Exploration 11: Polar Equations

By: Taylor Adams

We want to investigate the equation

When a=1 and b=1, i.e. a and b are equal, let’s look at changes when k = 1, 2, 3, 5, 7, 10 (the graphs are in order respectively)

One observation that can be made is that k tells us the number of “petals” the graph will have, when a and b are equal.

Let’s look at the equation

When b=1, let’s look at changes when k= 1, 2, 3, 5, 7, 10 (the graphs are in order respectively)

The variable k relates to this equation a little differently than in the previous equation. Similarly to the last equation, k tells us the number of “petals” the graph will have when k is an odd number. When k is an even number, however, there are double the amount of “petals” that we would assume from the variable k.

Let’s see what happens when we replace cos(θ) with sin (θ). Will the same relationships hold with k?

First, let’s look at

When a=1 and b=1, i.e. a and b are equal, let’s look at changes when k = 1, 2, 3, 5, 7, 10 (the graphs are in order respectively).

Similarly to the r=a+bcos(k)
graphs, when a and b are the same, k will determine the number of “petals” the
graph will have.

Let’s
now look at

When b=1, let’s look
at changes when k= 1, 2, 3, 5, 7, 10 (the graphs are in order respectively)

Again, the relationship that cosine help with the variable k holds true for sine as well. When a=0 and b=1, the variable k determines the number of “petals” the graph will have when k is an odd number. When k is an even number, the number of “petals” is equivalent to two times the number k.