## Parametric Equations

## by Morgan Guest

In this exploration, we will consider the parametric equation with t ranging from 0 to 2π:

x = a cos(t)

y = b sin(t)

Let's first look at the graph when a = 1 and b = 1. As expected, we see that the graph is a circle with the center at the origin and radius 1.

Below are two examples where a = b. We see that if a = b, the graph will be a circle with center at the origin and radius = a = b. This will hold true even if a and b are negative.

Example 1: a = 2, b = 2

Example 2: a = 4, b = 4

What happens if a does not equal b? First let's consider when a > b. The picture below shows the parametric graph when a = 2 and b = 1. Notice that the graph is not a circle but an oval with the domain from -2 to 2 and the range from -1 to 1.

Now look at the graph when a = 5 and b = 2. The graph is still an oval shape and the domain is from -5 to 5 and the range is -2 to 2.

Now we can consider what happens to the graph when b > a. For example, the first picture below shows the parametric equation where a = 1 and b = 3. The graph is an oval shape and the domain is from -1 to 1 and the range is from -3 to 3.

The next graph shows the parametric equation where a =1 and b = -3. It is the same graph as above, so we see that negative values for a and b do not change the graph.

From the graphs above, you can see that value of a appears to control the domain and the value of b appears to control the range. We see that the domain will always equal 2*|a| and the range will always equal 2*|b| for all values of a or b regardless of if the graph will be oval or circle shaped.