## Assignment # 11

Polar equations ...

We are to investigate the following equation:

Changes in 'a'

We will keep both 'b' and 'k' equal to 1. Let's look at what happens when we vary "a" ...

 As we go from our inner circle (in purple) to our outer one (in light blue), our equations are as follows:

So we get interesting behavior when a = 1 and when a = 2. The others produce circles always one more in diameter.

What happens if a is a fraction?

 Our curves will go inward. Starting with the purple curve (a = 1/2) to the blue curve (a = 1/4). Additional curves follow the pattern.

Changes in 'b'

Now let's keep 'a' and 'k' constant at 1 and see what happens when we vary 'b'.

 As we go from our inner circle (in red) to our outer one (in light blue), our equations are as follows:

So a pattern begins to form after b = 2. Now let's look at fractions:

 Curves: Purple: b = 1/2 Red: b = 1/3 Blue: b = 1/4

Changes in 'k'

Here we make 'a' and 'b' both equal to 1 and we see how changing k effects our graph:

 The number seems to be related to the number of leaflets (after k > 1). We get the standard circle when k = 0 and a heart when k = 1.

On the other hand, if we are to do away with the 'a' value altogether, here is what we come up with:

 The leaflets seem to double: Again, our last equation is in light blue.