Let's investigate the following polar equation:
Now let's see what happen by using different values for a, b, and k. We will let a =2 , b = 2 , and k = 2.
Let's hold a and k at two and change the values of b.
We will let b = 3, 4, 5 with the respective graphs.
Now we will keep the values of a and b =2 and vary the value for k. Let's let k = 3, 4, 5 respectively on the graphs.
This is sometimes considered the "n-leaf rose"
Now let's keep the value of 2 for b and k and change the value of a. We will now also let a = 3, 4, 5 respectively on the graphs.
Now we will consider the equation
We will keep the value of b = 1 and vary the values of k.
Let start with k = 2
Okay, that's interesting let's try some more values:
Maybe 3, 4, and 5.
I'm not sure if I see a pattern. Let's try some more.
Maybe 6, 7, and 8.
Well it looks like when k is an odd number then it equals the number of "leaves on the flower", but when k is even it equal the square of the number of "leaves on the flower".
Let's look at one last thing. What if you change the cos ( ) to the sin ( )? Let's see what we come up with now.
We will look at a few values of k and see what it gives us.
Maybe 1, 2, and 3.
These graphs are very similar to the first graph with only a slight rotation.