Graphing Calculator 3.2 and xFunction are suggested for these investigations. Some of them could be done with a TI-83 or similar.

Polar Coordinates are a way to describe the location of a point on a plane. A point is given by coordinates (r,theta). r is the distance from the point to the origin, and theta is the angle measured counterclockwise from the polar axis to the segment connecting the point to the origin.

Investigate

When a and b are equal, and k is an integer, this is one textbook version of the " n-leaf rose."
Compare with

for various k. What if . . . cos( ) is replaced with sin( )?

Let's look at r=a+bcos(ktheta) first.

This is the graph of a=1, b=1, k=1.

a=2, b=1, k=1

a=1, b=2, k=1

a=1, b=1, k=2

a=1, b=1, k=5

a=1, b=1, k=10

Now let us investigate our equation

for different values of b and k. First look at b=1 and various values for k.

k=1, k=2

k=3, k=5

Now let us look at various values for b when k remains equal to 1. Notice that larger values for b produce larger circles.

b=1, b=2

b=3, b=5

Now what if we replace cos with sin in our second equation? Let's see what happens.

b=1, b=2

b=3, b=5

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