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.