Feeling a bit Bi-Polar

Presented By

Dana TeCroney

 

 

The purpose of this investigation is to explore polar equations.  Polar equations are represented with a radius (r) and an angle from the x-axis (q).

 

 

 

The equation I would like to consider is of the form

 

What does a graph of this equation look like as the coefficients vary?

 

If a = b = c = 1, then the graph results in the following (q = 0…2p):

 

This graph should make sense if you consider the denominator of the fraction.  As q approaches either p/2 or 3p/2 the denominator approach zero, and hence the entire fraction (r) will be very large.  If our equation was , then the denominator would be confined to values between, or including, 1 to 2.  This implies that our r will range from 1 to ½ as seen below.

 

Changing our equation to  drastically changes the graph.  Now the denominator is undefined when cos(q) = 1/2.  Intuitively this should imply some asymptotic behavior and indeed there is…

 

Enough about the denominator, what about the numerator?  Well, changing the numerator dilates the graph, notice the scale: