7.6 Modeling with trigonometric equations (Page 6/14)

Precalculus

Page 6 / 14

Bounding curves in harmonic motion

Harmonic motion graphs may be enclosed by bounding curves. When a function has a varying amplitude , such that the amplitude rises and falls multiple times within a period, we can determine the bounding curves from part of the function.

Graphing an oscillating cosine curve

Graph the function $f (x) = \cos (2 π x) \cos (16 π x) .$

The graph produced by this function will be shown in two parts. The first graph will be the exact function $f (x)$ (see [link] ), and the second graph is the exact function $f (x)$ plus a bounding function (see [link] . The graphs look quite different.

Graph of f(x) = cos(2pi*x)cos(16pi*x), a sinusoidal function that increases and decreases its amplitude periodically.

Graph of f(x) = cos(2pi*x)cos(16pi*x), a sinusoidal function that increases and decreases its amplitude periodically. There is also a bonding function drawn over it in red, which makes the whole image look like a DNA (double helix) piece stretched along the x-axis.

Got questions? Get instant answers now!

Media

Access these online resources for additional instruction and practice with trigonometric applications.

Visit this website for additional practice questions from Learningpod.

Key equations


Standard form of sinusoidal equation	$y = A \sin (B t - C) + D or y = A \cos (B t - C) + D$
Simple harmonic motion	$d = a \cos (ω t) or d = a \sin (ω t)$
Damped harmonic motion	$f (t) = a e^{- c}^{t} \sin (ω t) or f (t) = a e^{- c t} \cos (ω t)$

Key concepts

Sinusoidal functions are represented by the sine and cosine graphs. In standard form, we can find the amplitude, period, and horizontal and vertical shifts. See [link] and [link] .
Use key points to graph a sinusoidal function. The five key points include the minimum and maximum values and the midline values. See [link] .
Periodic functions can model events that reoccur in set cycles, like the phases of the moon, the hands on a clock, and the seasons in a year. See [link] , [link] , [link] and [link] .
Harmonic motion functions are modeled from given data. Similar to periodic motion applications, harmonic motion requires a restoring force. Examples include gravitational force and spring motion activated by weight. See [link] .
Damped harmonic motion is a form of periodic behavior affected by a damping factor. Energy dissipating factors, like friction, cause the displacement of the object to shrink. See [link] , [link] , [link] , [link] , and [link] .
Bounding curves delineate the graph of harmonic motion with variable maximum and minimum values. See [link] .

Section exercises

Verbal

Explain what types of physical phenomena are best modeled by sinusoidal functions. What are the characteristics necessary?

Physical behavior should be periodic, or cyclical.

Got questions? Get instant answers now!

What information is necessary to construct a trigonometric model of daily temperature? Give examples of two different sets of information that would enable modeling with an equation.

Got questions? Get instant answers now!

If we want to model cumulative rainfall over the course of a year, would a sinusoidal function be a good model? Why or why not?

Since cumulative rainfall is always increasing, a sinusoidal function would not be ideal here.

Got questions? Get instant answers now!

Explain the effect of a damping factor on the graphs of harmonic motion functions.

Got questions? Get instant answers now!

Algebraic

For the following exercises, find a possible formula for the trigonometric function represented by the given table of values.


$x$	$y$
$0$	$- 4$
$3$	$- 1$
$6$	$2$
$9$	$- 1$
$12$	$- 4$
$15$	$- 1$
$18$	$2$

$y = - 3 \cos (\frac{π}{6} x) - 1$

Got questions? Get instant answers now!


$x$	$y$
$0$	$5$
$2$	$1$
$4$	$- 3$
$6$	$1$
$8$	$5$
$10$	$1$
$12$	$- 3$

Got questions? Get instant answers now!


$x$	$y$
$0$	$2$
$\frac{π}{4}$	$7$
$\frac{π}{2}$	$2$
$\frac{3 π}{4}$	$- 3$
$π$	$2$
$\frac{5 π}{4}$	$7$
$\frac{3 π}{2}$	$2$

$5 \sin (2 x) + 2$

Got questions? Get instant answers now!


$x$	$y$
$0$	$2$
$\frac{π}{4}$	$7$
$\frac{π}{2}$	$2$
$\frac{3 π}{4}$	$- 3$
$π$	$2$
$\frac{5 π}{4}$	$7$
$\frac{3 π}{2}$	$2$

Got questions? Get instant answers now!


$x$	$y$
$0$	$1$
$1$	$- 3$
$2$	$- 7$
$3$	$- 3$
$4$	$1$
$5$	$- 3$
$6$	$- 7$

$4 \cos (\frac{x π}{2}) - 3$

Got questions? Get instant answers now!


$x$	$y$
$0$	$- 2$
$1$	$4$
$2$	$10$
$3$	$4$
$4$	$- 2$
$5$	$4$
$6$	$10$

Got questions? Get instant answers now!

Questions & Answers

how do you get the 2/50

Abba Reply

number of sport play by 50 student construct discrete data

Aminu Reply

width of the frangebany leaves on how to write a introduction

Theresa Reply

Solve the mean of variance

Veronica Reply

Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ... Step 2: Find each score's deviation from the mean. ... Step 3: Square each deviation from the mean. ... Step 4: Find the sum of squares. ... Step 5: Divide the sum of squares by n – 1 or N.

kenneth

what is error

Yakuba Reply

Is mistake done to something

Vutshila

anas

What is the life teble

anas

Jibrin

statistics is the analyzing of data

Tajudeen Reply

what is statics?

Zelalem Reply

how do you calculate mean

Gloria Reply

diveving the sum if all values

Shaynaynay

let A1,A2 and A3 events be independent,show that (A1)^c, (A2)^c and (A3)^c are independent?

Fisaye Reply

what is statistics

Akhisani Reply

data collected all over the world

Shaynaynay

construct a less than and more than table

Imad Reply

The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.

Aschalew Reply

Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400 a. what is the probability of getting more than 12,000 hits? b. what is the probability of getting fewer than 9,000 hits?

Akshay Reply

Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400. a. What is the probability of getting more than 12,000 hits

Akshay

Bright

Sorry i want to learn more about this question

Bright

Someone help

Bright

a= 0.20233 b=0.3384

Sufiyan

Shaynaynay

How do I interpret level of significance?

Mohd Reply

It depends on your business problem or in Machine Learning you could use ROC- AUC cruve to decide the threshold value

Shivam

how skewness and kurtosis are used in statistics

Owen Reply

yes what is it

Taneeya

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

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 2

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

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

Notification Switch

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

Ask

	9 Key concepts By OpenStax Read Online Course
	7 Bounding curves in harmonic motion By OpenStax Read Online Course
	Financial Intelligence Quiz By Yasser Ibrahim Start Quiz
	Negotiations Conflict Management BUS403 By Charles Jumper Start Quiz
	Anthropology Language Culture By Richley Crapo Start Assignment
	Introductory statistics By OpenStax Read Online Course
	2 Physiotherapy Flashcards Set 2 By Rhodes Start Flashcards
	11 Sociology 11 Race and Ethnicity MCQ By OpenStax Start Quiz
©flickr: Elliott	Spanish Verbs Subject Pronouns By Mariah Hauptman Start Quiz
	28 AP 28 Development Inheritance MCQ By OpenStax Start Quiz
	5 AP Key Terms 05 The Integumentary System By OpenStax Start Key Terms
	Fireside Poets Quiz By Alec Moffit Start Quiz