1.4 Parametric Equations
Mt. Washington Cog Railway, NH
Photo by Greg Kelly, 2005
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
ENTER
PARAMETRIC
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
y t
y  x2
x y
x2  y
x0
x0
parabolic function

Circle:
If we let t = the angle, then:
t
x  cos t
Since:
y  sin t
0  t  2
sin 2 t  cos 2 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 )
2
Use a [-4,4] x [-2,2]
window.
WINDOW
GRAPH

Ellipse:
x  3cos t
y  4sin t
x
 cos t
3
y
 sin t
4
2
2
x  y
2
2


cos
t

sin
t
   
3  4
2
2
x  y
    1
3  4
This is the equation
of an ellipse.
