Section 6.3: Parametric

Equations and Motion

March 13, 2015

Parametric Equations

We can define both elements of the ordered pair, (x, y), in terms of another variable, t, called a parameter.

Example: Given and ,a) Find the points determined by t = -3, -2, -1, 0,

1, 2, & 3.b) Find a direct relationship between y and x and

determine whether the parametric equations determine y as a function of x.

c) Graph the relationship in the xy-plane.

1x t 2 2y t t

Graphing in Parametric Mode

In order to graph parametric equations, you must change your mode from functions to parametric. Hit the “Mode” key. On the 4th line, shift to PAR.

Example: Given and ,a) Use a graphing calculator to find the points

determined by t = -3, -2, -1, 0, 1, 2, & 3.b) Use a graphing calculator to graph the relation in

the xy-plane.c) Is y a function of x?d) Find an algebraic relationship between x and y.

x t 2 5y t

Graphing Parametric Equations

Example: Graph and on the following intervals…a)

b)

23x t 2y t

10 0t

2 4t

Eliminating Parameters

Solve for one variable. Substitute into 2nd equation. Simplify.

Example: Eliminate the parameter and identify the graph of the parametric curve:

a)

b)

2x t1y t

5cosx t5siny t

Find the Parametric Equation.

Write a parametric equation for a circle with radius of 7 and center at (-3, 5).

Finding Parametric Equations of a Line

Example: Find a parametrization for the line through the points (4,1)when t = 0 and (19, 26) when t = 5.