Exploring Parametric Equations
of the form:
,
,
, etc.
By: Lauren Wright
Let's first look at what happens when
a = b for
.
![](image7.gif)
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
.
![](image9.gif)
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
.
![](image16.gif)
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.
![](image24.gif)
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
.
![](image20.gif)
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
.
![](image30.gif)
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.
![](image26.gif)
This gives us a diamond-like shape
with x-intercept = y-intercept = +a = +b.
Next, let's look at
for a < b.
![](image28.gif)
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.
![](image32.gif)
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.
![](image35.gif)
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.
![](image37.gif)
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.
![](image39.gif)
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 of
or
and 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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