For assignment 11 I have chosen to investigate the polar equation

Here is the polar equation simply graphed with a = 1 , b = 1, and k an integer.

Now that we know what the basic equation looks like let's play with the values of a, b and k and see how they each affect the graph.

I. First let a and b be equal to each other and let k be any integer!

For this graph we let a, b = 1, a, b = 2 and a, b = 3/4.

With a little observation we can conclude that the value of a and b determines how big our graphed equation gets. It also appears that whatever our value of a and b is, the graph hits the y axis (both positive and negative) at that number. For example, when we let a and b be equal to two it hits the y axis at -2 and 2. Why do you think that is?

II. Now let's see when happens when we let a and b be different values but still keep k an integer.

For this particular graph we let a be less than b and we kept our original expression in our graph so that we could see the difference better:

We can see that when we let our a be less than our b the only thing that really changes is the fact that our loop gets bigger when there is a greater difference between the two. Now let's see when would happen if we let a be greater than a.

Now we can see the affect of a being greater than b. When this happens it almost seems as if our equation is trying to unfold into a circle and we no longer have the loop. Do you think that we could ever get the graph to be a circle?

III. Now that we know the effect of a and b let's change our k value while we keep a and b constant.


It appears that the value of k determines the number of "leaves" that we have. So when our k=2 we have 2 leaves and so on! But what do you think our graphs would look like if we also changed a and b? First let's see when our a is less than our b. Will we still have the same number of leaves as our value of k? We will still get a loops like our first equation?

YES!!!! We still get equal leafs! Our loop inside also gets the same number of leaves as our value of k! Now let us look at when our a is greater than our b. Will we still not have any loops and will it still try to form a circle? Let's see...

YES,we still do not get loops like before, but we can see that the higher the value of k the more loops try to form!

IV. One other option we have is to take away our a value. What do you think will happen?


We now get a more circular picture. It no longer dips in! What about our b value. What do you think happens when we change it?

We can see that it almost has the same affect as it did on our original equation. Our graph simply gets larger as our value of b grows. Do you think we will have the same affect for our value of k?

Here we can see that yes we still get equal number of leaves as our value of k. But notice that we only have the one set of pedals!Interesting....


Another look that we can take on this equation is if we change our cos to sin. So now we have the equation:

Let's do some of the same explorations on our new equation. Here is the equation simply graphed:

What can you notice about how this differs from our original equation? It seems to have flipped axis. Do you think that all of the correlations we found out about our original equation will be the same just on a different axis? Let's chose a few to test our theory.

I. Let us once again let k be any integer and let a and b be equal to each other.

We can see that we get exactly the same results as we did in our first equation but like before it is around the other axis.

II. Let's try to combine them all, seeing that we think the only thing that will change is the axis rotation.

Once again we are right. The only thing that will change is the rotation on the axis. The value of k still determines our number of k, when a is less than b we still get loops and when a is greater than be we get graphs that do not quite form loops. I leave it to you to see what would happen when we take our a variable away!