Exploring Parametric Equations of the form:

,

,

, etc.

By: Lauren Wright

Let's first look at what happens when a = b for

.

We see that parametric equations of this form with a = b will form circles with radius =

.

Now let's look at what happens when a < b for

.

We see that parametric equations of this form with a < b will form ellipses with the major axis equal to the greater of

or

and the minor axis equal to the smaller.

Again, when

, we have a circle.

Now let's look at what happens when a > b for

.

As we can see, the same is true for a < b and a > b.

Let's look now at equations of the form

for a = b.

We see that parametric equations of this form with a = b will form segments with a slope of -1 and y-intercept = x-intercept = a = b.

Now let's look at what happens for a < b with

.

Here we see that parametric equations of this form with a < b will form segments with slope =

, y-intercept = b, and x-intercept = a.

Lastly, let's look at what happens for a > b with

.

Once again for the parametric equation of the form

, the graphs when a < b have the same characteristics of the graphs when a > b.

Now, let's take a look at

for a = b.

This gives us a diamond-like shape with x-intercept = y-intercept = +a = +b.

Next, let's look at

for a < b.

This time, we get more diamond-like shapes, but they are not all similar. When a < b, x-intercept = +a and y-intercept = +b.

Lastly for

, we will look at the graph when a > b.

Again, the same is true for a < b and a > b when the parametric equation is of the form

.

Now, let's examine one more parametric equation of the form

so that we can be sure when we make our generalizations about parametric equations of this form.

This is similar to the parametric equation of the form

, only now we have curves instead of segments. But, the fact that y-intercept = x-intercept = a = b remains the same.

Let's now take a look at

for a < b.

This is also similar to

for a < b. Again, we have curves instead of segments, but y-intercept = b, and x-intercept = a.

Lastly, we observe

for a > b.

Again, we see the similarity to

.

CONCLUSION

After observing these graphs, I would like to make some generalizations:

1. Parametric equations of the form have the following characteristics:

M = 2 : A Special Case:

Segments will form with slope = , y-intercept = b, and x-intercept = a.

M Even:

Curves will from with y-intercept = b, and x-intercept = a.

M = 1 : A Special Case:

When a = b circles will form with radius = .
When , ellipses will form with the major axis equal to the greater oforand the minor axis equal to the smaller.

M Odd:

Diamond-like shapes form with curved lines and x-intercept = +a and y-intercept = +b.