Polar Equations

Investigate the graph of:

This is a special case of the graphs of . In theses cases, a = 0.


Begin by investigating the graphs when b=1 and k varies among the integers.

, k =1


, k = 2


, k = 3


, k = 4


To view other values of k, open this animation on Graphing Calculator.


We can already note some similarities and differences if we compare these graphs to those with a constant of a added to them.

First, the graphs where k = 3, 5, 7, etc. (odd integers) have the same number of leaves as those of the form . Thus, when k is odd, the rose has k leaves. The angle between the leaves is the same as before as well, i.e., equal to . When k is odd, the graph is symmetric with respect to the x-axis.

When k is even, however, the number of leaves is equal to 2k. In addition, the angle between each leaf is equal to . When k is even, the graph is symmetric with respect to both the x-axis and the y-axis.

When k = 1, the graph is no longer a leaf of a rose, but a circle with a radius one-half of k.

Next, investigate the graphs of as we vary b and keep k constant.

Below are examples where k = 4 and b = 1, 2, and 3. Notice how the length of the leaves changes in each case. The length of each leaf is equal to b. This makes sense since the length of each leaf for the equations is equal to and we are considering the special case where a = 0. In general, the length of each leaf of the graph of is equal to .

Negative b?

Since the graph is symmetric with respect to the y-axis when k is even, a negative value of b will produce the same graph as the positive value of b. Thus, we need only to investigate negative b-values for equations when k is odd. Below is an illustration of the graphs when k = 3 and b = -3, -2, -1, 1, 2, and 3. When b is negative, the graph is rotated radians from the graph when b is positive. This is the smallest angle of rotation that will produce the graph for a negative b-value. The graph could also be rotated by radians. In other words, the graph is reflected across the y-axis. This same phenomemnom occurs with the graphs of . Notice that the length of each leaf is equal to .

To summarize my findings for the graphs of :

Click here to view the graph of and change the values of b and k for yourself!

Return to the graphs of .

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