POLAR EQUATIONS

**The
following is an investigation of the polar equation:**

**If
we let a, b and k equal 1 and graph the function, we have the
following:**

**Now,
we will let a and b remain as 1 and vary k as an integer to see
what the graph produces.**

**is graphed in purple**

**is graphed in red**

**is graphed in blue**

**We
can begin to see that 'k' possibly produces the number of 'leaves'
of the graph. How many leaves will **

**Now
we will leave k constant, say 3, and look at a graph of**

**where
a < b. A graph of 1 +2 cos 3is shown
below**

**so
it appears that if a < b, we have a new inner leaf design,
and as b increases the size of the leaves increase.**

**We
will now look at what happens to the graph when b > a. Let
us look at**

**It
appears that the leaves are losing their definition. Let's look
at**

**and
finally**

**so
as a increases we loose our leaf shape completely.**