**Megan Langford**

** **

**LetŐs investigate the graph activity for functions
involving **

First, let us examine the graph of the equation

This is a commonly-known graph for the equation of a
circle. It is completely centered
at the origin, with a radius of 1.

Now, letŐs examine the graph of the equation

We can now notice that this graph has a similar shape
to the last one in the first quadrant.
However, from there, the lines simply extend out in the second and third
quadrants to form almost a straight line.

LetŐs go ahead and take a look at the graph for the
equation

This graph is more similar to the first graph, and we
can attribute this to the fact that both equations include even exponents. The main difference in this graph
compared to the first one is that it has extended out to include four smooth
corners, one in each quadrant.

LetŐs compare another graph to these results. We will now take a look at the graph
for the equation . Since this
equation includes odd exponents, I am going to predict that its shape will be
more similar to the graph.

As predicted, this graph is much more similar to the
graph of our second equation. The
main difference is that the corners form a slightly steeper slope than they did
in the previous graph.

To view these outcomes more easily, we will now show a
graph including all four equations:

As we can see here, the four corners of the shape form
a sharper curve as we increase the exponents.

To take this behavior to another level, letŐs examine
the graphs for and . I am going to
predict that the graph with the even exponents will have almost a perfect
square shape, and the graph with the odd exponents will form an elongated ŇWÓ
shape along the same diagonal.

And now we graph

We can now see that our prediction was fairly
accurate. If we were to continue
to increase the function to any greater even integer value, then we would form
sharper and sharper corners on the square figure of the graph.

Now, letŐs go ahead and look at the graph of

Again, our prediction of this graph was fairly
accurate. If we were to continue
to increase the exponents in the equation to any greater odd integer, we now
know that the corners in the figure would continue to make a sharper shape.