Problem 43 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The amplitude of the function \(y=\cos \left(x-\frac{\pi}{2}\right)\) is 1, the period is \(2\pi\), and the phase shift is \(\pi / 2\) to the right. The function graph starts from \(\pi / 2\) and extends for a period of \(2\pi\) with maximum and minimum values of 1 and -1 respectively.

Step 1: Determine the Amplitude

The amplitude of a cosine function is the absolute value of the coefficient of the cosine term. Here, the coefficient of the cosine term is 1, so the amplitude of the function \(y = \cos (x - \frac{\pi}{2})\) is \(|1|\), or 1.

Step 2: Determine the Period

The period of a cosine function is given by \(T = 2\pi / |b|\), where b is the coefficient of x inside the cosine function. Here, the coefficient of x is 1, so the period of the function \(y = \cos (x - \frac{\pi}{2})\) is \(T = 2\pi / |1|\), or \(2\pi\). Hence our function repeats after an interval of \(2\pi\).

Step 3: Determine the Phase Shift

The phase shift of a cosine function is given by \(c / |b|\), where b is the coefficient of x, and c is the constant added or subtracted from x. Here, -\(\pi / 2\) is subtracted from x, so the phase shift of the function \(y = \cos (x - \frac{\pi}{2})\) is \(-(\pi / 2) / |1|\), or \(\pi/2\). This represents a shift of \(\pi / 2\) units to the right.

Step 4: Graph the Function

Plot the cosine function starting from the phase shift of \(\pi / 2\) and extending for one period of \(2\pi\). The graph will start at \(y = 1\) at \(x = \pi / 2\), reach its minimum at \(x = \pi + \pi / 2\), return to its maximum at \(x = 2\pi + \pi / 2\), and repeat this pattern every \(2\pi\) units.