Polar Equations

by

Lizzy Shaughnessy

Assignment 11

In this write-up, we will discuss polar equations and the effects parameters have on the shape of the graph.

We will consider the following equations:

We will look at the graphs of these equations for varying values of a, b, c, and k.

First let's look at how the parameters a and k will affect the graph of .

So when we change the value of a it appears the size or length of a loop changes. This is shown by the fact that an a value of 1/2 has the smallest loop and when a is 6 the loop is always the biggest loop. This seems to make sense after Assignment 1 where we found that the leading coefficient of a sine or cosine function affect the amplitude or height of the graph.

We also notice that as k increases, the number of loops also increases. When k = .1 the graphs did not even complete 1 loop but when k was up to 10 the number of loops appears to be 20. Again, this information aligns with what we found in Assignment 1 where we found that the coefficient of the angle affect the period, or how often the graph repeats.

Now we will look at how the same parameters, a and k, affect the polar equation of cosine.

Just as with the graph of the sine equations, the value of a affects how wide or tall the graph is but the value of k affects the number of loops that the graph has. We do notice that there is a difference between the orientation fo the sine graph and the orientation of the cosine graph.

As we can see, each loop for the cosine graph is symmetric to the axes but the sine graph appears to be symmetric to the line y = x. The shape of the loops and the number of loops is the same in both graph but the sine graph is rotated.

Finally, we will look at the graphs of .

It is easier to see how the value of b affects the sine and cosine graphs if we use a movie. In both videos n varies from -5 to 5.

Here are the labels for the sine equations: