Problem 25 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The amplitude of the function is 2. The period of the function is \(\pi\). The phase shift of the function is \(-\frac{\pi}{4}\).

Step 1: Identify the Amplitude

The amplitude of a sine function is given by the absolute value of the coefficient of the sine term. That is, in this case, the amplitude of the function \(y = -2 \sin(2x + \frac{\pi}{2})\) is \(|-2|\), which means the amplitude is 2.

Step 2: Determine the Period

The period of a sine function is found by dividing \(2\pi\) by the absolute value of the coefficient of \(x\), which in this case is 2. So the period of \(y = -2 \sin(2x + \frac{\pi}{2})\) is \(2\pi / 2\), thus the period is \(\pi\).

Step 3: Compute the Phase Shift

The phase shift of a sine function is found by focusing on the term within the parentheses. In this case, it's \(2x + \frac{\pi}{2} = 0\). Solving this for \(x\) will give us the phase shift, which is \(-\frac{\pi}{4}\).

Step 4: Graph the Function

Graphing the function will require you to incorporate all of these elements: the amplitude, which determines the height of the function's peaks and troughs; the period, which determines the width of each cycle of the function; and the phase shift, which determines where along the x-axis the function begins its cycle. However, graphing this accurately would be beyond the scope of text description and would best be done through using a graphing tool, a piece of graph paper or a digital platform like Desmos or GeoGebra.