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Chapter 5: Problem 22

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=3 \sin \left(2 x-\frac{\pi}{2}\right)$$

Short Answer

Expert verified
The amplitude of the function \(y=3 \sin \left(2 x-\frac{\pi}{2}\right)\) is 3, the period is \(\pi\), and the phase shift is \(\frac{\pi}{4}\) units to the right.

Step by step solution

01

Identify the amplitude

The amplitude is the absolute value of the coefficient of the sine function. In this case, the amplitude of the function \(y=3 \sin \left(2 x-\frac{\pi}{2}\right)\) is 3.
02

Identify the period

The period of the function is derived from the coefficient of x inside the sine function, B. The period is found by using the formula \(Period = \frac{2\pi}{B}\). For the function \(y=3 \sin \left(2 x-\frac{\pi}{2}\right)\), B is 2, thus the period is \(\frac{2\pi}{2}= \pi\).
03

Identify the phase shift

The phase shift of the function is calculated from the term inside the sine function subtracted or added. To calculate the phase shift, we solve \(Bx - C = 0\). In this case, for the function \(y=3 \sin \left(2 x-\frac{\pi}{2}\right)\), when solving \(2x - \frac{\pi}{2} = 0\), x equals \(\frac{\pi}{4}\). Thus, the phase shift is \(\frac{\pi}{4}\) units to the right.
04

Graph the function

Steps 1 to 3 have described the parameters needed to graph a period of the function. To construct the graph, mark the amplitude on the y-axis, mark the period on the x-axis, and shift the graph \(\frac{\pi}{4}\) units to the right. Start with a basic sine graph, stretch it vertically by a factor of 3 (the amplitude), compress it horizontally by a factor of 2 (the period) and shift it to the right by \(\frac{\pi}{4}\) (phase shift). For a more detailed graph, it may be necessary to plot several key points based on those parameters and connect them smoothly.

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