## Parametric equations

We often define a curve by expression `$x$`

as a function of `$y$`

:

`$$y = f(x)$$`

Using *parametric equations* we define the `$x$`

and `$y$`

coordinates of the points
on the curve in terms of an *independent variable*, which we will call `$t$`

:

```
$$
\begin{align}
x = g(t)\\
y = h(t)
\end{align}
$$
```

For any value of `$t$`

, a value of `$x$`

and `$y$`

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
point `$(x, y)$`

will move, tracing the curve. But this is just an aid to understanding, the
parameter `$t$`

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.