Sinje J. Butler

We begin by looking at the definition of polar
coordinates. The position of point
**P** is given by

**()**.

The position of a point is described with reference to a fixed point, called the origin, and a fixed positively directed line called the polar axis.

The following will investigate various polar equations in the form

.

Lets begin by taking a look at the simple case of

where

**a = 1**

**b =1**

and

**k = 1.**

** **

** **

This graph is called a cardiod. Notice what takes place when the value of **b** is changed to higher numbers.** **Thus,
we have

for

**a = 1**

**b =1, 2, **and **3****.**

and

**k =1.**

** **

** **

As **k** increases,
it appears that the inner loop is increasing and the outer loop is increasing.

When the following equation,

°

with

**a = 1**

**b = 1**

**k = 1**

** **

is
changed to

°

with

**a = 1**

**b = 1**

**k = 1**

** **

we have the following
graph.

When **cos** is changed to **sin **in the equation, the cardiod appears to rotate counter clockwise around
the origin 90 degrees.

The graph below is

for

**a = 1**

**b = 1**

and

**k = 2,
3, **and
**4****.**

This is what is referred
to as the “n-leaf rose”.
As **k** is increased the flower like graph will have a number
of petals that is equal to **k. **Therefore, if **k **is increased to **25, **the graph will have **25** petals and looks
as follows.

If

°

is changed to

°

the petals appear to
change positions.

Let us take a look at
when

is changed to

If **k** is an even number it appears that the number of
petals double.

The graphs below are

°

°.

If **k** is an odd number it appears that the number of
petals remain the same. The
following graphs are

°

°.

** **

** **

** **