In
this section we will study situations in which *three* variables are used
to represent a curve in the plane.

Suppose
that *x* and *y* are both given as continuous functions of a third
variable *t* (called a **parameter**) by the equations

(called
**parametric equations**). Each value of *t* determines a point ,
which we can plot in the coordinate plane. As *t* varies, the point varies and traces out a curve *C*, which
we call a **plane curve** or **parametric curve**.

Parametric
graphing is like watching the curve draw itself over *time*. It’s
spiritual, man.

As
an **example**, consider the path followed by an object that is propelled
into the air at an angle of .
If the initial velocity of the object is 48 feet per second, the object travels
the parabolic path given by

By
writing *x* and *y* as functions of time *t*, we can obtain
parametric equations

When will the object reach maximum height?

As
another **example**, sketch and identify the curve defined by the parametric
equations

What if we restrict so that ?

As
further **examples**, what curves are represented by the parametric
equations below?

**Use
a graphing device** to graph the curve .
(Note that in general, we can graph curves of the form parametrically by letting and .)

Finding
a rectangular equation that represents the graph of a set of parametric
equations is called **eliminating the parameter**. The range of *x* and
*y* implied by the parametric equations may be altered by the change to
rectangular form.

As an **example**, sketch the curve represented by
the equations below by eliminating the parameter and adjusting the domain of
the resulting rectangular equation.

Find a set of parametric equations to represent the graph of , using each of the following parameters.

As you can see, a parametric expression need not be unique.

The
curve traced out by a point *P* on the circumference of a circle as the
circle rolls along a straight line is called a **cycloid**. Find parametric
equations for the position for the cycloid.

**Smooth Curves**

A
curve *C* represented by and on an interval *I* is called **smooth** if and are continuous on *I* and not simultaneously 0, except possibly at the endpoints of *I*.