1.4 Parametric Equations

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Greg Kelly, 2005

Mt. Washington Cog Railway, NH

There are times when we need to describe motion (or a curve) that is not a function.

We can do this by writing equations for the x and y coordinates in terms of a third variable (usually t or ).

x f t y g t These are calledparametric equations.

“t” is the parameter. (It is also the independent variable)

Example 1: 0x t y t t

To graph on the TI-89:

MODE Graph……. 2 ENTER

PARAMETRIC

Y= xt1 t

yt1 t2nd T ) ENTER

WINDOW

GRAPH

Hit zoom square to see the correct, undistorted curve.

We can confirm this algebraically:

x t y t

x y

2x y 0x

2y x 0x

parabolic function

t

Circle:

If we let t = the angle, then:

cos sin 0 2x t y t t

Since: 2 2sin cos 1t t

2 2 1y x

2 2 1x y We could identify the parametric equations as a circle.

Graph on your calculator:

Y=

xt1 cos( )tyt1 sin( )t

WINDOW

GRAPH

2

Use a [-4,4] x [-2,2] window.

Ellipse: 3cos 4sinx t y t

cos sin3 4

x yt t

2 22 2cos sin

3 4

x yt t

2 2

13 4

x y

This is the equation of an ellipse.