Explorations with

by: Lauren Wright

In this assignment, we explore the effects of changing a on the graph of

First, let's examine this graph for a > 0.

Looking at these graphs we can draw the conclusion that the bigger a is (or the further a is from zero), the more contracted the parabola becomes.

Now, let's look at the graph for a < 0.

It appears as though we can draw the same conclusion when a < 0. The smaller a gets (or the further a is from zero), the more contracted the parabolas become.

Now, let's look at the relationship between and . Looking at the previous two graphs, we could guess that making a negative simply reflects the parabola about the x-axis. Let's investigate this a bit further.

Well, it looks as though our conjecture was true!

From these graphs, we can see that the parabolas open up when a > 0.

They open down when a < 0.

But, what happens if -1 < a < 1 ? Let's look at a few of those.

First, we'll look at -1 < a < 0.

WOW! If we look at this graph on the same scale as the others, many of the parabolas are too wide for us to see. So, let's change the scale of the graph and get a better look at them.

That's better. Now we can see just how wide the parabolas really are. It appears that when

-1 < a < 0, the parabolas expand as a gets closer to zero.

Now, let's look at 0 < a < 1.

Well, it looks like the same thing is true for 0 < a < 1. The parabolas expand as a gets closer to zero.

We can explore a bit further with a movie that lets a vary between -5 and 5.

Let's summarize what we've found.

When working with the graph of :

1. If a > 0, then the parabola will open up. If a < 0, the parabola will open down.

2. The further a is from zero, the more contracted the parabola will become; OR, the closer a is to zero, the more expanded the parabola will become.