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.