Polar Equations

By: Diana Brown

(1) Investigate

When a and b are equal (in this case a and b both equal one), and k is an integer, this is one textbook version of the “ n-leaf rose.”

K = 1                                                                                                K = 2

K = 3                                                                                                K = 4

K = 5                                                                                                K = 6

K = 7                                                                                                K = 8

K = 9                                                                                                K = 10

Notice that k determines the number of “leaves/pedals” on the “rose.”

Click here to open a Graphing Calculator file to animate the graph as k varies.

(2) Investigate and compare

for various k.

K = 3                                                                                           K = 4

K = 5                                                                                           K = 6

Notice that if k is odd then the number of pedals is equal to k, but if k is even then the number of pedals is 2k.

(3) Investigate the same equation as # 2, but replace cosine with sine.

K = 3                                                                                           K = 4

K = 5                                                                                                    K = 6