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.
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