# 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 oforand
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.

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