Assignment 11

Jonathan Lawson

In this exploration we will graph equations in the polar coordinate system.  The polar coordinate system looks like a cartesian plane but points are measured by the radian measure and the radius of the endpoint (e.g. the distance the point is from the origin).

We will begin our exploration by examining the equation

r = a + b·cos(kP)

First we restrict the equation to a = 0 and b = 1 so we can understand how the constant k affects the equation.

Ex.

r = cos(2P)

graph1

Ex.

r = cos(3P)

graph2

The constant k affects the number of "petals" the graph has.  If k is even then there are 2k "petals" and if k is odd then there k "petals".

Now we will let a and b both be positive numbers and let a < b.  Here are some examples:

Ex.

r = 1+ 2·cos(4P)

graph3

Ex.

r = 1 + 3·cos(3P)

graph4

When a < b, there are 2k "petals" despite whether k is odd or even, but half of the "petals" appear to be of a diminished size.

Now we will let a and b both be positive numbers and let b < a.  Here are some examples:

Ex.

r = 2 + 1·cos(3P)

graph5

Ex.

r = 3 + 2·cos(4P)

graph6

When b < a, there are k "petals" despite whether k is odd or even, but none of the petals go through the origin.  The is no seperation of each individualized "petals" that appeared before in the previous problems.

Now we will look at the equation when the cosine is replaced with the sine function.  The first graph will be shown again with the new equation overlaying it.

Ex.

r = cos(2P) (in blue)

r = sin(2P) (in red)

graph7

The sine function is a translation of the cosine function by .5Ø so that

cos(2P) = sin(2P + .5Ø)

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