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Chapter 5: Problem 24

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=\frac{1}{2} \sin (x+\pi)$$

Short Answer

Expert verified
The amplitude of the function \( y=\frac{1}{2} \sin (x+\pi) \) is \(\frac{1}{2}\), the period is \(2\pi\), and the phase shift is \(-\pi\). Please consider all these factors when sketching the one-period graph of the function on your graph paper.

Step by step solution

01

Determine the Amplitude

The amplitude is the absolute value of the coefficient of the sin function. In this case, it is \( |\frac{1}{2}| = \frac{1}{2} \).
02

Determine the Period

The period of a sine function in the form \( y=A \sin(Bx+C) \) is \(\frac{2\pi}{|B|}\). Here, our B is 1 (since x coefficient is 1), so our period would be \(\frac{2\pi}{1} = 2\pi\).
03

Determine the Phase Shift

The phase shift is equivalent to the value of C in the standard sine function with negative sign i.e \(-C\). Here, \(C=\pi\) so the phase shift is \(-\pi\).
04

Drawing the Graph

The standard approach to drawing the graph is to mark the amplitude, period and phase shift on the graph, and then draw a sine curve accordingly. Here, the curve begins at -蟺 (phase shift), goes up to amplitude of 0.5, down through -0.5 and repeats every 2蟺 (period). Please plot this on your graph paper accordingly.

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