Open main menu

Wikibooks β

Calculus/Parametric Introduction

< Calculus


Parametric equations are typically defined by two equations that specify both the   coordinates of a graph using a parameter. They are graphed using the parameter (usually  ) to figure out both the   coordinates.

Example 1


Note: This parametric equation is equivalent to the rectangular equation   .

Example 2


Note: This parametric equation is equivalent to the rectangular equation   and the polar equation   .

Parametric equations can be plotted by using a  -table to show values of   for each value of   . They can also be plotted by eliminating the parameter though this method removes the parameter's importance.

Forms of Parametric EquationsEdit

Parametric equations can be described in three ways:

  • Parametric form
  • Vector form
  • An equality

The first two forms are used far more often, as they allow us to find the value of the component at the given value of the parameter. The final form is used less often; it allows us to verify a solution to the equation, or find the parameter (or some constant multiple thereof).

Parametric FormEdit

A parametric equation can be shown in parametric form by describing it with a system of equations. For instance:


Vector FormEdit

Vector form can be used to describe a parametric equation in a similar manner to parametric form. In this case, a position vector is given:



A parametric equation can also be described with a set of equalities. This is done by solving for the parameter, and equating the components. For example:


From here, we can solve for   :


And hence equate the two right-hand sides:


Converting Parametric EquationsEdit

There are a few common place methods used to change a parametric equation to rectangular form. The first involves solving for   in one of the two equations and then replacing the new expression for   with the variable found in the second equation.

Example 1




Example 2


Isolate the trigonometric functions


Use the identity