# Maths IA

**Topics:**Complex number, Quadratic equation, Polynomial

**Pages:**30 (2882 words)

**Published:**December 11, 2013

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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