The Product of Two Quadratic Functions and the Tangents They Create
with Their Product.

By

Tonya C.
Brooks

University of
Georgia-Athens

In this assignment I want to examine what happens when
we take two quadratic functions and multiply them together to get another
fourth degree quadratic function, and how these all relate to one another. The goal for this project is to find
two functions that will each have two points in common with their product and
that the tangents be the same at these points.

Let us begin by looking at two very simple
second degree quadratic functions x^{2} and -x^{2}.

As you can see, the
product of the two – the yellow x^{2} and the violet –x^{2}
– gives us the red (x^{2})(-x^{2}). Looking at the red, it looks as though
it Ňbubbles outÓ a little on each side.

Let us take a look at
what happens to the product if we have:

As you can see, we made the top one skinnier. However we werenŐt able to get it any closer to the
productŐs graph in order to create a common tangent, so letŐs try something
else.

How about:

It is starting to look like we might be
getting closer on our graph for –x^{2}, so letŐs try something
with that and leave our first equation as is since that is what caused brought
our second and third mappings closer.

How about it we try:

That looks like it took us farther away
from our goal for both equations, so letŐs try:

That is really starting to look like
something! LetŐs make that change
again and see what happens with:

That looks pretty close, if not
exact. LetŐs go in for a closer
look at the left side.

As you can see, it looks like both
equations brush against the product and it looks like they have the same
tangent line if we were to draw it in.
If we take a close look at the right side, we would see a reflection of
this.

From this, letŐs take a look and see what
happens if we change our multipliers for our x variable. LetŐs try:

That doesnŐt seem to work, so letŐs try
making the multipliers the same.
As in:

So it seems as though
we need to have our x multipliers be opposites in order for this to work.

How about shifting
the graphs to the left and right?
Is this okay or must we be centered on the y – axis? I would think that we would need to
move both functions in the same direction the same amount but letŐs look at
what happens if we donŐt follow this assumption. LetŐs try:

Whoa! We got some pretty wicked stuff but
nothing that helps us with our problem so letŐs try something else. How about:

Okay,
so it looks like we can shift the graph left and right without changing
anything as far as sharing tangents is concerned. The next question we might ask is: Can we shift the graph up and down if we keep the distance
between the critical points the same?

Click here to create your
own set of equations to test your hypothesis.

Now, we have found
two equations that do what we want, but if you notice, neither one of the
equations deals with any x to the first power. LetŐs look at what degree one does when we factor that in.

Look at what
the relation is between x^{2} +x - 1 and x^{2} – 1.

It looks as though
that x in our first equation shifts our graph a little to the left and
down. We might be able to create
another set of equations with our desired property if we counteract the first x
with a –x in our second equation.
LetŐs try with:

So, it is possible to create two second degree quadratic equations that
are both tangent to their product in two places. The trick seems to be having opposite x^{2}
multipliers with the positive having a -1 y-intercept and the negative having a
y-intercept value of 2, and opposite x multipliers.

_{ }