For this assignment I will be investigating various polar equations and their graphs.

I. I will begin by looking at the equation:

for various values of a, b, and k.

a. Let's look at a simple graph. When a=b and k is an integer.

This is an example of an "n-leaf rose". k determines the number of petals and |a|+|b| determines the length of each petal. The graph intercepts the y-axis at 1, -1 , and 0. The graph intercepts the x-axis at 2 and 0. Let's look at one more. I will fix k for comparisons.

By changing the signs of a and b, we see that this changes the orientation of the flower. Now the graph intercepts the x-axis at -2 and 0 and intercepts the y-axis in the same places 1,-1, and 0.

b. What about the case a>b and k is still an integer?

When a>b the graph will not intercept (0,0) which means that the picture will have an open bloom. The graph is beginning to look more like a starfish than a rose!!!

c. What about the case a<b and k is still an integer?

When a<b it produces a graph of two flowers.

II. Let's consider the equation:

for various values for b. I will fix k.

In this example there are still 5 petals but the petals are half as long as r*= 1+1 cos (5*theta) and not nearly as wide. Also, this graph does not have as many intercepts as r*. It intercepts the y-axis at 0 and the x-axis at 1 and 0.