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.