In this investigation we are going to look at polar functions and how these graphs change as we change different variables in the function, so let's get started.

The first function we are looking at is:

Now try playing with different values for a, b, and k. Graphing Calculator.

As the value for k increases so does the number of leaves. If it is not an integer then the leaves are not fully formed. As a and b are different the leaves do not touch. The closer they are to each other the closer they get to touching, but they also determine the length of the leaves. As a and b increase so does the length of the leaves. Also when b is larger than a then a second row of leaves are formed inside the larger ones.

Now we are going to look at if we change cosine to sine:

What happens to this as we change the a, b, and k values?

When k=1 this graph is only changed by the center looping get bigger as a and b get bigger. When k gets larger and a and b change the graph looks similar to the cosine graph lets check this out.

In the first one: a=6, b=8, k=3 And in the second: a=10 b=8, k=7.

We see that k still tells how many leaves, and when b is larger than a then the leaves repeat inside each other.

Second let's look at this:

We're gonna split these up and look at the first two first.

This is when a=1 and k=1 it looks like a normal circle.

But when we change the a and k values a little we might get something more like this.

when a=8 and k=6

The same trend is seen except the number of leaves is 2k instead of k.

This is just using cosine instead of sine. The circle has shifted, instead of the diameted being the y-axis it is now the x-axis. So what happens when we change a and k?

We see here that this looks similar to the sine graph except it is also rotated so that the x and y axes are bisecting the leaves, but there are still 2k leaves.

Now what about when we take the same equations and add b.

This is the same cosine equation but with b added. All a, b, and k equal one here. How do you think it will change as we change a, b, and k?

Here a=8, b=8 and k=6. We notice a difference this time. There are still 2k leaves but we have large and small leaves.

Here is a comparison of all the equations when a=b=k=1:

Right now they are all different but when we change the values for a, b, and k they look more alike.

If you look closely you can see how the red and purple are the same except for a rotation, the same with the blue and green. All four of these have four leaves, but the red and purple are all the same size and the blue and green have leaves of two different sizes. The black basically forms a line with point where it is undefined.

So we can see how we can change functions by just changing a few things like whether it is a sine or cosine function. The values being added, multiplied to the trig. function or being multiplied times the angle change each graph uniquely.