Precalculus
Precalculus
9th Edition
ISBN: 9780321716835
Author: Michael Sullivan
Publisher: Addison Wesley
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Chapter 6.6, Problem 5AYU

In Problems 3-14, find the amplitude, period, and phase shift of each function. Graph each function. Be sure to label key points. Show at least two periods.

y  = 2cos ( 3 x   +   π 2 )

Expert Solution & Answer
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To determine

To find: The amplitude, period, phase shift of the function, graph the function.

Answer to Problem 5AYU

Solution:

The amplitude = 2 , period =  2π 3 , phase shift =  π 6 .

Explanation of Solution

Given:

y = 2cos( 3x +  π 2 )

Calculation:

Compare y = 2cos( 3x +  π 2 ) to y = Acos( ωx  ϕ ) , note that A = 2, ω = 3 and ϕ =  π 2 . The graph is a cosine curve with amplitude | A | = 2 , period T =  2π ω  =  2π 3 and phase shift =  ϕ ω  =  π 2 3  =  π 6 .

The graph of y = 2cos( 3x +  π 2 ) will lie between 2 and 2 on the y-axis .

One cycle will begin at x =  ϕ ω  =  π 6 and end at x =  ϕ ω  +  2π ω  =  π 6  +  2π 3  =  π 2 .

To find five key points, divide the interval [ π 6 ,  π 2 ] in to four sub intervals, each of length π 2  +  π 6 4  =  4π 6 4  =  π 6 .

Use the values of x to determine the five key points on the graph:

( π 6 , 2 ), ( 0, 0 ), ( π 6 , 2 ), ( π 3 , 0 ), ( π 2 , 2 )

Plot these five points and fill in the graph of the sine function.

Precalculus, Chapter 6.6, Problem 5AYU

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