Problem 22 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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 \(\frac{2\pi}{2}= \pi\).

Step 3: 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.

Step 4: 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.