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Parametric functions

A parametric function expresses the coordinates of a point as a function of another variable called the parameter . This is used mainly in physics (mechanics) where the parameter is time.

In the plane, the usual notation is $$$\big(x(t),y(t)\big)$$$ for the function and $$t$$ for the parameter.

A function $$f(x)$$ is a parametric function with $$x(t)=t$$ and $$y(t) = f(t)$$.

The parametric equation for the circle is $$$ x(t) = \cos t,\quad y(t) = \sin t,\quad t\in\R.$$$ This is seen immediately as $$x^2(t)+y^2(t) = 1$$ for all $$t$$.