10.5 Parametric Equations

          With parametric equations, a new variable is added to the two variables of rectangular equations, x and y. This third variable is time or 't' which is called the parameter. This third variable allows you to not only know where something being plotted has been, but when it was there.
       
          For example, if t represents time in seconds, and the position (x,y) at t seconds is
                    x= 3t + 2,     y = t^2,     t  ≥ 0,
then the table would be:

t	_________x	_______y_
0	2	0
1	5	1
2	8	4
3	11	9
4	14	16
5	17	25
6	20	36
7	23	49
8	26	64
9	29	81
10	32	100

Graph of Example:

How To Find the Rectangular Equation from a Parametric Equation
          Using the example problem from above, you start out by setting each variable, x and y, equal to t, and then setting the two equations equal to each other.

     x = 3t + 2

     x - 2 = 3t
     t = (x - 2) / 3
     y = t^2
     t = √(y)

     
(x - 2) / 3 = √(y)

     ( x - 2)^2 = ( 3√(y))^2

     x^2 - 4x + 4 = 9y

     y = (x^2 - 4x + 4) / 9

     The Rectangular Equation

Hope this helps,

Alex H.