# Lisa Brock

Assignment 11

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.