Assignment # 11

Polar Equations

by

Megan Dickerson

The first thing that I examined in this investigation of Polar Equations, was the Equation:

I looked at multiple values for a,b and k.

 a = 1, b = 1, k = 1 a = 1, b = 1, k = 2 a = 2, b = 1, k = 1 a = 1, b = 2, k = 1 a = 2, b = 2, k = 4 a = 5, b = 2, k = 7 a = 5, b = 5, k = 7 a = 0, b = 1, k = 1 a = 0, b = 1, k = 2 a = 0, b = 4, k = 6

From this set of data I observed that when a = b, the center of the "flower" comes to a point at the origin.

k is the number of "leaves" or "petals" except for when a = 0 and k is even, then 2k is the number of "leaves."

The length of the leaves is (a+b).

When a and b are not equal there are still k loops, but there is not a pointed center.

When a = 0 and k = 1, a circle with diameter b is formed.

Then, I examined the Equation:

I looked at multiple values of a,b, and , but since I showed so many with cos I will only show a few for sin.

 a = 1, b = 1, k = 1 a = 3, b = 3, k = 7 a = 0, b = 4, k = 6

Many of the properties were the same for these sin graphs compared to the cos graphs.

The first graphs are different in that the domain and range were switched. The curve was on the left side of the cos graph but on the base of the sin graph.

Here are the animations of the cos and sin graph with a and b fixed as 1, but k animated:

It can be seen that when k is an even number these 2 graphs appear the same, but when k is an odd number, they are different. The domain and range have swapped.

Next, I looked at the equations:

I looked at multiple values for a,b,and k.

The top row is for sin and the second row is cos.

 a = 1, b = 0, k = 1 a = 1, b = 0, k = 3 a = 1, b = 0, k = 6 a = 3, b = 3, k = 6 a = 4, b = 4, k = 3

When b = 0, the graph is a circle with radius a for both cases, but for the sin graph the diameter is along the y-axis, and with cos the diameter is along the x-axis.

Once again, these graphs are similar but the "leaves" are in different positions.

Here is an animation of the graph of

with a and b set at 1 and k animated:

This graph is different from because it has double the amount of "leaves." The new "leaves" are small and inside the original ones.

Last I looked at this equation:

I looked at multiple values for a,b,c,and k.

 a = 1, b = 1, c = 1, k = 1 a = 1, b = 1, c = 4, k = 1 a = 4, b = 1, c = 1, k = 1 a = 4, b = 2, c = 4, k = 6 a = 6, b = 5, c = 8, k = 5

- a/b is the slope of the line when k = 1. This graph behaves very different from the ones above.