Problem 6.

Graph

What do you expect for the graph of

or

First, let's look at the graphs separately:

Looking at the graphs we observe a few trends. The first thing observed is that when the exponent is 2 or 4 the graphs are closed and when the exponent is 1, 3, or 5 the graph is continuous. This leads to the assumption that if the exponent is even the graph will be closed and if the exponent is odd the graph will be continuous. The second thing observed is that when the exponent equals 2 the graph is a circle and as the exponent increases the graph becomes more square-like.

Using the observations from the graphs, we can assume that the graph of will be closed and square-like and that the graph of will be continuous and square-like.

Let's look at the graphs and see if our assumptions are correct:

The graphs look like what we expected.

Here is a graph of all six equations. This allows you to see the graphs becoming more square-like as the exponent increases.