3091 WordsJan 27, 201313 Pages

Higher Level Mathematics
Internal Assessment Type I
Shadow Functions
Contents
Introduction: Functions/Polynomials 3
Part A: Quadratic Polynomials 4
Part B: Cubic Polynomials 12
Introduction:
In mathematics, function is defined as a relationship, or more of a correspondence between the set of input values and the set of output values. Also, a rule is involved, or as it may be referred to, a ‘set of ordered pairs’ that assigns a unique output for each of the input. The output correspondence is usually defined as f and the output is x. The correspondence is denoted as f(x). All functions are mainly defined by two factors, as was mentioned before, set of inputs - which are called arguments; and outputs - which*…show more content…*

So it can be said that:
-(x+2)2-9+18 = -Y1+2Ym = Y2
At this point, the real and imaginary components of the complex zeros of Y1 can be found using the values for zeros of Y2. As it is known, Y1 has complex zeros of a form of a±ib and Y2 has zeros of a±b. With this information available, I will show you how to obtain value for complex zeros of Y1.
Consider a function Y1, with a general statement of:
Y1 = (x-a)2+b2
As it has been found earlier, this function has zeros of a±ib. It also has a shadow function Y2:
Y2 = -(x-a)2+b2
With zeros of a±b. If we say that a = 2 and b = -3, then this function has zeros of 2±(-3). In this form of zeros, we can say that 2 is the x value of the vertex coordinates, lying on the axis of symmetry of Y2 on the x-axis, and ±3 are just the distances between the mid point 2 to the points where Y2 intersects x-axis. It is clearly shown on the graph below.
As we know the zeros of Y2, it will be very easy to determine the complex zeros of Y1. 2 will equal to the real part of the complex zeros of Y1, and subsequently, ±3 will equal to the imaginary part of the complex roots. Although, as we know that y-axis is the imaginary axis, the imaginary parts of the complex roots obtained, of the Y1, will be plot according to the y-axis.
As you may have observed, in order to get the complex zeros of

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