Investigation of the product of linear functions

by

Kristy Hawkins

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. Discuss and illustrate he method and the results.

Let f(x) = ax+b and g(x) = cx+d

We know that for linear function to be tangent to h(x) then the slope of h(x) must be equal to that function.

Clearly the slope of f(x) = a and the slope of g(x) = c, since they are both linear equations.

In order to find the arbitrary equations f(x) and g(x) for which this relationship holds true, we will solve h'(x) = a and h'(x) = c. This will perhaps give us some relationships that we can use to define any linear functions for which this relationship will hold true.

Wow! This f(x) and g(x) are tangent to their product at 2 distinct points.

But we also know how to find ANY function for which this is true by using our derived equations. To generalize our equations we will write:

where a does not equal 0

Here is a GSP file to demonstrate all of the possible functions for which this is true. Click here!

Click on the button in the top left hand corner of the file marked a to see what happens when a is varried. Then press it again to make the variable stop. Do the same for both b and x.

It is interesting so see that varying the value of a will change the shape of the parabola, while varying the value of b only moves it side to side.

Why to you think this is so?

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