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Shadow Functions Maths IB HL Portfolio

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Introduction

 June 252012

Polynomials represent a large area of mathematics. The word itself is a combination of the Greek poly and Latin binomium, meaning "many" and "binomial".

A polynomial function is in the form: where

is a non-negative integer and are constants.

We define roots or zeros of a polynomial the x-values for which these equal to zero;

Real roots of a polynomial are graphically represented by their y-intercepts, but how can complex roots be identified?

In this task, I will investigate the method of shadow functions and their generators, which help identify the real and imaginary components of complex zeros from key points along the x-axis.

To this effect, I will make use of the software Geogebra, a graphing program.

Part A: Quadratic Polynomials

Quadratic polynomials are in the form where

and

.

Their roots can be calculated by using the so-called discriminant .

And are calculated to be Let us consider the quadratic function: , where

Where the vertex has coordinates We can find the zeros of this quadratic by expanding the function: Where the discriminant ,

thus has two complex roots: The "shadow function" to is another quadratic which shares the same vertex but has opposite concavity and two real roots.   Let us proceed and use various values for

and

Middle

Giving Allowing us to express in terms of and .     I will now investigate similar cubic functions to see if the relationship between and remains the same.

Let us consider and Graphed: The shadow generating function is given by the equation

, Expressing in terms of and :    Which is the same relationship as with our previous cubic. Let us see if this pattern holds true with another similar cubic.  Graphed: The shadow generating function can be read to be

, Again, expressing in terms of and :    It seems that, just like in the case of the quadratics, the relationship between , and is given by the equation: Let us try to prove this statement, by generalising the functions , and .

We have:  .

These two cubics intersect where ,   The points of intersection are thus

and .

And the equation of the line passing through both points .

Where  Giving The relationship between , and :     My assumption has been proven correct.

The relationship between , and in the case of cubic polynomials is exactly the same as for quadratics. Perhaps, the zeros of can also be used in the same way to graphically determine the real and imaginary components of the complex zeros of .

Taking into consideration the function The roots are

and

, as proven in part A.

Conclusion will still work.

To show this I will use the function with roots

and

.

The shadow function will have roots

and

with equation: Graphed: As we have two pairs of conjugate roots in this case, we will also have to draw two circles, with centre A and C (representing the

and

parts of the real roots under the form of

and

and

respectively. The lines passing through A and C and parallel to the y-axis intercept the circles at E, F and G, H. The new points represent, as expected, the complex roots

and

when interpreted on the Argand plane.

Conclusion:

In this assessment, I have investigated and proven the relationship between quadratics and cubics with complex roots and their shadow-functions: .  I also showed that, when knowing the equation of the shadow function, you can easily, graphically identify the complex roots of the original function, with my "circle"-method.

However, the case of quartics is more complicated. First of all, there are two different cases to consider:

The first case with two complex roots  allowed me to define the shadow-generating function as , maintaining the above stated relationship.

But for the second case with four complex roots I failed to identify the shadow-generating function.

Yet, both cases still permitted me to graphically determine the complex roots of the original function from key-points along the x-axis of the shadow-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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