for a=b, a<b, and a>b

Let us first look at integer values of k. The following shows the general case when a,b, and k=1.

Now let us look at various integer values for k less than 6.28318530718.

We can see that we seem to be drawing a flower each time, adding one more flower petal or leaf. The value of k gives us the number of petals or leaves. This is called the "n-leaf rose".

Next let us look at when a<b. For this part of then investigation we will let k = 6. (This is so we can see what happens to the pretty flower.)

We can see from the table above that first for different values of a and b, where a<b, gives us two distinct curves. It is an appearance of a flower with two sets of petals. Also notice the difference in the size of the flowers when the values of a and b are changed.

We will now use the same method as above to look at when a<b.

Here we can see that we keep a six petal flower. The only difference is that the flower petals do not intersect at the origin as seen in the previous cases.