Understanding Parametric Equations

Hi Kristen,

Only relatively few curves in the plane can be described as the graph
of an equation y = f(x). A circle can't be described in this way, just
parts of it. Parametric equations can be used to describe circles,
and much more.

Parametric equations are like this: one gives the x and y coordinates
of points on the curve in separate equations. For example, the
parametric equations:

x = cos(t)
y = sin(t)

describe a circle, as t varies over [0,2pi]. For each value of t,
plot the point (x,y) = (cos(t),sin(t)). The moving point will trace a
circle. Notice, for example, that x^2+y^2 = cos^2(t)+sin^2(t) = 1.
Hence, (x,y) is on a circle with radius 1, centered at the origin.

Another example:

x = t*cos(t)
y = t*sin(t)

is a spiral, and:

x = t-cos(t)
y = 1-sin(t)

is a cycloid curve.

Parametric curves are used in three dimensions as well. The
parametric equations:

x = cos(t)
y = sin(t)
z = t

describe a helix, curling around the z-axis.

The variable t is a parameter, often describing time. The position of
a particle at any time t might be:

x = 4t-3
y = -t^2+7t+1

which is a parabolic path. Perhaps I've said enough for you to get the
idea.