College Algebra 9th txtbk

SECTION B–3 Parametric Equations A-25
Is there more than one parametric representation for a plane curve? The answer is yes.
In fact, there is an unlimited number of parametric representations for the same plane curve.
The following are two additional representations of the parabola in Figure 1.
x t 3
y t 2 2t
x t
y t 2 4t 3
6 t 6
6 t 6
The concepts introduced in the preceding discussion are summarized in Definition 1.
(2)
(3)
Z DEFINITION 1 Parametric Equations and Plane Curves
A plane curve is the set of points (x, y) determined by the parametric equations
x f (t)
y g (t)
where the parameter t varies over an interval I and the functions f and g are both
defined on the interval I.
Why are we interested in parametric representations of plane curves? It turns out that
this approach is more general than using equations with two variables as we have been
doing. In addition, the approach generalizes to curves in three- and higher-dimensional
spaces. Other important reasons for using parametric representations of plane curves will
be brought out in the discussion and examples that follow.
EXAMPLE 1 Eliminating the Parameter
Eliminate the parameter and identify the plane curve given parametrically by
x 1t
0 t 9
y 19 t
(4)
y
5
5
5
Z Figure 3
SOLUTION To eliminate the parameter t, we solve each equation (4) for t:
5
x
t 9 y 2
Equating the last two equations, we have
x 2 9 y 2
x 2 y 2 9
x 1t y 19 t
x 2 t
y 2 9 t
A circle of radius 3 centered at (0, 0)
As the parameter t increases from 0 to 9, x will increase from 0 to 3 and y will decrease
from 3 to 0.
So the graph of the parametric equations in (4) is the quarter of the circle of radius 3
centered at the origin that lies in the first quadrant (Fig. 3).
MATCHED PROBLEM 1
Eliminate the parameter and identify the plane curve given parametrically by x 14 t,
y 1t, 0 t 4.