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