College Trigonometry, 2011a

11.10 Parametric Equations 1049
object moves around the circle. In particular, the equations { x = 2960 cos ( π
12 t) ,y= 2960 sin ( π
12 t)
that model the motion of Lakeland Community College as the earth rotates (see Example 10.2.7 in
Section 10.2) parameterize a circle of radius 2960 with a counter-clockwise rotation which completes
one revolution as t runs through the interval [0, 24). It is time for another example.
{ x = t
Example 11.10.1. Sketch the curve described by
2 − 3
for t ≥−2.
y = 2t − 1
Solution. We follow the same procedure here as we have time and time again when asked to graph
anything new – choose friendly values of t, plot the corresponding points and connect the results
in a pleasing fashion. Since we are told t ≥−2, we start there and as we plot successive points, we
draw an arrow to indicate the direction of the path for increasing values of t.
t x(t) y(t) (x(t),y(t))
−2 1 −5 (1, −5)
−1 −2 −3 (−2, −3)
0 −3 −1 (−3, −1)
1 −2 1 (−2, 1)
2 1 3 (1, 3)
3 6 5 (6, 5)
y
5
4
3
2
1
−2−1
−1
1 2 3 4 5 6
−2
−3
−5
x
The curve sketched out in Example 11.10.1 certainly looks like a parabola, and the presence of
the
{
t 2 term in the equation x = t 2 − 3 reinforces this hunch. Since the parametric equations
x = t 2 − 3, y=2t − 1 given to describe this curve are a system of equations, we can use the
technique of substitution as described in Section 8.7 to eliminate the parameter t and get an
equation involving just x and y. To do so, we choose to solve the equation y =2t − 1fort to
( ) 2
get t = y+1
2 . Substituting this into the equation x = t2 − 3 yields x = y+1
2 − 3 or, after some
rearrangement, (y +1) 2 =4(x + 3). Thinking back to Section 7.3, we see that the graph of this
equation is a parabola with vertex (−3, −1) which opens to the right, as required. Technically
speaking, the equation (y +1) 2 =4(x + 3) describes the entire parabola, while the parametric
equations { x = t 2 − 3, y=2t − 1fort ≥−2 describe only a portion of the parabola. In this case, 3
we can remedy this situation by restricting the bounds on y. Since the portion of the parabola we
want is exactly the part where y ≥−5, the equation (y +1) 2 =4(x+3) coupled with the restriction
y ≥−5 describes the same curve as the given parametric equations. The one piece of information
we can never recover after eliminating the parameter is the orientation of the curve.
Eliminating the parameter and obtaining an equation in terms of x and y, whenever possible,
can be a great help in graphing curves determined by parametric equations. If the system of
parametric equations contains algebraic functions, as was the case in Example 11.10.1, then the
usual techniques of substitution and elimination as learned in Section 8.7 canbeappliedtothe
3 We will have an example shortly where no matter how we restrict x and y, we can never accurately describe the
curve once we’ve eliminated the parameter.