How do you find the parametric equation of a parabola?

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How do you find the parametric equation of a parabola?

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Answer 1 · 2014-08-15T07:15:41

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

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: $x^{2} + y^{2} = r^{2}$ $x^2+y^2=r^2$ . 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: $y = \pm \sqrt{r^{2} - x^{2}}$ $y=+-sqrt(r^2-x^2)$ . Again this is not a function, it is 2 functions combined.

When parameterizing a circle, we have:
$x = r cos t$ $x=r cos t$
$y = r sin t$ $y=r sin t$
$t in RR$

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

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How do you find the parametric equation of a parabola?

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