Exploring Parameters of a Polar Function

By: Sydney Roberts

For this exploration we will be comparing the graph of  with the graph of  for various values of k.

Let’s first look at different graphs of  for different integers b and k.

When b=1, k=1                                               When b=1, k=2

When b=1, k=3                                               When b=1, k=4

When b=1, k=5                                               When b=1, k=6

It appears as if k determines the number of petals the graph will have. Notice that when k is odd, then the number of total petals is k, and when k is even then the number of total petals is 2k. To see how the parameter b affects the graph, we will not vary b.

When b=2, k=2                                      When b=3, k=2

When b=4, k=3                                      When b=5, k=4

From this we can see that b determines the length of each of the petals. Therefore, the parameter b will not change the overall “look” of the graph. Hence, to compare  with the graph of  we will only consider when b=1. Below are different graphs comparing the two.

When a=1, k=1                                               When a=2, k=1

When a=3, k=1                                                          When a=1, k=2

When a=2, k=3                                                          When a=3, k=3

Notice that a changes two things. First, a determines the maximum distance  will be away from  and that distance is |a|. Secondly, as a increases,  becomes a smoother curve. For a better view of how the parameter a affects the graph, view the animation below which shows k=2 and a varying from -5 to 5. Notice that as a becomes negative, the maximum distance from  to  shifts from lying on the x-axis to lying on the y-axis.

Now, what would happen if  instead of  ?

Well, consider the graphs below.

Hopefully you noticed that sine instead of cosine only rotates the graph, but doesn’t change the shape of it. Now let’s look at the same graphs except this time compared to .

Again, notice that this only rotates the graph.

Return