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Shadow Functions

Polynomial comes from the Greek poly, which means many. While the word originated from Greece, it was first introduced in Latin by Franciscus Vieta. In this investigation, it involves 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.

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)

In mathematics, the word polynomial is an expression made with constants, variables and exponents, which are combined using addition, subtraction and multiplication. A quadratic polynomial is a polynomial of degree two. This means that the variable(s) in the expression can be raised to a maximum power of two (x^2 where x∈Z). A general quadratic polynomial is usually given in a form of ax^2+bx+c where a≠0

When dealing with quadratic polynomials, a shadow function is a function that is reflected on the horizontal plane that is tangent to the vertex of the original function. Therefore, the horizontal plane acts as a mirror. For example, consider a quadratic function, f1(x)=2x^2-8x+10 and a horizontal function f2(x)=2 which intersects at f1(x)’s vertex, when both functions are graphed using technology (GDC), it will look like this:
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