An investigation of the product of two linear functions such that the tho functions are tangent to the product.


Nathan Wisdom


In this Exploration we want to find two linear functions f(x) and g(x) such that their product

is tangent to each of f(x) and g(x) at two distinct points.

Without using any algebraic manipulation let us consider the function f(x) =x and g(x) = -x. Now plot on the same graph f, g, and h(x) = f(x) g(x)

The graph is shown


This produce a graph with roots {-1,0,1}and no tangent line.

Suppose we modify f(x) so that f(x) = (x+1) then graph the h(x) . The graph is shown here

This appeas to be the graphs we are looking for, but how do we know that the points (-1,0) and (0,0) are tangents to the curve?

We can now look at the parabola and calculate the the tangent to any point using calculus. We know that the slope at any point on the curve is equal to the slope of the tangent line to that point. The slope for the curve is given by the equation h'(x) = -2x-1. Now we can test the points (-1,0) and (0,0) for tangency to the curve.

If we evaluate h'(-1) we get 1. The equation of the line with slope 1, passing through (-1, 0) is y= x+1. This equation is f(x).

Similarly the equation of the linewith slope -1 and passing through (0, 0) is g(x). Thus the two lines are indeed tangents and we are done.

Is this the only solution? Can we get a family of solutions by simple transformation? Click here to see one such transformations and click here to see another.

How is the parabola related to the lines from which it was formed?