Parametric Equations - PowerPoint

							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

y t

x y
x2  y

y  x2

x0


x0

parabolic function

Circle:
If we let t = the angle, then:
t

x  cos t

y  sin t

0  t  2

Since:

sin 2 t  cos2 t  1
y2  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  sin 2 t   3  4  x  y      1 3  4
2 2

This is the equation of an ellipse.

