This is the 2nd part of my write-up of Assignment #11
Brian R. Lawler
EMAT 6680

Polar Equations

The Problem

Part I.
Part II.
Part III.
Investigate varying a, b, c, and k.

Analysis - Part II.

Recall that the discussion and summary of observations assumes the polar coordinate system. Recall that any ordered pairs (m, n) are first a radius (or length) than a clockwise rotation from the positive x-axis. Click on any graph or equation to view and manipulate the function in Graphing Calculator 3.0.
First, I considered the graph of with a = 1 and k = 1. Immediately I began to vary b.

with a = 1 and k = 1

For even number k, the decimal values of b between -2 and 2 is where both sets of petals occur (recall from Part I. that even k values appeared to have 2k petals). When b is 0 is when they appear the same size.

with a = 1 and k = 5

For odd number k, the decimal values of b between -2 and 2 is again where two sets of petals occur. However, in this case the petals emerge along the same radii, so when b is 0 is when they are not only the same size, but also in the same place.
And finally, the general curve, no matter the value of k, approaches a circle of radius b centered at the origin.
Next, I looked at the graph of with a = 1 and k = 1. I predicted the behavior would be the same. Of course, I began to vary band observe the results. As you can see below, the effect of b is the same as above.

With a = 1, k = 1, and b = 1

with a = 1, k = 4, and b = 1

click to continue to Part III.

Comments? Questions? e-mail me at

Last revised: December 28, 2000

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