hill
354
Glencoe Algebra 2
Chapter 6 Assessment Answer Key
Form 1 Page 355
1. 2. 3.
Page 356
12.
Form 2A Page 357
1.
B B C
D
C
13.
D
2.
B
4.
A
14.
A
3.
A
15. 5.
C
4.
B
B
16. 17.
D B
6.
B
5.
A
7. 8.
C B
18.
C
6.
D C
7.
9.
B
19.
B
8.
C
20. 10. 11.
C 1 and 7; 14
9. 10.
A B
B:
D A
(continued on the next page)
© Glencoe/McGraw-Hill
A23
Glencoe Algebra 2
Answers
Chapter 6 Assessment Answer Key
Form 2A (continued) Page 358
11. 12.
Form 2B Page 359
1.
Page 360
11. 12.
B D
B
C B
2. 13.
C
13.
C
3.
B
D
14.
D
14.
A
4. 15.
D
15.
D
5.
B
16.
A
C
16.
C
17.
B
6.
A D
17.
D
7. 18.
A
8.
18.
B
B
19.
19.
C
A
9. 20.
A C
D 16x 2 3 0
10.
20.
A 9x 2 2 0
B: Sample answer:
B: Sample answer:
© Glencoe/McGraw-Hill
A24
Glencoe Algebra 2
Chapter 6 Assessment Answer Key
Form 2C Page 361
1.
f (x )
5x
2
Page 362 f (x )
10x (1, 5) 1
10. 11. x { 2 2, 1
2 3
13}
x
(3, 0) O
12. 2.
i 31 10
maximum; 4
13. 0; 1 real, rational root
3.
2, 4 y 14. 33; 2 real, irrational roots 15. ( 5, 7); x
5; down
O
x
16. 17.
y y
3 (x 2
2)2 3)2
1.5)2
1 1
(x
16(t
between
2 and y 1;
4. between 1 and 2
18. h(t )
51; 51 ft
O
x
19.
y
5. 6.
3, 2
5
O
x
9 in. by 16 in. x x
1 or x 2
2 7. 4x
21x
18
0
20. B:
3
9x 2
7
0
8. 9.
{ 8, 2}
5 2 7
© Glencoe/McGraw-Hill
A25
Glencoe Algebra 2
Answers
Chapter 6 Assessment Answer Key
Form 2D Page 363
1.
f (x ) x2 f (x )
(0, 3) 4x 3 O (2,
Page 364
8. 9.
{3, 11}
2 3 6
x 1)
2
x
10. 11.
{4
2} 1, 2
3 41 4
2.
minimum;
17
9
12.
3.
1, y 3
13. 0; 1 real, rational root 14. 8; 2 complex roots 5); x
1 (x 4
O
x
15. (6, 16. y
6; down 4) 2 2
between 1 and 0; 4. between 1 and 2 y 17.
y
(x
2)2
2) 2
4
16(t 18. h(t) 76; 76 ft x O
19.
y
5. 6.
1, 4
3
O
x
8 in. by 18 in.
20.
x
3 2
x 5
5 0
7.
2x 2
5x
12
B:
16x 2
© Glencoe/McGraw-Hill
A26
Glencoe Algebra 2
Chapter 6 Assessment Answer Key
Form 3 Page 365
1.
f (x ) f (x )
3x 2 2x 3
Page 366
10.
{ 0.35, 0.85}
( x – 1, 8
3 3 1 3
)
O
(0, 3)
x
5
11. 12. 13. 14.
i 39 8
{ 3.5, 1} 6 4 2
2. 3. 4.
minimum; 22 25 $8.00; $6400 3, 6 y 1.2; two real, irrational roots
2 O 2
15. x 2 y k
2
16. between 3 and 2; between 4 and 5 5. y 2 1 3 x 7 ; 5 2 2 7 7 , 1 ;x ; 2 2 2
down
2 O 2
x
6. between 1 and 2 y 18.
y
29 (x 200 y 9)2
29 2
O
x
19.
7.
2 8. 12x
1 5 , 2 3
13x 14 0 20. x x
O
x
7 or x 2
1
9.
x
1
2 B: 16x
24x
29
0
© Glencoe/McGraw-Hill
A27
Glencoe Algebra 2
Answers
h(t) 9.1(t 32.5) 2 17. 30,000; 30,000 ft
Chapter 6 Assessment Answer Key
Page 367, Open-Ended Assessment Scoring Rubric
Score 4 General Description Superior A correct solution that is supported by welldeveloped, accurate explanations Specific Criteria • Shows thorough understanding of the concepts of graphing, analyzing, and finding the maximum and minimum values of quadratic functions; solving quadratic equations; and solving inequalities. • Uses appropriate strategies to solve problems. • Computations are correct. • Written explanations are exemplary. • Goes beyond requirements of some or all problems. • Shows an understanding of the concepts of graphing, analyzing, and finding the maximum and minimum values of quadratic functions; solving quadratic equations; and solving inequalities. • Uses appropriate strategies to solve problems. • Computations are mostly correct. • Written explanations are effective. • Satisfies all requirements of problems. • Shows an understanding of most of the concepts of graphing, analyzing, and finding the maximum and minimum values of quadratic functions; solving quadratic equations; and solving inequalities. • May not use appropriate strategies to solve problems. • Computations are mostly correct. • Written explanations are satisfactory. • Satisfies the requirements of most of the problems. • Final computation is correct. • No written explanations or work is shown to substantiate the final computation. • Satisfies minimal requirements of some of the problems. • Shows little or no understanding of most of the concepts of graphing, analyzing, and finding the maximum and minimum values of quadratic functions; solving quadratic equations; and solving inequalities. • Does not use appropriate strategies to solve problems. • Computations are incorrect. • Written explanations are unsatisfactory. • Does not satisfy requirements of problems. • No answer may be given.
3
Satisfactory A generally correct solution, but may contain minor flaws in reasoning or computation
2
Nearly Satisfactory A partially correct interpretation and/or solution to the problem
1
Nearly Unsatisfactory A correct solution with no supporting evidence or explanation Unsatisfactory An incorrect solution indicating no mathematical understanding of the concept or task, or no solution is given
0
© Glencoe/McGraw-Hill
A28
Glencoe Algebra 2
Chapter 6 Assessment Answer Key
Page 367, Open-Ended Assessment Sample Answers
In addition to the scoring rubric found on page A28, the following sample answers may be used as guidance in evaluating open-ended assessment items. 1. Student responses should indicate that using the Square Root Property, as Mi-Ling’s group did, would take less time than the other two methods since the equation is already set up as a perfect square set equal to a constant. To solve using either of the other two methods, the binomial would need to be expanded and the constant on the right brought to the left side of the equal sign. 2a. Jocelyn had trouble because the problem is impossible. No such parabola exists. 2b. Student responses will vary. One of the three conditions must be omitted or modified. Sample answer: Delete “...and passes through ( 1, 0).” 2c. Answers will vary and depend on the answer for part b. For example, for the sample answer in part b above, a possible equation is: y 2(x 3)2 4. 3a. Answer must be of the form y a(x h)2 8 where h is any real number and a 0. 3b. Answers must be of the form y a[x (h n)]2 8 where h and a represent the same values as in part a. The student choice is for the value of n. The student should indicate that the graph will shift to the left n units if his or her value of n is negative, but will shift the graph to the right n units if the chosen value of n is positive. 4. Students should indicate that Joseph’s answer is not correct. In Step 2, when he completed the square by inserting 9 inside the parentheses, he actually added 2(9) 18 to the right side of the equation, so he must subtract 18 from the constant on the same side, rather than add 9, to keep the statements equivalent. The correct solution is f(x) 2(x 3)2 23. 5a. ; The graph is strictly above the x-axis for all values of x other than 2. 5b. ; The graph is never below the x-axis. 5c. ; The graph is always on or above the x-axis.
© Glencoe/McGraw-Hill
A29
Glencoe Algebra 2
Answers
Chapter 6 Assessment Answer Key
Vocabulary Test/Review Page 368
1. false; Zero Product
Quiz (Lessons 6–1 and 6–2) Page 369
1. 2.
Quiz (Lessons 6–5 and 6–6) Page 370
1. 2. 3.
Property
2. false; constant term 3. false; quadratic
2
5
3; x f (x ) x
1
1;
1
96; 2 complex roots
inequality
4. false; roots 5. true 6. false; minimum
y x
2
O
x f (x) x2
2x 3
( 1,
4)
(0,
3)
O
(2,
x 1)
value
7. false; quadratic
3.
minimum; 1
4. y
term
8. false; (the) 4.
3(x y 2(x
2)2 5)2
6
3,
1
5.
Quadratic Formula
9. true 10. false; discriminant 11. Sample answer:
between 1 and 2; between 6 and 5 5.
Quiz (Lesson 6–7) Page 370 Quiz (Lessons 6–3 and 6–4) Page 369
1. 2. 3.
2 4. x
2 5. 3x
A parabola is a smooth curve that is the graph of a quadratic function.
12. Sample answer:
1.
y
5, 2
3
O
An axis of symmetry is a line along which you can fold a graph and get matching parts on both sides of the line.
x
{ 9, 5} B 4x
10x
2.
{x 1 y x
5}
12
8
0
0
3.
6. 7. 8. 9. 10.
© Glencoe/McGraw-Hill
{ 10, 2} {1
2 5
3 5}
3
O
x
{ 1, 11} {2 i
A30
{x x 4.
1 or x
3}
14}
all reals
Glencoe Algebra 2
Chapter 6 Assessment Answer Key
Mid-Chapter Test Page 371
1.
Cumulative Review Page 372
1. 2.
B
17
3 4
2.
B
3.
inconsistent
( 2, 0), ( 2, 8), 4. (0, 2), (8, 2) A
5. 6. 4.
3.
92 (2, 2x 3 4 x2
8 x 3
3) 2x
C
7.
5.
D
8. 9.
5.599
6.
1, 3 y 10.
136 ft; 1.5 s
O
x
11.
1, 3 y 7. 8. 9. 10.
minimum;
91
2
O
{ 2, 9}
0, 1
4 5 3 1
x
12. 13.
{ 2, 3}
3, 2 complex roots
14.
© Glencoe/McGraw-Hill
y
x
7 2 2
29 4
Glencoe Algebra 2
A31
Answers
51
Chapter 6 Assessment Answer Key
Standardized Test Practice Page 373
1.
A B C D
Page 374
11.
4 9 / 4
. / . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 . 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
12.
1 1 0
. 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 . 0 1 2 3 4 5 6 7 8 9
2.
E
F
G
H
3.
A
B
C
D
13. 4.
E F G H
1 5 4
. 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 . 0 1 2 3 4 5 6 7 8 9
14.
8 0
. 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 / . 0 1 2 3 4 5 6 7 8 9 . 0 1 2 3 4 5 6 7 8 9
5.
A
B
C
D
6.
E
F
G
H
7.
A
B
C
D
15.
A
B
C
D
8.
E
F
G
H
16.
A
B
C
D
9.
A
B
C
D
17. 10.
E F G H
A
B
C
D
© Glencoe/McGraw-Hill
A32
Glencoe Algebra 2
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