Summary of Parametric Equations

Parametric equations provide a convenient way to describe a curve. A parameter can represent time or some other meaningful quantity.
It is often possible to eliminate the parameter in a parameterized curve to obtain a function or relation describing that curve.
There is always more than one way to parameterize a curve.
Parametric equations can describe complicated curves that are difficult or perhaps impossible to describe using rectangular coordinates.

Glossary

cycloid: the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage

orientation: the direction that a point moves on a graph as the parameter increases

parameter: an independent variable that both x and y depend on in a parametric curve; usually represented by the variable t

parametric curve: the graph of the parametric equations $x\left(t\right)$ and $y\left(t\right)$ over an interval $a\le t\le b$ combined with the equations

parametric equations: the equations $x=x\left(t\right)$ and $y=y\left(t\right)$ that define a parametric curve

parameterization of a curve: rewriting the equation of a curve defined by a function $y=f\left(x\right)$ as parametric equations