Graphs of Cosine

By Tonya C. Brooks

     In this investigation, I decided to look at the graph of r = a + b cos(kq).  LetÕs look at the graph when a, b and k are all equal to 1.

     This doesnÕt look very exciting to we will look at other values of a, b and k and letÕs let them all be the same for a time.

     Here, you can see several different cases of when a, b and k are all equal.  What do you notice?  Do you see why this set of equations might be called an n-leaf rose?

     Now, letÕs take a look at what happens when we keep a and b the same and let k differ.  Here is a collection of several cases.

     Do you see any similarities from before?  How many leaves are there for the different k-values?

     LetÕs look at a few others that might be interesting.

     What do you notice for cases when our a is larger than our b?  How can you see this from the equation?

     Take a look at the last graph.  What is happening?  Any guesses as to why this might happen?  LetÕs look a little more in depth on this case by looking at a few other graphs.

     Notice the difference between even and odd k-values.  What can you say about this?

     If we wanted to look at sine instead of cosine, what do you think would happen?  Would we see the same graphs?  No we wouldnÕt but we would see something similar due to the relationship between sine and cosine.  What would be the difference?