### Polar Equation Investigation

#### Helene Chidsey and Lou Ann Lovin

In this writeup we investigated the following equations for different values of p, for e > 1, e = 1, and e < 1.

First notice, that when p or e = 0, r = 0, thus we eliminated these possibilities from the investigation. We can also disregard negative values for p since both cosine and sine oscillate between 1 and -1. (For the same reason we can ignore negative values for e.)

Therefore we began our investigation allowing p to vary, and e=1. Consider the case when p = 1 and e = 1.  Notice that the first cosine graph is just the negative of the second cosine graph. Now look at the sine graphs for p = 1 and e = 1.  The same relationship occurs for the two sine functions. Furthermore, the first sine function is actually the inverse of the first cosine function and the second sine function is the inverse of the second cosine function. In other words, is the inverse of and is the inverse of .

Thus, we restricted our investigation to just one of the functions, since they are all either negatives, inverses, or negative inverses of each other. We continued the investigation with .

Using this function, consider p = 5 , 10, 15, 20, and e = 1.      Examining the graphs, one notices that as p increases, the parabola expands. So, as p decreases, let's see what happens to the parabola. Letting p = 0.75, 0.5, 0.25, 0.1 and e = 1, we obtain the following graphs.      Thus as expected, as p decreases, the parabola contracts.

Now consider the function as p varies, and e > 1. Let p = 1, 5, 10, 15 and e = 1.5.      When e > 1, the graph becomes a hyperbola, and as p increases the graph again expands.
Next examine the graphs as p decreases. Setting p = 1, 0.75, 0.5, 0.25 and e = 1.5, produces the following pictures.      Again as expected, when p decreases, the graphs of the hyperbolas contract.

Lastly, look at e < 1 as p varies. Let p = 1, 2, 4, 5 and e = 0.5.      When e < 1, the graph becomes an ellipse and as p increases the graph again expands (look closely at the range of r). Consider the graphs as p decreases, p= 1, 0.75, 0.5, 0.25 and
e = 0.5. We obtained the following pictures.      Once again for e < 1, we have an ellipse, and as p decreases the ellipse becomes smaller.

In summary, we eliminated p = 0 and e = 0 (since the results are trivial) as well as negative values for both (since cosine and sine both oscillate between 1 and -1). For e = 1 the graph is a parabola; for e > 1 the graph is a hyperbola; and for e < 1 the graph is an ellipse. Finally, when p varies, we found that the graphs expand for larger values of p, and contract for smaller values of p.