Polar Equations

By Colleen Garrett

In this assignment I will investigate .

LetŐs first look at  when a=b and k is some integer.

These shapes we have created are called cartiods.  Notice that as the value of a and b increase the cartiod gets larger.  Now look at the equations  and . They are the same graph because cosine is an even function. You should also notice that when we change a and b to –a and –b the cartiod is reflected across the y-axis.

Now letŐs look at the shapes we get when a<b and a>b!

These shapes along with the cartiod are called limacons.

Now letŐs explore what happens when a=0, b=1 and k is not equal to one.

We no longer get a cartiod.  We have a rose!  We see that when k is odd  represents the number of petals the graph will have.

What happens if k is even?

Now it appears that 2 represents the number of petals the graph will have.

What if we substitute sine in for cosine?

The same holds for sine as for cosine in regards to the number of petals. When k is odd  represents the number of petals the graph will have and when k is even 2 represents the number of petals the graph will have.