Parametric Equations

Introduction to Parametric Equations

Parametric equations are a method of expressing a set of related quantities as explicit functions of an independent parameter, usually denoted as t. This approach is especially useful for describing curves and geometric figures in the plane, such as lines, circles, and ellipses.

• Definition: A parametric equation expresses both x and y as functions of a third variable, t (the parameter):

• Purpose: Parametric equations allow for more flexibility in describing curves that may not be functions in the traditional sense (i.e., those that fail the vertical line test).

Eliminating the Parameter

To convert parametric equations back to a rectangular (Cartesian) equation, eliminate the parameter t by solving one equation for t and substituting into the other.

• Example: Given and , solve for t in terms of x and substitute into y:

Writing Parametric Equations

To write parametric equations for a given rectangular equation, introduce a parameter (often t or x = t) and solve for the other variable in terms of t.

• Example: For , two possible sets of parametric equations are:

Set 1: , Set 2: ,

How To: Writing Parametric Equations

1. Define x, unless given.

2. Let (or , or as appropriate).

3. Solve for t.

4. Plug into the original equation for y.

5. Solve for .

Equations of Circles and Ellipses

Parametric Equations for Circles

The standard equation for a circle centered at the origin with radius r is . The parametric equations are:

where

Parametric Equations for Ellipses

The standard equation for an ellipse centered at the origin is . The parametric equations are:

where

How To: Parametrize Circles and Ellipses

1. Write the equation in the form .

2. Set , .

3. Let vary from $0.

Example: Parametric Equations for a Circle

• Given , the parametric equations are:

where

Example: Parametric Equations for an Ellipse

• Given , rewrite as :

where

Practice Problem

• Write parametric equations for :

where

Summary Table: Parametric Forms