4th Edition

ISBN: 9781337516853

Author: Larson

Publisher: CENGAGE CO

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Question

Chapter 4.5, Problem 50E

To determine

To sketch : the graph of the function including two full periods

Expert Solution & Answer

Explanation of Solution

Given information: $y=−3sin(6x+π)$

Concept Involved:

Amplitude:

The constant factor $a$ in $y=asinx$ and $y=acosx$ as a scaling factor − a vertical stretch or vertical shrink of the basic curve. When $|a|>1,$ the basic curve is stretched, and when $0<|a|<1,$ the basic curve is shrunk. The result is that the graphs of $y=asinx$ and $y=acosx$ range between $–a$ and $a$ instead of between $−1$ and $1$ . The absolute value of $a$ is the amplitude of the function. The range of the function for $a>0$ is $−a≤y≤a$ .

The amplitude of $y=asinx$ and $y=acosx$ represents half the distance between the maximum and the minimum values of the function and is given by $Amplitude=|a|$ .

Period:

Let $b$ be a positive real number.

The period $y=asinbx$ and $y=acosbx$ is given by $Period=2π/b$

Note that when $0<b<1,$ the period of $y=asinbx$ is greater than $2π$ and represents a horizontal stretch of the basic curve. Similarly, when $b>1,$ the period of $y=asinbx$ is less than $2π$ and represents a horizontal shrink of the basic curve. These two statements are also true for $y=acosbx$ . When b is negative, rewrite the function using the identify $sin(−x)=−sinx or cos(−x)=cosx$ .

Horizontal Translation:

The constant c in the equation $y=asin(bx−c)$ and $y=acos(bx−c)$ results in horizontal translations (shifts) of the basics curves. For example, compare the graphs of $y=asinbx$ & $y=asin(bx−c)$ . The graph of $y=asin(bx−c)$ completes one cycle from $bx−c=0$ to $bx−c=2π$ . Solve for x to find that interval for one cycle is $c/b ︷Left endpoint≤x≤c/b + 2π/b ︸Period ︷Right endpoint$ . This implies that the period of $y=asin(bx−c)$ is $2π/b$ , and the graph of $y=asinbx$ is shifted by an amount $c/b$ . $Phase Shift=c/b$ .

Vertical Translation:

The constant $d$ in the equation $y=asin(bx−c)+d$ and $y=acos(bx−c)+d$ results in vertical translations of the basic curves. The shift is $d$ units up for $d>0$ and $d$ units down for $d<0.$ In other words, the graph oscillates about the horizontal line $y=d$ instead of about the x- axis.

Calculation:

Rewrite the given function $y=−3sin(6x−(−π))$ in the form $y=asin(bx−c)+d$

$y=−3sin(6x−(−π))+0$

Identify $a, b, c, & d$ of our given function by comparing it with $y=asin(bx−c)+d$

$a=−3$ ; $b=6$ ; $c=−π$ ; $d=0$

Identify Amplitude, Period, Phase shift {Horizontal Shift}, Vertical shift& mid-line

Amplitude is $\|a\|$	Period is $2π/b$	Phase shift is $c/b$	Vertical Shift is $d$	Mid-line of function
$\|−3\|=3$	$2π6=π3$	$cb=−π6$	$d=0$	$y=0$

Graph:

EBK PRECALCULUS W/LIMITS, Chapter 4.5, Problem 50E

Interpretation:

The graph of the function $y=−3sin(6x+π)$ has one complete cycle in the interval $(−π/6, π/6)$ .
It has a midline $y=0$ , has amplitude as 3, maximum height of the function 3 andminimum height of the function is -3.
Since a is negative, the function reflected over the x-axis.
The start point {Left end} of the graph would be $−π/6$ & the end point of one complete cycle is $π/6$

Chapter 4.5, Problem 49E

Chapter 4.5, Problem 51E

Chapter 4 Solutions

EBK PRECALCULUS W/LIMITS

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Prob. 47E Ch. 4.7 - Prob. 48E Ch. 4.7 - Prob. 49E Ch. 4.7 - Prob. 50E Ch. 4.7 - Prob. 51E Ch. 4.7 - Prob. 52E Ch. 4.7 - Prob. 53E Ch. 4.7 - Prob. 54E Ch. 4.7 - Prob. 55E Ch. 4.7 - Prob. 56E Ch. 4.7 - Prob. 57E Ch. 4.7 - Prob. 58E Ch. 4.7 - Prob. 59E Ch. 4.7 - Prob. 60E Ch. 4.7 - Prob. 61E Ch. 4.7 - Prob. 62E Ch. 4.7 - Prob. 63E Ch. 4.7 - Prob. 64E Ch. 4.7 - Prob. 65E Ch. 4.7 - Prob. 66E Ch. 4.7 - Prob. 67E Ch. 4.7 - Prob. 68E Ch. 4.7 - Prob. 69E Ch. 4.7 - Prob. 70E Ch. 4.7 - Prob. 71E Ch. 4.7 - Prob. 72E Ch. 4.7 - Prob. 73E Ch. 4.7 - Prob. 74E Ch. 4.7 - Prob. 75E Ch. 4.7 - Prob. 76E Ch. 4.7 - Prob. 77E Ch. 4.7 - Prob. 78E Ch. 4.7 - Prob. 79E Ch. 4.7 - Prob. 80E Ch. 4.7 - Prob. 81E Ch. 4.7 - Prob. 82E Ch. 4.7 - Prob. 83E Ch. 4.7 - Prob. 84E Ch. 4.7 - Prob. 85E Ch. 4.7 - Prob. 86E Ch. 4.7 - Prob. 87E Ch. 4.7 - Prob. 88E Ch. 4.7 - Prob. 89E Ch. 4.7 - Prob. 90E Ch. 4.7 - Prob. 91E Ch. 4.7 - Prob. 92E Ch. 4.7 - Prob. 93E Ch. 4.7 - Prob. 94E Ch. 4.7 - Prob. 95E Ch. 4.7 - Prob. 96E Ch. 4.7 - 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Prob. 16E Ch. 4.8 - Prob. 17E Ch. 4.8 - Prob. 18E Ch. 4.8 - Prob. 19E Ch. 4.8 - Prob. 20E Ch. 4.8 - Prob. 21E Ch. 4.8 - Prob. 22E Ch. 4.8 - Prob. 23E Ch. 4.8 - Prob. 24E Ch. 4.8 - Prob. 25E Ch. 4.8 - Prob. 26E Ch. 4.8 - Prob. 27E Ch. 4.8 - Prob. 28E Ch. 4.8 - Prob. 29E Ch. 4.8 - Prob. 30E Ch. 4.8 - Prob. 31E Ch. 4.8 - Prob. 32E Ch. 4.8 - Prob. 33E Ch. 4.8 - Prob. 34E Ch. 4.8 - Prob. 35E Ch. 4.8 - Prob. 36E Ch. 4.8 - Prob. 37E Ch. 4.8 - Prob. 38E Ch. 4.8 - Prob. 39E Ch. 4.8 - Prob. 40E Ch. 4.8 - Prob. 41E Ch. 4.8 - Prob. 42E Ch. 4.8 - Prob. 43E Ch. 4.8 - Prob. 44E Ch. 4.8 - Prob. 45E Ch. 4.8 - Prob. 46E Ch. 4.8 - Prob. 47E Ch. 4.8 - Prob. 48E Ch. 4.8 - Prob. 49E Ch. 4.8 - Prob. 50E Ch. 4.8 - Prob. 51E Ch. 4.8 - Prob. 52E Ch. 4.8 - Prob. 53E Ch. 4.8 - Prob. 54E Ch. 4.8 - Prob. 55E Ch. 4.8 - Prob. 56E Ch. 4.8 - Prob. 57E Ch. 4.8 - Prob. 58E Ch. 4.8 - Prob. 59E Ch. 4.8 - Prob. 60E Ch. 4 - Prob. 1RE Ch. 4 - Prob. 2RE Ch. 4 - Prob. 3RE Ch. 4 - Prob. 4RE Ch. 4 - Prob. 5RE Ch. 4 - Prob. 6RE Ch. 4 - Prob. 7RE Ch. 4 - Prob. 8RE Ch. 4 - Prob. 9RE Ch. 4 - Prob. 10RE Ch. 4 - Prob. 11RE Ch. 4 - Prob. 12RE Ch. 4 - Prob. 13RE Ch. 4 - Prob. 14RE Ch. 4 - Prob. 15RE Ch. 4 - Prob. 16RE Ch. 4 - Prob. 17RE Ch. 4 - Prob. 18RE Ch. 4 - Prob. 19RE Ch. 4 - Prob. 20RE Ch. 4 - Prob. 21RE Ch. 4 - Prob. 22RE Ch. 4 - Prob. 23RE Ch. 4 - Prob. 24RE Ch. 4 - Prob. 25RE Ch. 4 - Prob. 26RE Ch. 4 - Prob. 27RE Ch. 4 - Prob. 28RE Ch. 4 - Prob. 29RE Ch. 4 - Prob. 30RE Ch. 4 - Prob. 31RE Ch. 4 - Prob. 32RE Ch. 4 - Prob. 33RE Ch. 4 - Prob. 34RE Ch. 4 - Prob. 35RE Ch. 4 - Prob. 36RE Ch. 4 - Prob. 37RE Ch. 4 - Prob. 38RE Ch. 4 - Prob. 39RE Ch. 4 - Prob. 40RE Ch. 4 - Prob. 41RE Ch. 4 - Prob. 42RE Ch. 4 - Prob. 43RE Ch. 4 - Prob. 44RE Ch. 4 - Prob. 45RE Ch. 4 - Prob. 46RE Ch. 4 - Prob. 47RE Ch. 4 - Prob. 48RE Ch. 4 - Prob. 49RE Ch. 4 - Prob. 50RE Ch. 4 - Prob. 51RE Ch. 4 - Prob. 52RE Ch. 4 - Prob. 53RE Ch. 4 - Prob. 54RE Ch. 4 - Prob. 55RE Ch. 4 - Prob. 56RE Ch. 4 - Prob. 57RE Ch. 4 - Prob. 58RE Ch. 4 - Prob. 59RE Ch. 4 - Prob. 60RE Ch. 4 - Prob. 61RE Ch. 4 - Prob. 62RE Ch. 4 - Prob. 63RE Ch. 4 - Prob. 64RE Ch. 4 - Prob. 65RE Ch. 4 - Prob. 66RE Ch. 4 - Prob. 67RE Ch. 4 - Prob. 68RE Ch. 4 - Prob. 69RE Ch. 4 - Prob. 70RE Ch. 4 - Prob. 71RE Ch. 4 - Prob. 72RE Ch. 4 - Prob. 73RE Ch. 4 - Prob. 74RE Ch. 4 - Prob. 75RE Ch. 4 - Prob. 76RE Ch. 4 - Prob. 77RE Ch. 4 - Prob. 78RE Ch. 4 - Prob. 79RE Ch. 4 - Prob. 80RE Ch. 4 - Prob. 81RE Ch. 4 - Prob. 82RE Ch. 4 - Prob. 83RE Ch. 4 - Prob. 84RE Ch. 4 - Prob. 85RE Ch. 4 - Prob. 86RE Ch. 4 - Prob. 87RE Ch. 4 - Prob. 88RE Ch. 4 - Prob. 89RE Ch. 4 - Prob. 90RE Ch. 4 - Prob. 91RE Ch. 4 - Prob. 92RE Ch. 4 - Prob. 93RE Ch. 4 - Prob. 94RE Ch. 4 - Prob. 95RE Ch. 4 - Prob. 96RE Ch. 4 - Prob. 97RE Ch. 4 - Prob. 98RE Ch. 4 - Prob. 99RE Ch. 4 - Prob. 100RE Ch. 4 - Prob. 101RE Ch. 4 - Prob. 102RE Ch. 4 - Prob. 1CT Ch. 4 - Prob. 2CT Ch. 4 - Prob. 3CT Ch. 4 - Prob. 4CT Ch. 4 - Prob. 5CT Ch. 4 - Prob. 6CT Ch. 4 - Prob. 7CT Ch. 4 - Prob. 8CT Ch. 4 - Prob. 9CT Ch. 4 - Prob. 10CT Ch. 4 - Prob. 11CT Ch. 4 - Prob. 12CT Ch. 4 - Prob. 13CT Ch. 4 - Prob. 14CT Ch. 4 - Prob. 15CT Ch. 4 - Prob. 16CT Ch. 4 - Prob. 17CT Ch. 4 - Prob. 18CT Ch. 4 - Prob. 19CT Ch. 4 - Prob. 20CT Ch. 4 - Prob. 1PS Ch. 4 - Prob. 2PS Ch. 4 - Prob. 3PS Ch. 4 - Prob. 4PS Ch. 4 - Prob. 5PS Ch. 4 - Prob. 6PS Ch. 4 - Prob. 7PS Ch. 4 - Prob. 8PS Ch. 4 - Prob. 9PS Ch. 4 - Prob. 10PS Ch. 4 - Prob. 11PS Ch. 4 - Prob. 12PS Ch. 4 - Prob. 13PS Ch. 4 - Prob. 14PS

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