Summary of Parametric Equations

Essential Concepts

  • 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.


a pointed end or part where two curves meet
the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage
the direction that a point moves on a graph as the parameter increases
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 [latex]x\left(t\right)[/latex] and [latex]y\left(t\right)[/latex] over an interval [latex]a\le t\le b[/latex] combined with the equations
parametric equations
the equations [latex]x=x\left(t\right)[/latex] and [latex]y=y\left(t\right)[/latex] that define a parametric curve
parameterization of a curve
rewriting the equation of a curve defined by a function [latex]y=f\left(x\right)[/latex] as parametric equations