Polar Equations

Assignment 11

Investigate

When a, b and k equals 1, the graph looks like this:

Let's keep a and b equal to one and vary k.  When k = 2 the graph looks like this:

We get two leaves.  Let's try k = 3, 4 and then 5.

It seems as k increases the number of leaves increases.  In fact, the number of leaves equals k.

Compare with

When b and k equals 1, the graph looks like this:

Now the graph looks like a circle.  Let's see what happens when we change k to 2.

Four leaves!  Let's look at the graph when k = 3, 4, and 5, respectively.

It looks as if the number of leaves equals k when is is odd, but doubles k when it is even.

What if . . . cos ( ) is replaced with sin( )?

When a, b and k equals 1, the graph looks like this:

It  rotated! Let's keep a and b equal to one and vary k.  When k = 2 the graph looks like this:

Same graph as cos, but rotated again!   Let's try k = 3, 4 and then 5.

Again, all the same graphs as cos, but rotated.  Let's compare with b sin (k0).

When b and k equals 1, the graph looks like this:

Let's see what happens when we change k to 2.

Let's look at the graph when k = 3, 4, and 5, respectively.

Just as I suspected, the same graphs as above, but, rotated!