Feeling a bit Bi-Polar
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: