This write up is for problems #2 in Assignment
11.

**Investigating Polar
Equations**
**by**
**Kimberly Burrell
and Brad Simmons**

Investigation of
After we investigated the polar equation
by varying "a" and "k", were given a similar
polar equation except for the addition of b. At this point in
our investigation, we will vary the "b" using the mentioned
values for "a" and "k". Again the sine equation
will remain symmetric about the y-axis.
When all values of a, b, and k are
1, we have the following graph.
The graph now has a distinct shape.
There appears to be a smaller curve inside the larger curve, which
occurs at the origin. Will the curve behave like the previous
curves if we change the value of "b" to 2? Let's see.
The smaller curve appears to disappear.
However, there is still a noticeable bend at the origin of the
graph.
Now, let us vary both "k"
and "b".
When a =1, k = 2, and b = 2, we get
the following graph.
Here, we can observe two pedal-like
curves that are connected at the origin. Now, let's vary "a"
to be 2, also.
When a = 2, we get 4 pedal-like curves
that join each other at the origin. Two of these pedals are smaller
than the original two pedals. Now let's see what occurs when we
change a = 5.
The pedals have expanded as with the
previous changes of "a". Therefore, if we change "k",
then the number of pedals should change if the same principles
for "k" apply in this polar equation.
Let's try k = 5 while a = 2 and b =
2.
By observation one can see that there
are 5 large pedals and 5 small pedals which are inside the larger
ones.
Finally, we will vary all the values
of a, b, and k.
Here a = 2, b = 1, and k = 5. What
conclusions can you draw from this?

Improvement on the
Theory of the Number of Pedals
The number of pedals
is based upon "k". After careful investigation and research,
we found the following information.
- If k is even, then the number
of pedals is 2k.
- If k is odd, then the number
of pedals is also 2k. This is the case although it appears that
only one set of pedals exist in some cases. These pedals are
actually laying one on top of the other. Furthermore, this is
more apparent when you add and vary "b" in both the
sine and cosine equations. One set of pedals shows up smaller
than the other set of pedals.

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