10.7 Parametric equations: graphs

Algebra and trigonometry Page 1 / 4

In this section you will:

Graph plane curves described by parametric equations by plotting points.
Graph parametric equations.

It is the bottom of the ninth inning, with two outs and two men on base. The home team is losing by two runs. The batter swings and hits the baseball at 140 feet per second and at an angle of approximately $45°$ to the horizontal. How far will the ball travel? Will it clear the fence for a game-winning home run? The outcome may depend partly on other factors (for example, the wind), but mathematicians can model the path of a projectile and predict approximately how far it will travel using parametric equations . In this section, we’ll discuss parametric equations and some common applications, such as projectile motion problems.

Photo of a baseball batter swinging. — Parametric equations can model the path of a projectile. (credit: Paul Kreher, Flickr)

Graphing parametric equations by plotting points

In lieu of a graphing calculator or a computer graphing program, plotting points to represent the graph of an equation is the standard method. As long as we are careful in calculating the values, point-plotting is highly dependable.

How To

Given a pair of parametric equations, sketch a graph by plotting points.

Construct a table with three columns: $t, x (t), and y (t) .$
Evaluate $x$ and $y$ for values of $t$ over the interval for which the functions are defined.
Plot the resulting pairs $(x, y) .$

Sketching the graph of a pair of parametric equations by plotting points

Sketch the graph of the parametric equations $x (t) = t^{2} + 1, y (t) = 2 + t .$

Construct a table of values for $t, x (t),$ and $y (t),$ as in [link] , and plot the points in a plane.

$t$	$x (t) = t^{2} + 1$	$y (t) = 2 + t$
$- 5$	$26$	$- 3$
$- 4$	$17$	$- 2$
$- 3$	$10$	$- 1$
$- 2$	$5$	$0$
$- 1$	$2$	$1$
$0$	$1$	$2$
$1$	$2$	$3$
$2$	$5$	$4$
$3$	$10$	$5$
$4$	$17$	$6$
$5$	$26$	$7$

The graph is a parabola with vertex at the point $(1, 2),$ opening to the right. See [link] .

Graph of the given parabola opening to the right.

Got questions? Get instant answers now!

Try It

Sketch the graph of the parametric equations $x = \sqrt{t}, y = 2 t + 3, 0 \leq t \leq 3.$

Graph of the given parametric equations with the restricted domain - it looks like the right half of an upward opening parabola.

Got questions? Get instant answers now!

Sketching the graph of trigonometric parametric equations

Construct a table of values for the given parametric equations and sketch the graph:

\begin{array}{l} \begin{array}{l} x = 2 \cos t \\ y = 4 \sin t \end{array} \end{array}

Construct a table like that in [link] using angle measure in radians as inputs for $t,$ and evaluating $x$ and $y .$ Using angles with known sine and cosine values for $t$ makes calculations easier.

$t$	$x = 2 \cos t$	$y = 4 \sin t$
0	$x = 2 \cos (0) = 2$	$y = 4 \sin (0) = 0$
$\frac{π}{6}$	$x = 2 \cos (\frac{π}{6}) = \sqrt{3}$	$y = 4 \sin (\frac{π}{6}) = 2$
$\frac{π}{3}$	$x = 2 \cos (\frac{π}{3}) = 1$	$y = 4 \sin (\frac{π}{3}) = 2 \sqrt{3}$
$\frac{π}{2}$	$x = 2 \cos (\frac{π}{2}) = 0$	$y = 4 \sin (\frac{π}{2}) = 4$
$\frac{2 π}{3}$	$x = 2 \cos (\frac{2 π}{3}) = - 1$	$y = 4 \sin (\frac{2 π}{3}) = 2 \sqrt{3}$
$\frac{5 π}{6}$	$x = 2 \cos (\frac{5 π}{6}) = - \sqrt{3}$	$y = 4 \sin (\frac{5 π}{6}) = 2$
$π$	$x = 2 \cos (π) = - 2$	$y = 4 \sin (π) = 0$
$\frac{7 π}{6}$	$x = 2 \cos (\frac{7 π}{6}) = - \sqrt{3}$	$y = 4 \sin (\frac{7 π}{6}) = - 2$
$\frac{4 π}{3}$	$x = 2 \cos (\frac{4 π}{3}) = - 1$	$y = 4 \sin (\frac{4 π}{3}) = - 2 \sqrt{3}$
$\frac{3 π}{2}$	$x = 2 \cos (\frac{3 π}{2}) = 0$	$y = 4 \sin (\frac{3 π}{2}) = - 4$
$\frac{5 π}{3}$	$x = 2 \cos (\frac{5 π}{3}) = 1$	$y = 4 \sin (\frac{5 π}{3}) = - 2 \sqrt{3}$
$\frac{11 π}{6}$	$x = 2 \cos (\frac{11 π}{6}) = \sqrt{3}$	$y = 4 \sin (\frac{11 π}{6}) = - 2$
$2 π$	$x = 2 \cos (2 π) = 2$	$y = 4 \sin (2 π) = 0$

[link] shows the graph.

Graph of the given equations - a vertical ellipse.

By the symmetry shown in the values of $x$ and $y,$ we see that the parametric equations represent an ellipse . The ellipse is mapped in a counterclockwise direction as shown by the arrows indicating increasing $t$ values.

Got questions? Get instant answers now!

Try It

Graph the parametric equations: $x = 5 \cos t, y = 3 \sin t .$

Graph of the given equations - a horizontal ellipse.

Got questions? Get instant answers now!

Graphing parametric equations and rectangular form together

Graph the parametric equations $x = 5 \cos t$ and $y = 2 \sin t .$ First, construct the graph using data points generated from the parametric form . Then graph the rectangular form of the equation. Compare the two graphs.

Construct a table of values like that in [link] .

$t$	$x = 5 \cos t$	$y = 2 \sin t$
$0$	$x = 5 \cos (0) = 5$	$y = 2 \sin (0) = 0$
$1$	$x = 5 \cos (1) \approx 2.7$	$y = 2 \sin (1) \approx 1.7$
$2$	$x = 5 \cos (2) \approx −2.1$	$y = 2 \sin (2) \approx 1.8$
$3$	$x = 5 \cos (3) \approx −4.95$	$y = 2 \sin (3) \approx 0.28$
$4$	$x = 5 \cos (4) \approx −3.3$	$y = 2 \sin (4) \approx −1.5$
$5$	$x = 5 \cos (5) \approx 1.4$	$y = 2 \sin (5) \approx −1.9$
$−1$	$x = 5 \cos (−1) \approx 2.7$	$y = 2 \sin (−1) \approx −1.7$
$−2$	$x = 5 \cos (−2) \approx −2.1$	$y = 2 \sin (−2) \approx −1.8$
$−3$	$x = 5 \cos (−3) \approx −4.95$	$y = 2 \sin (−3) \approx −0.28$
$−4$	$x = 5 \cos (−4) \approx −3.3$	$y = 2 \sin (−4) \approx 1.5$
$−5$	$x = 5 \cos (−5) \approx 1.4$	$y = 2 \sin (−5) \approx 1.9$

Plot the $(x, y)$ values from the table. See [link] .

Graph of the given ellipse in parametric and rectangular coordinates - it is the same thing in both images.

Next, translate the parametric equations to rectangular form. To do this, we solve for $t$ in either $x (t)$ or $y (t),$ and then substitute the expression for $t$ in the other equation. The result will be a function $y (x)$ if solving for $t$ as a function of $x,$ or $x (y)$ if solving for $t$ as a function of $y .$

\begin{array}{l} x = 5 \cos t \\ \frac{x}{5} = \cos t & Solve for \cos t . \\ y = 2 \sin t \begin{matrix}  \end{matrix} & Solve for \sin t . \\ \frac{y}{2} = \sin t \end{array}

Then, use the Pythagorean Theorem .

\begin{array}{r} \cos^{2} t + \sin^{2} t = 1 \\ {(\frac{x}{5})}^{2} + {(\frac{y}{2})}^{2} = 1 \\ \frac{x^{2}}{25} + \frac{y^{2}}{4} = 1 \end{array}

Got questions? Get instant answers now!

Questions & Answers

summarize halerambos & holbon

David Reply

the Three stages of Auguste Comte

Clementina Reply

what are agents of socialization

Antonio Reply

sociology of education

Nuhu Reply

definition of sociology of education

Nuhu

what is culture

Abdulrahim Reply

shared beliefs, values, and practices

AI-Robot

What are the two type of scientific method

ogunniran Reply

I'm willing to join you

Aceng Reply

what are the scientific method of sociology

Man

what is socialization

ogunniran Reply

the process wherein people come to understand societal norms and expectations, to accept society's beliefs, and to be aware of societal values

AI-Robot

scientific method in doing research

ogunniran

defimition of sickness in afica

Anita

Cosmology

ogunniran

Hmmm

ogunniran

list and explain the terms that found in society

REMMY Reply

list and explain the terms that found in society

Mukhtar

what are the agents of socialization

Antonio

Family Peer group Institution

Abdulwajud

I mean the definition

Antonio

ways of perceived deviance indifferent society

Naomi Reply

reasons of joining groups

SAM

to bring development to the nation at large

Hyellafiya

entails of consultative and consensus building from others

Gadama

World first Sociologist?

Abu

What is evolutionary model

Muhammad Reply

Evolution models refer to mathematical and computational representations of the processes involved in biological evolution. These models aim to simulate and understand how species change over time through mechanisms such as natural selection, genetic drift, and mutation. Evolutionary models can be u

faruk

what are the modern trends in religious behaviours

Selekeye Reply

what are social norms

Daniel Reply

shared standards of acceptable behavior by the group or appropriate behavior in a particular institution or those behaviors that are acceptable in a society

Lucius

that is how i understood it

Lucius

examples of societal norms

Diamond

Discuss the characteristics of the research located within positivist and the interpretivist paradigm

Tariro Reply

what is Industrialisation

Selekeye Reply

industrialization

Angelo

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask

	2 Parametric equations: graphs By OpenStax Read Online Course
	Chemistry Practice Test By Sandhills MLT Start Test
	19 Biology 19 The Evolution of Populations MCQ By OpenStax Start Quiz
	6 AP 06 Skeletal System Essay By OpenStax Start Flashcards
	2 Timeshare 2 By Jams Kalo Start Quiz
	PE Power Enigeering Safety By Gerr Zen Start Quiz
	44 Biology 44 Ecology and the Biosphere MCQ By OpenStax Start Quiz
	18 Biology 18 Evolution the Origin of Species MCQ By OpenStax Start Quiz
	3 Business Law MCQ 3 By Maureen Miller Start Exam
	Anthropology Politics Culture By Richley Crapo Start Assignment
	4 Biology 04 Cell Structure MCQ By OpenStax Start Quiz