Parametric Equations

Karyn Carson

For
various **a **and **b**, investigate

LetÕs set
a and b equal to a range of values and see what happens and what relationships
can be seenÉ

LetÕs
begin with situations where a = b and they equal one:

This
equation results in a circle with center at the origin and a radius of one.

Now,
letÕs change a and b to 2:

ItÕs the
same circle. How can that be?

LetÕs
change a and b to 3:

Same
circleÉ

What
happens if we make a and b less than one?
How about 0.5?

Aha! We get a half-circle! I think this means that if a and b are more
than one then the same points are graphed, thereby just going over the previous
points.

What if a
and b are different?

a=1 and
b=2

This is a
much different graph! The limits
are still one, but now it looks like a bowÉWhat will happen if a=1 and b=3?

Does b
determine the number of sections if a and b arenÕt equal?

a=1, b=4

It would
appear soÉat least when a=1. Now
IÕd like to keep b constant and vary aÉ

a=2, b=1

The limit
appears still to be one. I need to
look at more graphsÉ

a=3, b=1

This
looks just like the graph where a=1 and b=3, but itÕs been turned and visually
appears to be centered on the y-axis as opposed to the other graph, which
appeared to be centered on the x-axis.
Now a seems to be determining the number of sections.