We often define a curve by expression
$x$ as a function of
$$y = f(x)$$
Using parametric equations we define the
$y$ coordinates of the points
on the curve in terms of an independent variable, which we will call
x = g(t)\\
y = h(t)
For any value of
$t$, a value of
$y$ can be calculated, and the point
$(x, y)$ will lie on the curve.
One way to think of this is to imagine
$t$ representing time. As the time changes, the
$(x, y)$ will move, tracing the curve. But this is just an aid to understanding, the
$t$ does not necessarily represent time.
In this section we will look at the parametric equations of parabolas and hyperbolas, and also see how to express them as Cartesian equations.