Example 2 from Algebra and Trigonometry, 10.6 Parametric Equations

Finding a Pair of Parametric Equations

Find a pair of parametric equations that models the graph of $y = 1 - x^{2},$ using the parameter $x (t) = t .$ Plot some points and sketch the graph.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution

If $x (t) = t$ and we substitute $t$ for $x$ into the $y$ equation, then $y (t) = 1 - t^{2} .$ Our pair of parametric equations is

\begin{array}{l} x (t) = t \\ y (t) = 1 - t^{2} \end{array}

To graph the equations, first we construct a table of values like that in Table 2. We can choose values around $t = 0,$ from $t = - 3$ to $t = 3.$ The values in the $x (t)$ column will be the same as those in the $t$ column because $x (t) = t .$ Calculate values for the column $y (t) .$

$t$	$x (t) = t$	$y (t) = 1 - t^{2}$
$- 3$	$- 3$	$y (- 3) = 1 - {(- 3)}^{2} = - 8$
$- 2$	$- 2$	$y (- 2) = 1 - {(- 2)}^{2} = - 3$
$- 1$	$- 1$	$y (- 1) = 1 - {(- 1)}^{2} = 0$
$0$	$0$	$y (0) = 1 - 0 = 1$
$1$	$1$	$y (1) = 1 - {(1)}^{2} = 0$
$2$	$2$	$y (2) = 1 - {(2)}^{2} = - 3$
$3$	$3$	$y (3) = 1 - {(3)}^{2} = - 8$

Table 2

The graph of $y = 1 - t^{2}$ is a parabola facing downward, as shown in Figure 4. We have mapped the curve over the interval $[−3, 3],$ shown as a solid line with arrows indicating the orientation of the curve according to $t .$ Orientation refers to the path traced along the curve in terms of increasing values of $t .$ As this parabola is symmetric with respect to the line $x = 0,$ the values of $x$ are reflected across the y-axis.

Graph of given downward facing parabola.

Figure 4

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Parameterize the curve given by $x = y^{3} - 2 y .$

How did it go?