1.4 Parametric Equations
Greg Kelly, Hanford High School, Richland, Washington
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 called parametric equations.
“t” is the parameter. (It is also the independent variable)
Example 1:
x t
y t
t 0
To graph on the TI-89:
MODE
Graph…….
2
PARAMETRIC
ENTER
Y=
xt1 t yt1 t
2nd
T
)
ENTER
WINDOW GRAPH
Hit zoom square to see the correct, undistorted curve.
We can confirm this algebraically:
x t
x y
x2 y
y t
y x2
x0
x0
parabolic function
Circle:
If we let t = the angle, then:
t
x cos t
y sin t
0 t 2
Since:
sin2 t cos2 t 1
y 2 x2 1
We could identify the parametric equations as a circle.
x2 y 2 1
Graph on your calculator: Y=
xt1 cos( t) yt1 sin( t)
Use a [-4,4] x [-2,2] window.
WINDOW GRAPH
2
Ellipse:
x 3cos t
y 4sin t
x cos t 3
2 2
y sin t 4
x y cos2 t sin2 t 3 4 x y 1 3 4
2 2
This is the equation of an ellipse.