Oh Say Can You Functions

Exploring Functions and Their Products

Assignment 1

By Amber Candela


The problem

Find two linear functions f(x) and g(x) such that their product, h(x)=[f(x)][g(x)] is tangent to each of f(x) and g(x) at two distinct points. 

            One example of two equations that will solve this problem is shown

f(x)= -2x + 2

g(x) = 2x -1

[f(x)][g(x)] = (-2x+2)(2x-1)

It will work for any

f(x) = (-ax +b)

g(x) = (ax + (-b+1))

f(x)g(x)= (-ax+b)(ax+(-b+1))

In finding the generalizations, I first plugged in examples of the equaitions that I thought would work.When looking for f(x) and g(x) I knew one coefficient of "x" would be positive and one would be negative. This is becuase I needed one positive slope and one negative slope. I then realized that the sum of the y- intercents needed to be 1. I realized that ( b + -b +1= 1) so in one equation the y-intercept = b and in the other the y-intercept = (-b+1). It does not matter with which y -intercept = b and which y -intercept = (-b+1). If we switch them the graph will look like the following:

f(x) = (-2x -1)

g(x) = (2x+2)

f(x)g(x) = (-2x - 1)(2x + 2)

For the sake of this write up, we will only be looking at the general case:

When looking at the graph, there are certain properties one will notice.

The equations of the lines are tangent to the parabola at the roots of the equation. For the equation y = (-ax+b)(ax+(-b+1) the roots are (-ax+b)=0 and (ax+(-b+1) = 0 with the roots being and .

Another noticing in the graph is that the two linear equations intersect at the point at which "x" is the vertex of the parabola and y is equal to 1/2. First solve for x

Then using the value found for x, find the value of y.

The vertex of the parabola can be found using the formula .

Thus expanding the parabola to the form y = -a² x² + (2ab-1) x-b + b where a = a² and b=2ab - a. This then leads to:

Thus the "x" value is the same at the vertex of the parabola and the intersection of the two linear equations.