## Cartesian equation of a parabola

We can convert the parametric equation of a parabola into a *Cartesian equation* (one involving only
`$x$`

and `$y$`

but not `$t$`

). Here are the parametric equations:

```
$$
\begin{align}
x = a t^2\\
y = 2 a t
\end{align}
$$
```

We can eliminate `$t$`

from these equations by first finding `$t$`

as a function of `$y$`

:

```
$$
\begin{align}
y = 2 a t\\
t = \frac{y}{2 a}
\end{align}
$$
```

Then we can substitute `$t$`

in the equation for `$x$`

```
$$\begin{align}
x &= a t^2\\
x &= a (\frac{y}{2 a})^2\\
x &= \frac{a y^2}{4 a^2}\\
x &= \frac{y^2}{4 a}
\end{align}$$
```

This can also be written as:

```
$$
y^2 = 4 a x
$$
```

## A parabola is a quadratic curve

As you can see, a parabola involves a term in `$y^2$`

which means it is a quadratic curve. It has the classic
quadratic curve shape. But because `$x$`

is a function of `$y^2$`

, the x-axis and y-axis. swap roles. More accurately,
the curve is mirrored about the line `$y = x$`

.

In the graph, the red curve is our parabola (for `$a = 1$`

), and for comparison the blue curve is the quadratic curve
`$y = x^2/4$`

. The green line is the line `$y = x$`

.