How do you find the parametric equation of a parabola?

Aug 15, 2014

If we have a parabola defined as $y = f \left(x\right)$, then the parametric equations are $y = f \left(t\right)$ and $x = t$.

In fact, any function will have this trivial solution.

It is more useful to parameterize relations or implicit equations because once parameterized, they become explicit functions.

For instance a circle can be defined as: ${x}^{2} + {y}^{2} = {r}^{2}$. You know that a relation is a function when it passes the vertical line test; a circle certainly does not.

When you try to define the circle explicitly, you get: $y = \pm \sqrt{{r}^{2} - {x}^{2}}$. Again this is not a function, it is 2 functions combined.

When parameterizing a circle, we have:
$x = r \cos t$
$y = r \sin t$
$t \in \mathbb{R}$

Both $x$ and $y$ are explicit functions, and we can easily plot, integrate, or differentiate them as necessary.