Exploring Polar Equations

By Princess Browne

** **

Investigate

Note:

- When a
and b are equal, and k is an integer, this is one textbook version of the Òn-leaf
rose."

- Compare with

For various k. What if .
. . cos ( ) is replaced with sin ( )?

**r = b sin (k**q**)**

** **

We will
start this investigation by graphing the function of

**r = a + b cos (k**q**)**

** **

a=b=k=1

The
following function gives us a cardiod, which is known as the textbook version
of the Ôn-leaf roseÕ. We want to graph the function leaving k as a constant and
change the values of a and b. Next, we will graph the function leaving a and b
constant and change the values of k. Finally, we will compare the equations by
using the different graphs.

Purple r
= a + b cos (kq), Red r = a + 2 cos (kq)

Purple: r
= a + b cos (kq), Red: r = a - 2 cos (kq)

Purple: a
= 1, Red: a = 2, Blue: a = 3

When a and b are constant and k
varies

Red: k =
2, Purple: k = 3, Blue: k = 4, and Green: K = 5

Next, we will look at r = b cos (kq)

Purple: K
= 2, Red: K = 3, Blue: k = 5, Green: k =4

Basally,
k indicates the number of leaf rose the graph creates. When k is an odd integer
the graph will create a rose with k petals. When k is an even integer, the
graph will create a rose with 2k petals. For instance, if k = 3 the rose will
produce a rose with 3 pedals and if k = 4 the rose will produce a rose with 8
pedals.

**Finally, we will graph the
equations using sine instead of Cosine!**

Teal: r
= a + b sin (kq)

Purple: r
= a + b sin (2q)

Red: r = a + b sin (3q)

Blue: r
= a + b sin (4q)

Green: r
= a + b sin (5q)

Teal: r = b sin (kq)

Purple: r
= b sin (2q)

Red: r = b sin (3q)

Green: r = b sin (4q)

Blue: r = b sin (5q)

The graphs
rotate towards the y-axis. The value of k also determines the number of pedals
created. We can see that the graphs with sine and cosine are the same with the
exception of the rotation.