College Algebra Notes

MIDTERM NOTES
Solve Linear Equations
0. Cancel all denominators by multiplying every term by the LCD.
1. Simplify LHS and RHS.
2. Eliminate variable term on RHS.
3. Eliminate variable term on LHS.
4. Eliminate the coefficient of the variable.
Solve Rational Equations
1. Find LCD.
2. Cancel all denominators by multiplying every term by the LCD.
3. Solve.
4. Omit those solutions that make LCD=0.
Complex Numbers: a + bi
Powers of i: i0 = 1, i1 = i, i2 = −1, i3 = −i in = ir where r is the remainder of n upon division by 4.
Addition of complex numbers:

(a + bi) + (c + di) = (a + c) + (b + d)i

Subtraction of complex numbers:

(a + bi) − (c + di) = (a − c) + (b − d)i

Multiplication of complex numbers:
1. FOIL.
2. Change i2 into -1.
3. Combine like terms.

(a + bi) · (c + di) = (ac − bd) + (ad + bc)i

Division of complex numbers a b a + bi
= + i : Separate the division. c c c a + bi
: Multiply top and bottom by an i. di a + bi
: Multiply top and bottom by the conjugate of the denominator, c − di. c + di
Quadratic Equations
Solve by Factoring
Solve by completing the square
1. Divide every term by the leading coefficient.
2. Isolate the x2 and the x terms. coeff of x 2
3. Add to both sides
.
2
4. Rewrite the LHS as a square.
5. Solve using square root property.
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Solve by quadratic formula
Quadratic formula:
Discriminant:

x=

−b ±

b2 − 4ac = 0 b2 − 4ac > 0 b2 − 4ac < 0

√

−→
−→
−→

b2 − 4ac
2a
one real solution two real solutions two complex solutions

Solve Polynomial Equations by Factoring
1. Set to 0.
2. Factor completely.
3. Solutions are zeros of each factor:
Ax + B = 0

−→

x=

−B
A

Solve Radical Equations
1. Isolate a radical.
2. Take n-th power of both sides to get rid of radical.
3. Repeat (2) until all radicals are eliminated.
4. Solve.
5. Check answers.
Solve Equations that are Quadratic in Form
1. Make the appropriate t-substitution so that you get a quadratic equation in t.
2. Solve the resulting quadratic equation in t.
3. Back-substitute the substitution in (1) and solve for the original variable.
4. Check answers if an even radical is involved in the original problem.
Solve Absolute Value Equations
1. Isolate the absolute value.
2. Get rid of absolute value:
|A| = B −→ A = B or A = −B
3. Solve.
Solve Linear Inequalities
0. Cancel all denominators by multiplying every term by the LCD.
1. Simplify LHS and RHS.
2. Eliminate variable term on RHS.
3. Eliminate variable term on LHS.
4. Eliminate the coefficient of the variable.
Solve Absolute Value Inequalities
1. Isolate the absolute value.
2. Get rid of absolute value:
|A| < B −→ −B < A < B
|A| ≤ B −→ −B ≤ A ≤ B
|A| > B −→ A < −B or A > B
|A| ≥ B −→ A ≤ −B or A ≥ B
3. Solve the resulting compound inequalities.

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Functions
Function: one input, one output
Domain: set of inputs
Range: set of outputs
Set representation:
Function: no repeated x-coordinates
Domain: set of x-coordinates
Range: set of y-coordinates
Evaluation: (input, output)
Graph representation:
Function: Vertical line test
Domain: set of x-coordinates
Range: set of y-coordinates
Evaluation: (input, output)
Function as a formula:
Function: f (x) = an expression in x
Domain: denominator = 0 inside square root ≥ 0
Evaluation: output = f (input)

Lines
Two points determine a line (to graph the line).
Find two points: x-intercept: Plug in y = 0 and solve for x. y-intercept: Plug in x = 0 and solve for y.
Given slope, m, and a point:
Use the slope to find the second point.
Slope formula:

m=

y2 − y1 x2 − x1

Slope and a point determine a line (to find an equation of the line).
Standard form:

Ax + By = C

slope from standard form: m =
Point-slope form:

−A
B

y − y1 = m(x − x1 )

Slope-intercept form: y = mx + b slope = m and y-intercept: (0, b)
Parallel lines: same slope
Perpendicular lines: opposite reciprocal slopes

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Symmetry; Even and Odd Functions
Symmetry: Given an equation in x and y.
Symmetry about x-axis: Replace y with −y, and get the same equation.
Symmetry about y-axis: Replace x with −x, and get the same equation.
Symmetry about the origin: Replace (x, y) with (−x, −y), and get the same equation.
Even and odd Functions: Given a function, f (x).
Even function (symmetry about y-axis): f (−x) = f (x).
Odd function (symmetry about the origin): f (−x) = −f (x).

Transformations of Functions
Given y = f (x).
Translations (shifts): y = f (x) + k : vertical shift by k ; (x, y) −→ (x, y + k) y = f (x − h) : horizontal shift by h ; (x, y) −→ (x + h, y)
Scalings:
y = cf (x) : y = f (cx) :

vertical scaling by c ; horizontal scaling by

1 c (x, y) −→ (x, cy)
; (x, y) −→ ( 1 x, y) c Reflections: y = −f (x) : reflection about x-axis ; (x, y) −→ (x, −y) y = f (−x) : reflection about y-axis ; (x, y) −→ (−x, y)

Algebra on Functions
Operations on functions: Given f (x) and g(x).
(f + g)(x) = f (x) + g(x)
(f − g)(x) = f (x) − g(x)
(f · g)(x) = f (x) · g(x) f (x) f (x) = where g(x) = 0 g g(x)
Domain:
dom(f + g, f − g, f · g) = dom(f ) and dom(g) f dom
= dom(f ) and dom(g) and g(x) = 0 g Composition of functions: Given f (x) and g(x).
(f ◦ g)(x) = f (g(x))
Domain:
dom(f ◦ g) = dom(g) and dom(f (g(x)))

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Inverse Function, f −1 (x)
One-to-one function: different inputs, different outputs.
One-to-one functions have inverse functions.
Given a one-to-one function, f (x), find the inverse function.
1. Replace f (x) with y.
2. Switch x and y.
3. Solve for y.
4. Replace y with f −1 (x).
Given f (x) and f −1 (x), verify that they are indeed inverses.
Show that: f −1 (f (x)) = x f (f −1 (x)) = x

Graph Quadratic Functions
Vertex form: f (x) = a(x − h)2 + k
Same shape as y = ax2
Vertex: (h, k)
Axis of symmetry: x = h
Max/min value: k
General form: f (x) = ax2 + bx + c
Same shape as y = ax2
Vertex: (h, k)
−b
and k = f (h) h= 2a
Axis of symmetry: x = h
Max/min value: k

Graph Polynomial Functions
1. End behavior determined by the leading term, axn : a > 0 and n = even −→ up, up. a < 0 and n = even −→ down, down. a > 0 and n = odd −→ down, up. a < 0 and n = odd −→ up, down.
2. Zeros with multiplicities:
Even multiplicity −→ bounce off.
Odd multiplicity −→ go through.

Intermediate Value Theorem
Given polynomial function, f (x), on [a, b].
There is a zero between a and b if one of f (a) and f (b) is positive and the other is negative.

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Graph Rational Functions
Domain: denominator = 0
Zeros with multiplicities: numerator = 0
Even multiplicity −→ bounce off.
Odd multiplicity −→ go through.
Vertical asymptotes: denominator = 0
Even multiplicity −→ same sides (i.e. up, up or down, down).
Odd multiplicity −→ opposite sides (i.e. down, up or up, down).
Horizontal asymptote: deg(num) < deg(denom) −→ H.A.: y = 0. deg(num) > deg(denom) −→ no H.A. leading coeff of numerator deg(num) = deg(denom) −→ H.A.: y =
.
leading coeff of denominator
Division of Polynomials
Long division
Synthetic division: divisor = x ± #

Zeros of Polynomials
Given a polynomial function, f (x) = an xn + an−1 xn−1 + ... + a1 x + a0 : f (x) has n complex zeros counting multiplicities.
The following are equivalent: f (c) = 0 : c is a zero of f (x).
Remainder Theorem: r = 0 for f (x) upon division by x − c.
Factor Theorem: x − c is a factor of f (x).
Complex zeros always come as a conjugate pair: a + bi and a − bi −→ factor: x2 − 2ax + (a2 + b2 ) bi and −bi −→ factor: x2 + b2
Given a polynomial function, f (x) = an xn + an−1 xn−1 + ... + a1 x + a0 :
Rational Zero Theorem:

p is a zero of f (x), then p is a factor of a0 and q is a factor of an . q Decarte’s Rule of Signs: possible number of positive zeros: number of sign changes in f (x) − even whole number. possible number of negative zeros: number of sign changes in f (−x) − even whole number.
Use synthetic division to find all the zeros.

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Solve Polynomial Inequalities
1. Set to 0.
2. Factor completely.
3. Boundary values are zeros of each factor:
< or >: open circle
≤ or ≥: closed circle
4. Pick a test value in each partition, and plug into the inequality:
If True, shade the partition.
If False, don’t shade.

Solve Rational Inequalities
1. Set to 0.
2. Combine into a single rational expression.
3. Boundary values are zeros of numerator and zeros of denominator:
Numerator: < or >: open circle
≤ or ≥: closed circle
Denominator: always open circle
4. Pick a test value in each partition, and plug into the inequality:
If True, shade the partition.
If False, don’t shade.