Problem 50 - Algebra and Trigonometry

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 of the function \(y=-4 \cos \left(2 x-\frac{\pi}{2}\right)\) is 4, the period is \(\pi\), and the phase shift is \(\pi/4\) unit to the right.

Step 1: Identify Amplitude

In the given function, which is of the form \(y = A \cos(B(x - C)) + D\), \(A\) corresponds to the amplitude. The amplitude represented in this expression is the absolute value of \(A\). The amplitude is \(-4\) in this case, so we take the absolute value to get the amplitude as 4.

Step 2: Calculate Period

The term \(B\) in the function influences the period of the function. The usual period of the function \(y = \cos(x)\) is \(2\pi\); however, this period is divided by \(|B|\) to obtain the actual period of the function. In this case, \(B = 2\). Therefore, the period of the function \(y=-4 \cos \left(2 x-\frac{\pi}{2}\right)\) is \(2\pi/|2| = \pi\).

Step 3: Find Phase Shift

The term \(C\) in the function represents the phase shift. This corresponds to a shift of the function to the left or right. The phase shift is given by \(C/B\). However, if the value inside the cosine function is represented as a positive quantity, such as \(2x - \pi/2\) in this case, the phase shift is given by \(C/B\), yet it occurs in the opposite direction. Therefore, in this case, the phase shift is \(\pi/4\) units to the right.

Step 4: Graphing the function

Start by plotting the midline of the function, which is \(y = 0\) for this function. Then, plot a point at the phase shift of \(\pi/4\) units to the right. From this point, draw one complete wave of the function over a distance of the function period \(\pi\) in both directions. Because the amplitude is 4, the function will reach a maximum and minimum of 4 and -4, respectively, and the function will have a negative amplitude because the leading coefficient is negative.