Tiffany N. KeysÕ Assignment 11:

Polar
Equations

Investigate:
r = a + b cos (kQ)

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

Compare
with b cos (kQ) for
various k. What if . . . cos( ) is replaced with sin( )?

In
r = a + b cos (kQ), the variables a, b, and k are held
constant and Q is between 0 and 2

When
k = 2, the graph is as follows:

r = a + b cos (2Q)

Observations:

á
When
k=2, there are two leaves on the graph.

á
These
leaves cross the x-axis at the origin and points (-2,0) and (2,0).

When
k = 3, the graph is as follows:

r = a + b cos (3Q)

Observations:

á
When
k=3, there are three leaves on the graph.

When
k = ½, the graph is as follows:

r = a + b cos (1/2Q)

Observations:

á
When
k = ½, the graph appears to be half of the graph when k = 1.

á
It
is safe to conclude that k determines the number of leaves a graph will
contain.

LetÕs
move on to investigate r = b cos (kQ)

Observations:

á
On
this graph, there is only one leaf.

á
We
cannot yet conclude that k determines the number of leaves of a graph because
this graph is different in shape from the graph of r = a + b cos (kQ).

When
k = 2, the graph is as follows:

Observations:

á
This
graph has six leaves.

When
k = 3, the graph is as follows:

Observations:

á
This
graph only has three leaves

á
When
k = 2, why does the number of leaves double?

When
k = 4, the graph is as follows:

Observations:

á
Again,
the number of leaves appear to have doubled.

á
Can
we conclude that when k is an even number the number of leaves will be 2k?

When
k = 6, the graph is as follows:

When
k = 8, the graph is as follows:

Observations:

á
When
k = 6, there are 12 leaves on the graph and when k = 8, there are 16 leaves on
the graph.

á
We
can conclude that when k is an even integer, the number of leaves on the graph
will be 2k.

LetÕs
investigate what the graphs will look like if cosine was replaced with sine . .
.

The
graph of the equation r = a + b sin (kQ) is as follows:

r = a + b sin(2Q) r = a + b sin(3Q)

Observations:

á
It
appears that when cosine is replaced with sine, the graph is rotated across the
x-axis

The
graph of the equation r = b sin (kQ) is as follows:

r = b
sin(2Q) r = b sin(3Q)

Observations:

á
Like
in the graph of the equation r = a + b sin(kQ), it appears that
when cosine is replaced with sine, the graph is rotated across the x-axis