College Algebra 9th txtbk

A-24 APPENDIX B SPECIAL TOPICS
10
y
Each value of t determines a value of x, a value of y, and therefore, an ordered pair (x, y).
To graph the set of ordered pairs (x, y) determined by letting t assume all real values, we
construct Table 1 listing selected values of t and the corresponding values of x and y. Then
we plot the ordered pairs (x, y) and connect them with a continuous curve, as shown in
Figure 1. The variable t is called a parameter and does not appear on the graph. Equations
(1) are called parametric equations because both x and y are expressed in terms of the
parameter t. The graph of the ordered pairs (x, y) is called a plane curve.
5
x
Table 1
Z Figure 1 Graph of x t 1,
y t 2 2t, t .
t 0 1 2 3 4 1 2
x 1 2 3 4 5 0 1
y 0 1 0 3 8 3 8
Technology Connections
Parametric equations can also be graphed on a graphing
calculator. Figure 2(a) shows the Parametric mode selected
on a Texas Instruments TI-84 calculator. Figure 2(b) shows
the equation editor with the parametric equations in (1) entered
as x 1T and y 1T . In Figure 2(c), notice that there are
three new window variables, Tmin, Tmax, and Tstep, that
must be entered by the user.
10
3
7
(a) (b) (c) (d)
Z Figure 2 Graphing parametric equations on a graphing calculator.
2
In some cases, it is possible to eliminate the parameter by solving one of the equations
for t and substituting into the other. In the example just considered, solving the first equation
for t in terms of x, we have
t x 1
Then, substituting the result into the second equation, we obtain
y (x 1) 2 2(x 1)
x 2 4x 3
We recognize this as the equation of a parabola, as we would guess from Figure 1.
In other cases, it may not be easy or possible to eliminate the parameter to obtain an
equation in just x and y. For example, for
x t log t
y t e t t 7 0
you will not find it possible to solve either equation for t in terms of functions we have
considered.