We use parametric equations to think of problems or equations with respect to time. These equations have many real life applications.I will begin my exploration by looking at the graphs of sin and cos from a parametric perspective. I will range t from 0 <= t <= 2pi. This will be one complete cycle for the curves.

This is the graph when a=b=1. The a-value tells you where the graph will intersect the x-axis and the b value tells you where the graph will intersect the y-axis.

To obtain this graph in Graphing Calculator we need the equation to be in a vector form

Now I will vary the variables a and b from my original equation. I will look at three different situations:

i. a = b

So it seems that when |a| = |b| it will always form a circle with a radius of |a| or |b| since they are equal.

ii. |a| > |b|

These graphs are elipse. These graphs will be longer than they are wider when |a| > |b|.

iii. |a| < |b|

When |a| < |b| then the graph will be taller than it will be long.

I am only taking looking at the absolute values of these graphs because the same conclusions will not apply if we look at a = b, a > b, and a < b. That would be another good topic to investigate.