Question

How do you find the parametric equation of a parabola?

Answer 1

If we have a parabola defined as `y=f(x)`, then the parametric equations are `y=f(t)` 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=+-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 RR`

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