The Effect of k on r=a+bcos(kq) and r=bcos(kq)
Lets investigate the polar curve r=a+b cos (kq). Let begin by looking at the graph when a=b=k=1.
Now lets hold a and b at 1 and vary k. Lets look at k=2, 3, 4
The number of petals on the graph is the value of k. Just to be sure, lets look at k=10 and k=100.
As you can see, this relationship holds for all integer values of k. This is called the n-leaf rose because there are n petals or leaves when k=n.
Lets see what happens when we eliminate a. Then our equation is r=bcos(kq). Lets begin by looking at the graph when b=k=1.
Now lets hold b at 1 and vary k. Lets look at k=2,3,4.
When k=2, there are 4 petals. In this case, the number of petals is 2k. Based on this, I expected that there would be 6 petals when k=3. But there were only 3 petals. In this case, the number of petals is k. When k=4, there are 8 petals. Again the number of petals is 2k. This leads me to believe that when k is even, the number of petals is 2k. When k is odd, the number of petals is k.
Lets look at k=5,6,7 to see if this theory is true.
The theory appears to hold true. So for the polar equation r=bcos(kq), the are k petals when k is an even integer and 2k petals when k is an odd integer.