- Level: International Baccalaureate
- Subject: Maths
- Word count: 1868
Shadow Functions Maths IB HL Portfolio
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Introduction
June 25 2012 |
Shadow Functions
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.
with shadow function
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 radius
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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