Parametric Equations

By Taylor Adams

In this exploration, we are going to
look at different values of a and b for

When a=1 and b=1, we get a circle of
radius 1.

When a=2 and b=2, we get a circle of
radius 2.

Let’s see if a=3 and b=3, if the
radius of the produced circle will be 3.

What happens if a
and b are negative?

When a=-1 and b=-1, it produces the
same circle as when a=1 and b=1.

Therefore, when a
and be are equal, their absolute value provides the radius of the circle
formed.

What happens when a
and b are not equal?

Let’s see what happens when a=2 and
b=1.

In this case, the parametric equation
produces an oval. The a-value gives us
the distance from the origin to the greatest positive x-value, and b gives us the
distance from the origin to the greatest positive y-value.

What happens when a=1 and b=2?

This parametric equation is also an
oval. The a-value gives us the distance
from the origin to the greatest positive x-value, and b gives us the distance
from the origin to the greatest positive y-value.

Let’s see if a=2 and b=6 will give us
the same relationship.

It, in fact, does. In this case, a=2, which is the greatest
x-value, and b=6, which is the greatest y-value.

Even though a and
b are not equal in these cases, a and b effect the parametric equation in the same
way as when they are equal. As I talked
about above, a gives us the distance from the origin to the greatest positive
x-value, and b gives us the distance from the origin to the greatest positive y-value,
which is true whether or not a and be are equal or which value is greater.