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