8.6 Parametric equations (Page 2/6)

Precalculus

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Graph of a circle in the rectangular coordinate system - the vertical line test shows that the circle r^2 = x^2 + y^2 is not a function. The dotted red vertical line intersects the function in two places - it should only intersect in one place to be a function.

However, if we were to graph each equation on its own, each one would pass the vertical line test and therefore would represent a function. In some instances, the concept of breaking up the equation for a circle into two functions is similar to the concept of creating parametric equations, as we use two functions to produce a non-function. This will become clearer as we move forward.

A General Note

Parametric equations

Suppose $t$ is a number on an interval, $I .$ The set of ordered pairs, $(x (t), y (t)),$ where $x = f (t)$ and $y = g (t),$ forms a plane curve based on the parameter $t .$ The equations $x = f (t)$ and $y = g (t)$ are the parametric equations.

Parameterizing a curve

Parameterize the curve $y = x^{2} - 1$ letting $x (t) = t .$ Graph both equations.

If $x (t) = t,$ then to find $y (t)$ we replace the variable $x$ with the expression given in $x (t) .$ In other words, $y (t) = t^{2} - 1.$ Make a table of values similar to [link] , and sketch the graph.

$t$	$x (t)$	$y (t)$
$- 4$	$- 4$	$y (- 4) = {(- 4)}^{2} - 1 = 15$
$- 3$	$- 3$	$y (- 3) = {(- 3)}^{2} - 1 = 8$
$- 2$	$- 2$	$y (- 2) = {(- 2)}^{2} - 1 = 3$
$- 1$	$- 1$	$y (- 1) = {(- 1)}^{2} - 1 = 0$
$0$	$0$	$y (0) = {(0)}^{2} - 1 = - 1$
$1$	$1$	$y (1) = {(1)}^{2} - 1 = 0$
$2$	$2$	$y (2) = {(2)}^{2} - 1 = 3$
$3$	$3$	$y (3) = {(3)}^{2} - 1 = 8$
$4$	$4$	$y (4) = {(4)}^{2} - 1 = 15$

See the graphs in [link] . It may be helpful to use the TRACE feature of a graphing calculator to see how the points are generated as $t$ increases.

Graph of a parabola in two forms: a parametric equation and rectangular coordinates. It is the same function, just different ways of writing it. — (a) Parametric $y (t) = t^{2} - 1$ (b) Rectangular $y = x^{2} - 1$

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Construct a table of values and plot the parametric equations: $x (t) = t - 3, y (t) = 2 t + 4; - 1 \leq t \leq 2.$


$t$	$x (t)$	$y (t)$
$- 1$	$- 4$	$2$
$0$	$- 3$	$4$
$1$	$- 2$	$6$
$2$	$- 1$	$8$

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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.

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 [link] . 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 $y (t) .$

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

The graph of $y = 1 - t^{2}$ is a parabola facing downward, as shown in [link] . 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.

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

$\begin{array}{l} x (t) = t^{3} - 2 t \\ y (t) = t \end{array}$

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Finding parametric equations that model given criteria

An object travels at a steady rate along a straight path $(−5, 3)$ to $(3, −1)$ in the same plane in four seconds. The coordinates are measured in meters. Find parametric equations for the position of the object.

The parametric equations are simple linear expressions, but we need to view this problem in a step-by-step fashion. The x -value of the object starts at $−5$ meters and goes to 3 meters. This means the distance x has changed by 8 meters in 4 seconds, which is a rate of $\frac{8 m}{4 s},$ or $2 m / s .$ We can write the x -coordinate as a linear function with respect to time as $x (t) = 2 t - 5.$ In the linear function template $y = m x + b, 2 t = m x$ and $- 5 = b .$

Similarly, the y -value of the object starts at 3 and goes to $−1,$ which is a change in the distance y of −4 meters in 4 seconds, which is a rate of $\frac{- 4 m}{4 s},$ or $- 1 m / s .$ We can also write the y -coordinate as the linear function $y (t) = - t + 3.$ Together, these are the parametric equations for the position of the object, where $x$ and $y$ are expressed in meters and $t$ represents time:

\begin{array}{l} x (t) = 2 t - 5 \\ y (t) = - t + 3 \end{array}

Using these equations, we can build a table of values for $t, x,$ and $y$ (see [link] ). In this example, we limited values of $t$ to non-negative numbers. In general, any value of $t$ can be used.

$t$	$x (t) = 2 t - 5$	$y (t) = - t + 3$
$0$	$x = 2 (0) - 5 = - 5$	$y = - (0) + 3 = 3$
$1$	$x = 2 (1) - 5 = - 3$	$y = - (1) + 3 = 2$
$2$	$x = 2 (2) - 5 = - 1$	$y = - (2) + 3 = 1$
$3$	$x = 2 (3) - 5 = 1$	$y = - (3) + 3 = 0$
$4$	$x = 2 (4) - 5 = 3$	$y = - (4) + 3 = - 1$

From this table, we can create three graphs, as shown in [link] .

Three graphs side by side. (A) has the horizontal position over time, (B) has the vertical position over time, and (C) has the position of the object in the plane at time t. See caption for more information. — (a) A graph of $x$ vs. $t,$ representing the horizontal position over time. (b) A graph of $y$ vs. $t,$ representing the vertical position over time. (c) A graph of $y$ vs. $x,$ representing the position of the object in the plane at time $t .$

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Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

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