Parametric Equations

By: Kimberly Young and Mindy Swain

Consider the following parametric equation:

, where

We have seen from prior investigations that different values of coefficients can an effect of the graphs of the functions. It is the same for parametric equations. Let's investigate and determine what effects a and b have on the above equation.

First consider when a=b. I began by graphing the equation when a=b=1, a=b=2, and a=b=-2. (As shown below. The colors correspond to the graph.

From looking at the graph, I noticed two things:


and are the same graph. This is why we cannot see the red graph. We can generalize that when a=b, positive values for a and b generate the same graph as negative values.


When a=b, it determines the radius of the circle that corresponds to the graph. For example if I had , I would expect to see a graph of a circle with radius 0.5, as seen below:

What happens when a is not equal to b?

Consider the following equations:

, , , and .


The following graph is the graph for all four equations.

The |a| is the distance from the origin on the x-axis and the |b| is the distance from the origin on the y-axis.