The purpose of this assignment is to explore polar equations. We begin by graphing the equation:

. We obtain the following graph when a=b=k=1.

We now hold a and b constant and vary k. The following graphs are when a=b=1 and k=2, k=3, k=4.

We can determine that as k increases the number of leaves increases. Therefore for the following graph when k=20, there will be 20 "leaves."

Now we can explore the equation when a<b, a=1 and k=2. The following graphs show when b=1, b=2, b=3 and b=4.

By comparing these graphs with the previous graph in red,we can see that when a<b, a new set of smaller leaves appears and the size of both sets of leaves increases by one. Therefore when b=1 the smaller new set of leaves was at zero and the original set was at one.

We now examine the graph when a varies. We graph the equation when a>b, b=1 and k=2. The following graph shows when a=1, a=2, a=3 and a=4.

As a increases the leaves lose their form and they get further away from the origin. Now we can examine what happens if a and b increase together. The following are graphs for a=b=1, a=b=2, a=b=3 and a=b=4 while k remains constant at two.

We can predict from these graphs that if a and b increase together then the leaves will not lose their form (they will continue to pass through the origin if a=b) and the size of the leaves will increase by 2.

We can also compare the graphs if cosine is replaced by sine. The following graph is for a=b=k=1 for sin and cosine.

We can see these graphs are the same except the sin graph rotated 90 degrees. We can compare the graphs of sin and cosine when a<b, a=1, b=2 and k=2.

Again these graphs are the same but this time the sin graph only rotated 45 degrees. We expect the graphs of sin and cosine to be the same when a>b, a=2, b=1 and k=2 but again the sin graph is rotated. The following is the graphs of sin and cosine with the stated conditions.

As we predicted the graphs are the same except the sin graph is again rotated about the origin.