Tonya C. Brooks
University of Georgia-Athens
Let us begin by looking at two very simple second degree quadratic functions x2 and -x2.
As you can see, the product of the two – the yellow x2 and the violet –x2 – gives us the red (x2)(-x2). 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.
It is starting to look like we might be getting closer on our graph for –x2, 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:
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 x2 +x - 1 and x2 – 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 x2 multipliers with the positive having a -1 y-intercept and the negative having a y-intercept value of 2, and opposite x multipliers.