The Department of Mathematics and Science Education

tiffani c. knight

Polar Equations….well, a polar equation is an equation for a curve written in terms of polar coordinates. What are polar coordinates you ask? Well, the polar coordinates, according to www.mathwords.com/p/polar_equation.htm , are r and . I don’t know why theta is raised and I can’t seem to get it to come down. : ) Oh well…we will continue.

First, I chose to explore the when a and b are equal and k is an integer.

Here is the graph of the original polar equation when a, b, and k equal 1. Oh, and theta is set between 0 and 2pi.

Now I will explore the same equation but replace k with 2, 3, 4, and 5.

When k = 2

When k = 3

When k = 4

When k = 5

It appears that k determines the number of “leaves” . So, when a and b are equal and k is an integer, you get the “n-leaf rose”. Thirteen is my favorite number, so I’ll try that:

When k = 13

Yep! It works. Ain’t it “purdy” : )

Second, I explored when b = 1 are and k is an integer.

Here is the graph of the polar equation when b, and k equal 1 and theta is between 0 and 2pi.

Next, I explored the equation when k equaled 2, 3, 4, 5, and 13.

When k = 2

When k = 3

When k = 4

When k = 5

When k = 13

Well, the pattern that emerged from is that when k is an odd integer, the number of “leaves” will be k. And when k is an even integer, the number of leaves will be 2k.

The final thing I explored was repeating the first two equations, but replacing cos with sin.

Using . Here is the graph of that equation when a, b, and k = 1 and theta is again between 0 and 2pi.

Next, we will graph When k = 2, 3, 4, 5, and 13.

When k = 2

When k = 3

When k = 4

When k = 5

When k = 13

Well, when the equation was replaced with sin, the graph still produced an “n-leaf rose”. However, the orientation of the rose was different.

On to the second equation . Here is the graph of this equation when b and k equal 1 and theta is between 0 and 2pi.

Next we will construct the graphs when b remains 1 and k equals 2, 3, 4, 5, and 13.

When k = 2

When k = 3

When k = 4

When k = 5

When k = 13

Well, when the equation was replaced with sin, it still gave you the same results as far as k determining the number of leaves. When k is odd, n = k. And when k is even, n = 2k. However, the shape of the leaves changed…I wonder what’s up with that….