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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