Exploring Parametric Curves
by
Rita Meyers

This exploration will deal with variations
of equations that form parametric curves
First we will start with graphing a
basic parametric curve where in all cases
0 __<__ t __<__ 50
To explore parametric equations further
we will include variable a and b where
First we will investigate some different
values of a...let's look at this equation where a = 3 and a = 6
Looking at our graph
we can see that when the value of a is positive that the curve
stretches along the x-axis the number of a units
What happens when a
is a negative value?
Let's set a = -3 and -6
We get the same graph.
Now let's explore what happens when
we change the value of b...lets try b = 3 and b = 6
Here is looks like the curve stretches
along the y-axis for the different values of b and again if you
make b negative values you would get the same graph.
What would happen if you set a and
b equal to each other...let's explore the following equations
the first being the red curve and the second being the blue curve
If you notice they are similar to our
very first graph...that is due to
being the same as if a and b were both
equal to 1 such as
In any case the curve stretches along
the x-axis for the value of plus or minus a and it stretches along
the y-axis for the value of plus or minus b.
Lastly let's look at what happen when
we graph the following equations; again the first being depicted
in red and the second in blue
Again this just shows that the graph
stretches along the different axis for the values of a and b.
So following all this investigations
we can assume the following
1. When a > b the curve is an ellipse
stretched across the x-axis
2. When a < b the curve is an ellipse
stretched across the y-axis
and
3. When a = b the curve is a circle
with a radius of the value of a or b

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