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?