How do you find the parametric equation of a parabola?

1 Answer
Amory W.
Aug 15, 2014

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.

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