Investigating Polar Equations

by Kristy Hawkins

What is a polar equation? It is an equation for a curve written in polar coordinates. But what are polar coordinates? They are a way to describe the location of a point on a plane. A point is given coordinates

where 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.

Let's investigate one of these polar equations.

This picture here shows our equation when a=b=k=1. I wonder what will happen as we change these coefficients...Let's try keeping a=b=1, but varying k to different integer values.

k=2,3,4

So it looks like for each positive integer increase of k, we get a new petal on our flower.

Look at how the petals form when k varies from zero to ten.

What happens when k is negative? Open this Graphing Calculator file to find out!

Another interesting investigation would be to see what happens when a is not equal to b. First let's try a<b for integer values. What do you think it will look like?

Wow, this looks like one petal inside each of the 3 petals. This happens for each integer value of k. What would happen if we keep increasing b? Explore.

Here are some more interesting situations to investigate as well:

Animation of b varying from 0 to 10.

Animation of a varying from 0 to 10.

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