Exploring Parametric Equations

By Mary Negley

For this exploration I will look at  and , where  and  and  are different values.  First I will look at the case when .

When , a circle of radius 1, centered at the origin, is formed.

How about if , but every other condition remains unchanged?  Will the graph change?

Clearly, the graph is unchanged.  Through own exploration, one may easily find that if  and , the graph will be identical to the two previous graphs.

Now letŐs explore when .  In this example, let  and .

The graph is now an ellipse, instead of a circle.  It is longer vertically than it is horizontally, and it is still centered at the origin, but it crosses the x-axis at -2 and 2 and it crosses the y-axis at -5 and 5.  Notice that in this example,  and , and the positive and negative values of those two numbers are also the points where the graph crosses the axes.  My prediction is that the graph will always cross the x-axis at and the y-axis at.  Now, letŐs see what happens if I graph the same parametric equations only this time  and .  If my prediction is correct, then the new graph should be an ellipse that is longer horizontally than it is vertically and centered at the origin. Moreover, the graph should cross the x-axis atand the y-axis at.

I was correct.

From looking at the previous four graphs, one would assume that if , then the graph would be longer vertically than horizontally, if , then the graph would be longer horizontally than vertically, and if , then the graph would be a circle.  LetŐs try a few more cases testing those assumptions.

First we will study when .  First letŐs look at when  and .

This graph is an ellipse that is longer horizontally than it is vertically.  Now letŐs look at when  and .

Again, this is an ellipse, which is longer vertically than it is horizontally.  Now letŐs look at the case when .  This time letŐs look at when  and .

Notice that this time we have an ellipse that is longer horizontally than it is vertically.  Now letŐs try when  and .

Again, we have an ellipse, which is longer horizontally than it is vertically.

Now I will investigate the graphs of the following equations:

for

where h is any real number.

First I want to look at the case when ,  and .

This graph appears to be similar to the ellipse from two examples ago, but it is skinnier and it has been rotated 45 degrees counter-clockwise about the origin.  LetŐs see what happens when .

This appears to be the same ellipse as the previous example only this time it is rotated clockwise instead of counter-clockwise.  What happens if ?

This appears to be an ellipse that it rotated 45 degrees counter-clockwise and it is wider than the previous ellipse.  If , I would suggest that it would be an ellipse identical to the previous one, only it is rotated 45 degrees clockwise.

My suggestion was correct.