Investigation of

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

Again,  we see that the equation is symmetric around the y-axis, however it has a distinctly different shape.  There appears to be a smaller curve inside the larger one...how will this change if we change the value of b.

To investigate further we will see what happens to the graph when we vary the value of b...therefore let's explore the graph of the equation when a = 1and k = 1 and b = 2.

It appears that the smaller curve has disappeared ...it now appears that there is a bend in the larger curve.

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

Not quite sure what is happening here...now we appear to have 2 pedals connected at the origin...let's look if we vary a as well; a = 2, k = 2 and b = 2.

Now we have 4 pedals with two of them smaller than the original two...what happens when we make a = 5.

In this case it just appears that the shape of the pedals expanded as it did in the previous investigation when we changed the value of a...so let's change the value of k...let's look at the equation when a = 2, k = 5, and b = 2.

Here it appears that we have 5 large pedals with 5 smaller pedals inside.
Lastly let's vary all of the values...a = 3, b = 1, k = 5

Playing around with different variables tends to lead that one can make the assumption that the variable b determines whether or not you have inside pedals

In the above graph, a = 1, k = 5, and I have set b equal to 1, 2, 3 and 4