7.6 Modeling with trigonometric equations (Page 6/14)

Precalculus

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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.

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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.

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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.

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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.

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

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Algebraic

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


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

$y = - 3 \cos (\frac{π}{6} x) - 1$

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

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

$5 \sin (2 x) + 2$

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

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

$4 \cos (\frac{x π}{2}) - 3$

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

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Questions & Answers

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?

ADNAN Reply

linear speed of an object

Melissa Reply

an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object

Melissa

test

Matrix

how to find domain

Mohamed Reply

like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.

Dan

define the term of domain

Moha

if a>0 then the graph is concave

Angel Reply

if a<0 then the graph is concave blank

Angel

what's a domain

Kamogelo Reply

The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.

Spiro

how fast can i understand functions without much difficulty

Joe Reply

what is inequalities

Nathaniel

functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2

Dan

what is set?

Kelvin Reply

a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size

Divya Reply

I got 300 minutes. is it right?

Patience

no. should be about 150 minutes.

Jason

It should be 158.5 minutes.

ok, thanks

Patience

100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes

Thomas

158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49

Dan

what is the importance knowing the graph of circular functions?

Arabella Reply

can get some help basic precalculus

ismail Reply

What do you need help with?

Andrew

how to convert general to standard form with not perfect trinomial

Camalia Reply

can get some help inverse function

ismail

Rectangle coordinate

Asma Reply

how to find for x

Jhon Reply

it depends on the equation

Robert

yeah, it does. why do we attempt to gain all of them one side or the other?

Melissa

how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2

Dan

whats a domain

mike Reply

The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.

Spiro

Spiro; thanks for putting it out there like that, 😁

Melissa

foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.

Churlene Reply

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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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