The Study of Polar Equations
By: Amanda
Sawyer
In this assignment, we are asked to investigate the polar equation 
. We want to
determine:
A) When a
and b are equal and k is an integer, what happens?
B) How is the original polar
equation related to 
for various k
values?
C) How is the original polar
equation related to 
for various k
values?
Polar coordinates are points defined in space that
are defined by its vector r from the origin with its angle
measurement from the positive x-axis.
From this information we can define the coordinate from distance from
the origin and it rotate from the positive x-axis
as the coordinates of the point. We can see this in the picture below.
A)
When a and
b are equal and k is an integer, what happens?
Let’s first determine what happens to our polar
equation when a=b and k is an
integer. We can view this infromation
from the table below.
|
A AND B VALUES |
K VALUE |
GRAPH |
|
1 |
1 |
|
|
2 |
|
|
|
3 |
|
|
|
4 |
|
|
|
2 |
1 |
|
|
2 |
|
|
|
3 |
|
|
|
4 |
|
|
|
4 |
1 |
|
|
2 |
|
|
|
3 |
|
|
|
4 |
|
A clear pattern is created from these 12 pictures. The k value represents the number of
“loops” in our graph. For example, when
our k
value is one, our graph has only one loop shape. When our graph’s k value is two, it
contains two loops. We also can notice
that the maximum x-value of our graph is determined by our values of a and b.
When a and b are equal to a value
lets name M, then the maximum x-value of the equation is 2M. For example, in the above picture we can see
that when a=b=2, then the maximum x-value is (4,0). This holds true for all values of a=b.
B) How is the original polar
equation related to for various k values?
Next,
let’s study the polar equation for various k
values. The table below gives the value
of k
and b
with its given graph.
|
B VALUE |
K VALUE |
GRAPH |
|
1 |
1 |
|
|
2 |
|
|
|
4 |
|
|
|
2 |
1 |
|
|
2 |
|
|
|
4 |
|
|
|
4 |
1 |
|
|
2 |
|
|
|
4 |
|
In
this equation, we can see that the maximum x-value of our graphs are obtained from the b value. For example, when b=2, the maximum x-value
on the graph was (2,0). When b=4, the maximum x-value of the
graph was (4,0).
The other observation that we can make is that the number of loops are
now a multiple of k. For any equation , the number of “loops” the graph will have is 2k
for values of 2 or more. We can see this when k is 4, the number of
loops the graph had was 8.
C) How is the original polar
equation related to for various k values?
Finally let’s study the polar equation for various k
values. The table below gives the value
of k
and b
with its given graph.
|
B VALUE |
K VALUE |
GRAPH |
|
1 |
1 |
|
|
2 |
|
|
|
4 |
|
|
|
2 |
1 |
|
|
2 |
|
|
|
4 |
|
|
|
4 |
1 |
|
|
2 |
|
|
|
4 |
|
This
equation acted very similar to the equation in part B, we can see that the
maximum x-value of our graphs is obtained from the b value. For example, when b=2, the maximum x-value
on the graph was (2,0). When b=4, the maximum x-value of the
graph was (4,0).
The other observation that we can make is that the number of loops are
now a multiple of k. For any equation , the number of “loops” the graph will have is 2k
for values of 2 or more. We can see this when k is 4, the number of
loops the graph had was 8. The biggest
difference between part B and part C is that the graph for the sine function
has 90-degree counterclockwise rotational symmetry to the cosine function’s
graph.
Again,
through the use of graphing calculator, we are able to graph out the functions
and determine different properties of each of our equations.