Parametric Equations

By Tonya Brooks

In this investigation, I want to take a look at one set of parametric equations in depth.  LetŐs take a look at what happens when

for - p  < t < p.  Our graph looks like this:

As I am sure that you have noticed, we have a big gap in our graph right around where x = - 1.  One of the things that might have happened is we might not have let t range far enough, so letŐs try letting t go from - 2p to 2p.  Making this change, we get:

Do you see the difference between the two?  We still have that gap, but it isnŐt as big now, so letŐs try letting t go from - 5p to 5p.

Unfortunately, we still have not closed our circle.  The only thing that we are doing is making the gap smaller and smaller.  Any ideas why?

LetŐs take a look at our x-values.  For our purposes, x = (1 - t2)/(1 + t2).  In order for us to close the gap in our circle, we have to have x go to -1.  What values for t will allow this to happen?  It might be helpful to look at x as though we wanted to take the limit.

If we divide both the numerator and denominator of x by t2, we get:

.

We need x to go to -1 and we can see that in order for this to happen, our t2 must go to infinity.  This means that t also has to get really large (or really small).  Another way to interpret this is that our time must go on forever in order for the gap to get closer and closer together.

If you notice, I didnŐt discuss much about the y values for this equation.  In this case, my y values cannot help me very much.  When the gap closes, our y-value goes to zero, but we already have a point where y is zero, so I chose to work with x since there is only one point where x is -1.