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# Quadratic Polynomials. Real and Imaginary components

Extracts from this document...

Introduction

MATHS IA                                                                                                             Alex Chen

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

.

From the function given,

The coordinates of the vertex is

where

Hence,

has zeros

, and

By subbing in different numbers of

into the equation:

For:

, it is given that

, which is equal to

:

:

 Value of a value of b value of y1 1 1 2 2 3 3 4 4
 Value of a value of b value of y1 1 1 2 2 3 3 4 4

For:

:

:

 Value of a value of b value of y1 1 1 2 2 3 3 4 4
 Value of a value of b value of y1 1 1 2 2 2 3 3 4 4

Middle

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

.

To express

in terms of

,

,

.

By substitution into this function:

As

Coordinates of the vertex,

Complex zeros:

The reflection

As the above results shown, in the complex zeros:

the imaginary components is

PART B (Cubic Polynomials)

The investigation

Conclusion

.

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

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.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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