Exploring Linear Equations whose Product is a Parabola

by

Julia Neal

The Problem: to 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.

The first step was to explore the product of two linear functions.

This graph shows y = 4, y = 2, and their product y = 8.

Unfortunately, the product of these functions is not a parabola. It is a linear equation, so we need be more judicious as to our choice of the two linear functions. It would be more efficient to choose two linear functions whose product IS a parabola. So, let's take a look at the equation for the graph of a parabola,

.

Notice in my first example, there was no x, let alone an x raised to the second power.

In order to acheive x raised to the second power, each of the two linear functions needs an x. So let us try y=x +4, y = x+2, and their product, y =(x+4)(2x+2).

Notice, this time we did get a parabola. Now we can begin to focus more on the way the lines intersect the parabola. We know that the lines will intersect the parabola at the parabola's ROOTS, or x intercepts, since the parabola is written as their product. We may wish to focus on the roots as we further this investigation. While we would also like the lines to be tangent to the parabola, we can tell that we have not achieved this yet because the red line (y=x+4) hits the parabola twice.

With our next try using the equations y=x =4, y = -x+2, and y=(x+4)(-x+2), we again notice the intersections at the roots, but we are no closer to tangency. We need to try to move the parabola down.

With this fourth try using the equations y=x-2, y=-x+2, and y=(x-2)(-x+2) we have come much closer. Unfortunately, we have only one root. It looks like the parabola will need two distinct roots, to achieve tangency.

With these equations, y=x-3, y=-x+5 , and y=(x-3)(-x+5), we do not seem to have made much progress, but comparing these last two graphs, notice that the vertex of the parabola is the same as the point of intersection of the two lines. We want the eliminate the second point of interection with the parabola which happens to be the point of intersection of the lines as well. We want to move the point of intersection ABOVE the vertex of the parabola. This would move the graph of the parabola down a little further, eliminating any extra points of intersection between the lines and the parabola.

To accomplish this, we need to decrease the distance between the two roots. In this example, the roots are 2 units apart. Lets try to get the roots only one unit apart. To do this we need to look at the difference of the xintercepts of the linear equations, .

So, let's try y=x-3, y=-x+4, and y=(x-3)(-x+4)

Success! The lines are tangent to the parabola.

We have discovered that in order for the two linear equations to intersect such that they will be tangent to the parabola which is their product, the lines must :

have slopes with the same magniude but different directions and

have x intercepts that are 1 unit apart.