Maths IA

Topics: Complex number, Quadratic equation, Polynomial Pages: 30 (2882 words) Published: December 11, 2013
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 of

, the following equation should be solved.
0

where a,b
where

3

Candidate Name: Emil Abrahamyan
Candidate Number: 006343-021

The shadow function to
in Diagram 2.

Mathematics Higher Level
Type 1: Shadow Functions

is another quadratic

which shares the same vertex as

as shown

x

Diagram 2

The properties of
Function

and

are illustrated in Table 1.
Equation

Coordinates of
Vertex

Zeros

Table 1

4

Candidate Name: Emil Abrahamyan
Candidate Number: 006343-021

Mathematics Higher Level
Type 1: Shadow Functions

With the purpose of finding any patterns between , and , various values of be used in order to generate pairs of , and
as shown below.
Values of

2

3

and

and

4

or

will

4
or

Diagram 3

Diagram 4

1

1

x

x
m

m

2

2

Comments

Comments

As seen from the graph
has downwards concavity
Diagram 4 illustrates that
cuts the x-axis at
and cuts the x-axis at
and
, which
and
, which means that it has zeroes
means that it has zeros
and .
and .
As (upwards concavity) doesn’t cross the x-axis, it
Again
doesn’t cross x-axis and it has
hasn’t any real zeros. It has imaginary zeroes
imaginary zeroes
.
The equation of shadow generating function is
.
The equation of shadow generating function is
.
, which means that the position of shadow
generating function depends on the positions of
and .
Table 2

5

Candidate Name: Emil Abrahamyan
Candidate Number: 006343-021

Now other values of

and

Mathematics Higher Level
Type 1: Shadow Functions

will be tested as shown in Table 3
Values of

-2

-5

and

6

or

8

or

Diagram 5

Diagram 6

1

x

x
m

1
2

m

2

Comments

Comments

As seen from the graph
has downwards
Diagram 6 illustrates that
cuts the x-axis at
concavity and cuts the x-axis at
and
, which means that it has
and
, which means that it has zeros and
zeroes and .
.
Again...


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
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