Parametric Equations

By Annie Sun

A parametric curve in the plane is a pair of functions: x = f(t) and y = g(t),
where the two continuous functions define ordered pairs (x,y). The two equations are usually called the parametric equations of a curve. The extent of the curve will depend on the range of t and your work with parametric equations should pay close attention the range of t . In many applications, we think of x and y "varying with time t " or the angle of rotation that some line makes from an initial location.

We will start the exploration by graphing

With 0 ≤ t ≤ 2π

This corresponds with the table of values for the equations:

 t x y 0 1 0 π/2 0 1 π -1 0 3π/2 0 -1

Let’s explore different graphs by plugging in numbers into the equations, starting in the form:

After using the same number for a in both equations, the resulting graph is the same:

So, instead, I will try when a < b and a > b, since we know we will get the unit circle when a = b.

Here is an example when a < b:

Also, there was a difference in graphs when b was odd and even; when a and b are even numbers, they did “overlap” each other:

The graph on the left was typical of a < b values and the right one for a > b

The graphs seem to be rotated 90 degrees.

And, there also seems to be a difference with odd and even values; and if a and b are divisible by each other:

Now let’s see what happens when we change the a and b values using this form of the equation:

Using a = b = 3, we get this:

Which corresponds to our table of values

 t x y 0 3 0 π/2 0 3 π -3 0 3π/2 0 -3

Next I tried a < b:

As I increased the values for b, the ellipse was stretched vertically.

And when a > b:

As the values for a increased, the ellipse was stretched horizontally.

Using some of the basic ideas for what the graph “does” for
and

We can put these characteristics together and see how they as:

In this case a < c, which means a stretch in the vertical direction and b < d, which means the graph “flows” vertically.