Clay Bennett

Assignment 11.1

In this investigation, I will be exploring polar
equations. Polar equations is
another subject that I have not had much experience with. We dabbled with polar equations a
little in my high school calculus class, but other than that I have not really
worked with polar equations. The
focus of my investigation will be on the equations…

I will begin by examining the first equation. Here is a picture of its graph when all
of the variables are equal to one.

In high school, we referred to this plot as the
butt-print (teacher included), but from this I assignment I have learned that
it has a formal name, the n-leaf rose.
My next step in this investigation was to began playing around for the
values of a. My conclusion was that
when an absolute value got larger and b and k stay at one its graph became a
circle with a radius equal to a and a center point of one. Here is a picture of the equation when
a = 10 and when a = -10 while b and k =1.

As you
can see, it doesn't matter if the value of a is negative or positive. Next, I began to explore a when it was
very close to zero. This did not
prove to be interesting because it just gave the graph of

Next, I began to look at the values of b when a and k
=1. When b became larger, the loop
within the circle became larger.

Graph for when b=5.

As it got extremely large, the inside loop would
almost be the same as the outside loop.
I would imagine if you took the limit of b -> infinity then the two
loops would be equal. Graph for
when b = 1000.

When I took b to be less than zero the same thing
happened, the only exception was that the graph was reflected across the
y-axis.

Graph of b = -5.

Graph of b = -1000.

Next, I looked at b when it approached
zero. I discovered that as b
approaches zero then the graph becomes the unit circle or r = 1, which makes
sense. Finally, I underwent the
same investigations with the k value.
I found k to be very interesting.
I've probably said that I've found something interesting a lot
throughout these assignments, but I really mean it this time. For every value of k there would be
abs(k) number of loops. This
really impressed me. Here is a
picture for when k= 3 and 8.

When k got really big, then the outline of the
n-leaf rose began to appear. Here
is a picture.

Lastly, I looked at k when it approached zero, which
would give us r = 1 + 1 => a circle with center (0, 0) and radius equal to
2.

I really feel I got something out of this
investigation. I have definitely spent
the most time on this one out of all the others. I will definitely continue to play with this equation.