**A parametric curve in the plane
is a pair of functions**

**
x=f(t)**

**
y=f(t)**

**where the two continuous functions
define ordered pairs (x,y). The two functions are normally called
parametric equations of a curve, and the curve is dependent upon
the range of t. We will be exploring the parametric equations**

**as well as variations of these
equations. In these cases, t is between o and 2(pi) inclusively.**

**First let's look at the case
when **

**This graph is one that is known
to most students as the unit circle. The center is at (0, 0) and
the radius is 1. I want to explore what happens to this graph
as values in the equation change.**

**Let's look at the case when something,
b is added to cos t. So we now have the situation of **

**The center of this circle is
at (1, 0) and the radius is still 1. So it appears that as b varies
in this situation the graph slides along the x-axis. That is the
center of the circle is (b, 0) a nd the radius is 1. After further
exploration we see that adding c to the sin t function, it slides
the circle up and down the y-axis. That is the center of the circle
is at (0, c) and the radius is still 1.**

**Therefore I feel that it is time
to generalize. **

**When we have the equations**

**then the graphs that we get are
circles where the center at (b, c) and radius 1.**

**More exploration can be done
to the graphs by investigating what would happen if we had the
functions**