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.