Investigation of

To start this investigation, again,  we will look at the graph of the above equation when a = 1 and k = 1.

In this graph we see that equation is symmetric around the y-axis and that it appears that the circle has a center of 1 on the polar scale.

To investigate further we will see what happens to the graph when we vary the value of a...therefore let's explore the graph of the equation when a = 3 and k = 1.

It appears that the circle has expanded ...it now appears that the center is equal to 3 on the polar scale.

Now let's see what happens when we vary the value of k...so let's look at what happens if a = 1 and k = 2

It seems that if we allow k > 1 we get a completely different type of graph...instead of a circle we have let's say pedals.

Since the graph has 4 pedals...we could possible assume that k times 2 would equal the number of pedals.
Let's look at another graph to check...how about when k = 5 and a is still equal to 1.

Here it looks like the number of pedals equals the value of k

Click on the links to see the graphs when k = 4 and k = 7.

After looking at these graphs we can assume that when k is an even number, the number of pedals will be twice the value of k.  However, when k is an odd number, the number of pedals is equal to the value of k.

Now what happens when we vary both the values of "a" and "k"?

Lets look when a = 3 and k = 5...if our assumptions stand true we should get a graph with 5 pedals and the center of the pedal being equal to 3.

Yep, what do you know...we have a graph where the center appears to be 3 and it does indeed have 5 pedals.