Let's examine some polar equations.

Investigate the graph of

This is a graph of a n-leaf rose. In this graph a, b, and k equal 1.

What if we change k to 3?

We can see that k determines the number of leaves.

Let's let k=5.

Let's compare this graph to

In this equation b and k equal 1.

Let's let k=3.

Once again k determines the number of leaves.

What if we replace cos with sin? What do you think will happen?

Before, our graph (circle) was centered at (0.5, 0). Now, the circle is centered at (0, 0.5).

Let's set k at 3.

Once again we get a three-leaf rose. And our rose has been rotated around the origin.