Maths IA
EUROPEAN SCHOOL
Mathematics Higher Level
Portfolio
Type 1
SHADOW FUNCTIONS
Candidate Name: Emil Abrahamyan
Candidate Number: 006343-021
Supervisor: Avtandil Gagnidze
Session Year: 2013 May
Candidate Name: Emil Abrahamyan
Candidate Number: 006343-021
Mathematics Higher Level
Type 1: Shadow Functions
SHADOW FUNCTIONS
The Aim of the Investigation:
The overall aim of this investigation is to investigate different polynomials with different powers and create shadow function for each one. Afterwards identify the real and imaginary components of complex zeros from the key points along the x-axis using the method of shadow functions and their generators.
Technology Used:
Technology that had been used is shown below
1)
Autograph (Version 3.3)
Graphing Display Calculator TI-84 Plus Texas Instruments
2)
Defining terms:i
Quadratic, cubic, quartic functions are members of the family of polynomials.
A quadratic function is a function of the form constants and
A cubic function is a function of the form are constants and
A quartic function is a function of the form where are constants and
Complex numbers is any number of the form
where ,
where
are
where
are real and
.
The vertex of parabola is point where the parabola crosses its axes of symmetry.
ii
nd
Urban, P., Martin, D., Haese, R., Haese, S., Haese M. and Humphries, M. (2008) Mathematics HL (Core). 2 ed.;
Adelaide Airport: Haese & Harris Publications
2
Candidate Name: Emil Abrahamyan
Candidate Number: 006343-021
Mathematics Higher Level
Type 1: Shadow Functions
Processing:
, where is the transformation of the graph
by a vector
as shown in the Diagram 1.
x
Diagram 1
As the coordinates of the vertex of will be
.
Diagram 1 clearly shows that
are
then the coordinates of the vertex of
doesn’t have any real solutions, as it doesn’t intersect x-axis.
In order to find the imaginary solutions
Mathematics Higher Level
Portfolio
Type 1
SHADOW FUNCTIONS
Candidate Name: Emil Abrahamyan
Candidate Number: 006343-021
Supervisor: Avtandil Gagnidze
Session Year: 2013 May
Candidate Name: Emil Abrahamyan
Candidate Number: 006343-021
Mathematics Higher Level
Type 1: Shadow Functions
SHADOW FUNCTIONS
The Aim of the Investigation:
The overall aim of this investigation is to investigate different polynomials with different powers and create shadow function for each one. Afterwards identify the real and imaginary components of complex zeros from the key points along the x-axis using the method of shadow functions and their generators.
Technology Used:
Technology that had been used is shown below
1)
Autograph (Version 3.3)
Graphing Display Calculator TI-84 Plus Texas Instruments
2)
Defining terms:i
Quadratic, cubic, quartic functions are members of the family of polynomials.
A quadratic function is a function of the form constants and
A cubic function is a function of the form are constants and
A quartic function is a function of the form where are constants and
Complex numbers is any number of the form
where ,
where
are
where
are real and
.
The vertex of parabola is point where the parabola crosses its axes of symmetry.
ii
nd
Urban, P., Martin, D., Haese, R., Haese, S., Haese M. and Humphries, M. (2008) Mathematics HL (Core). 2 ed.;
Adelaide Airport: Haese & Harris Publications
2
Candidate Name: Emil Abrahamyan
Candidate Number: 006343-021
Mathematics Higher Level
Type 1: Shadow Functions
Processing:
, where is the transformation of the graph
by a vector
as shown in the Diagram 1.
x
Diagram 1
As the coordinates of the vertex of will be
.
Diagram 1 clearly shows that
are
then the coordinates of the vertex of
doesn’t have any real solutions, as it doesn’t intersect x-axis.
In order to find the imaginary solutions
Bibliography: Books: 1. Urban, P., Martin, D., Haese, R., Haese, S., Haese M. and Humphries, M. (2008) Mathematics HL (Core). 2nd ed.; Adelaide Airport: Haese & Harris Publications Technology Used: 1. Autograph (Version 3.3) 2. Graphing Display Calculator TI-84 Plus Texas Instruments 23