:

After subbing different values for

and

From the above results,

by comparing with

, it can be seen that their values are opposite,

have negative results,

’s results are always a positive number or bigger than 0.

A graph of y1 and y2 is shown below when a= 3 and b=5,

We know that

has zeros

, while

has opposite concavity to

,which is in the form

.

From the graph, it can be seen that,

is a reflection of

,

Therefore, the equation of the quadratic

is :

.

When

As the graph shown above,

and

have point of intersection, The shadow generating function is

.

Therefore, the shadow generating function for this quadratic is

To express

in terms of

,

,

.

By substitution into this function:

As

Coordinates of the vertex,

Shadow function:

Shadow root:

Complex zeros:

The reflection

As the above results shown, in the complex zeros:

the imaginary components is

PART B (Cubic Polynomials)

The investigation is to find out if the findings is related the cubic polynomials and to determine the real and imaginary components of the complex zeros of

and

Consider the cubic function:

It is said that it has zeros which are

,

the roots of

are:

Therefore,

can be expressed as:

,

An expression for

is:

To find out the points of intersection between

and

,

When

,

Hence:

Considering the result that has got above:

When

,

When

,

The points of intersection between

are (

.

To determine the equation of the shadow generating function

,

When

Expand the cubic function of

can be expressed in general form

which can be written in the below format:

In general form, it can be written:

By using the equation which is derived above from the quadratic polynomials

It can be seen that the above result is the same as given:

Roots of

are:

The above graph shows the cubic function of

and

,

To find out the general form of

and

and the generating function

,

The intersection points should be found out first.

As the intersection points of the cubic function

and

are (-2, 0) and (3, 20).

By using the following formula:

we can find out the generating function

,

In the above diagram, it shows the cubic function

,

, and generating function

,

In conclusion,

The general form of

can be expressed as:

which can be written in the below format:

The general form of

is obtained from the formula:

Which is calculated as:

.

It can be written in the below format:

For the generating function of

Firstly, it’s slope needs to be found and the intersection point of

,

By using the equation:

And the generating formula is calculated as:

As the above graph shown, the cubic function

and

,

cuts through between the intersection points of

and

,

fits the result.

In general, the cubic function

is a reflection of

.

Considering the derived function,

,

the function

Again, considering the derived function,

,

As the above diagram shows,

from point E to D and point F to D is the distance of the quadratic function

,

therefore the derived function

works in a quadratic function.

As the above diagram shows,

from point a to b and point c to d is the distance of the

,

therefore the derived function

works in cubic function.