The Study of the Product of Two Quadratic Functions

By: Amanda Sawyer

In the given problem, we are to explore two quadratic functions with the product of these quadratic functions creating two different tangent points.  The two general quadratic functions will be named

The product of these two general quadratic functions will be named

As we can see, the product produces a fourth degree polynomial.  We can study this fourth degree polynomial and its behavior to the given two functions using Graphing Calculator.  It is known that the maximum number of turns that a fourth degree polynomial can have is one less than the degree.  Since, the problem asks for this function to be tangent to both of our given functions at two points, it will be necessary for the function h(x) to have the three changes in concavity. It is also known that we must study the relationship between h(x)=0 and  f(x)g(x)=0 to determine what criteria is needed for the intersection of these graphs.  To do this, we will set f(x)=g(x)=y, and study what happens when they create h(x):

Since , it is easy to see  (a+b) = 0, (b+g) = 0, and (f+c) = 1.  To show this property, we can calculate these values using graphing calculator as shown below.  It is set such that a=1, d=-1, b=1, g=-1, c=n, and f=1-n.  Through this investigation, we can study what is needed to create the two tangent points.   The next three pictures shows how n effects the intersection of h(x).

 The Values of n The Graph n = 1 n = 0.5 n = 0

As we can see, other restrictions are necessary to create this reaction.  We also need the two original functions to intersect at two points.  Therefore, we must set f(x)=g(x).  Since we already know some properties about our variables, we set d=-a, g=-b, f=1-c.  This creates the equation,

This gives us some restrictions on what our c variable must be to create these tangent points.   Therefore, for c<.5, the product of these two graphs will creates two tangent points from the given original functions. From this investigation, it has been shown that we can use mathematics and technology to study the effects of given equations.