Exploring Parametric Curves

by

Rita Meyers

This exploration will deal with variations of equations that form parametric curves

First we will start with graphing a basic parametric curve where in all cases

0 < t < 50

To explore parametric equations further we will include variable a and b where

First we will investigate some different values of a...let's look at this equation where a = 3 and a = 6

Looking at our graph we can see that when the value of a is positive that the curve stretches along the x-axis the number of a units

What happens when a is a negative value?

Let's set a = -3 and -6

We get the same graph.

Now let's explore what happens when we change the value of b...lets try b = 3 and b = 6

Here is looks like the curve stretches along the y-axis for the different values of b and again if you make b negative values you would get the same graph.

What would happen if you set a and b equal to each other...let's explore the following equations the first being the red curve and the second being the blue curve

If you notice they are similar to our very first graph...that is due to

being the same as if a and b were both equal to 1 such as

In any case the curve stretches along the x-axis for the value of plus or minus a and it stretches along the y-axis for the value of plus or minus b.

Lastly let's look at what happen when we graph the following equations; again the first being depicted in red and the second in blue

Again this just shows that the graph stretches along the different axis for the values of a and b.

So following all this investigations we can assume the following

1. When a > b the curve is an ellipse stretched across the x-axis

2. When a < b the curve is an ellipse stretched across the y-axis

and

3. When a = b the curve is a circle with a radius of the value of a or b