Problem 50 - Precalculus Essentials

Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=-4 \cos \left(2 x-\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

The amplitude is 4, the period is \(\pi\), and the phase shift is \(\pi/4\) to the right.

Step 1: Determine the Amplitude

The amplitude of a function is the absolute value of the coefficient of the cosine function. In the given function, the coefficient is -4. Therefore, the amplitude is \(|-4| = 4\).

Step 2: Evaluate the Period

The period is evaluated using the formula \(2\pi/|B|\), where \(B\) is the coefficient of \(x\) in the cosine function. In this case, \(B\) is 2, so the period is \(2\pi / 2 = \pi\).

Step 3: Calculate the Phase Shift

The phase shift is given by \(h\) and calculated as \(-C/B\) where \(C\) is the value subtracted from \(x\) in the cosine function. Here, \(C\) is \(-\pi/2\) and \(B\) is 2. Therefore, the phase shift is \(-(-\pi/2)/2 = \pi/4\), to the right.

Step 4: Draw the Graph

To draw the graph, we start at the phase shift, then draw the cosine function with the determined amplitude and period. The function starts at the phase shift (\(\pi/4\)) and rises and falls between the amplitude, covering a full cycle in the period (\(\pi\)). Due to the negative sign, the graph will be reflected on the x-axis.