– 4 – MATHL/PF/M12/N12/M13/N13
For final assessment in 2012 and 2013
SHADOW FUNCTIONS HL TYPE I
To the student: The work that you produce to address the questions in this task should be a report
that can stand on its own. It is best to avoid copying the questions in the task to adopt a “question
and answer” format.
While real zeros of polynomial functions may be easily read off the graph of the
polynomial, the same is not true for complex zeros. In this task, you will investigate the
method of shadow functions and their generators, which helps identify the real and
imaginary components of complex zeros from key points along the x-axis.
Part A (Quadratic Polynomials)
Consider the quadratic function 2 2
1 y xa b ! ( ) ! ” , where a b, # ! .
$ Write down the coordinates of the vertex.
$ Show that 1 y has zeros a b % i , where i 1 ! ! .
The “shadow function” to 1 y is another quadratic 2 y which shares the same vertex as 1 y .
However, 2 y has opposite concavity to that of 1 y and its zeros are in the form a b % .
$ Use various values for a and b to generate pairs of functions 1 y and 2 y .
$ Hence, or otherwise, express 2 y in terms of 1 y and my , where my is called the “shadow
$ On a labelled diagram, illustrate how the zeros of 2 y may be helpful in the determination
of the real and imaginary components of the complex zeros of 1 y . – 5 – MATHL/PF/M12/N12/M13/N13
For final assessment in 2012 and 2013
Part B (Cubic Polynomials)
Now consider the cubic function yx x x 1 # $ %$ %% ( 2) (3 2i) (3 2i) ! “! ” .
The shadow function in this case is another cubic function 2 y which shares two points with 1 y ,
has opposite concavity and its zeros are %2 and 3 2 % as depicted in the figure below. The shadow
generating function in this case passes through the two points of intersection.
& Write down an expression for 2 y and hence find the points of intersection between 1 y and 2 y .
& Hence, or otherwise, determine the equation of the shadow generating function my in this
& Once again, express 2 y in terms of 1 y and my . Use technology to investigate similar
& Write a general statement about the functions 1 y , 2 y and my and prove your statement.
& On a labelled diagram, illustrate how the zeros of 2 y may be helpful in the determination
of the real and imaginary components of the complex zeros of 1 y .
& How can the findings with quadratics and cubics be applied to quartics?
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