Problem 44 - 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

Amplitude of the function is 1, the period is \(2\pi\) and the phase shift is \(-\pi/2\). The function graph will look like a regular cosine wave shifted to the left by \(\pi / 2\).

Step 1: Find the Amplitude

The amplitude is the absolute value of A. In this case, the \(A=1\). Therefore, the amplitude of the function is 1.

Step 2: Calculate the Period

The period is calculated as \(2\pi / B\). Here, B is 1. So, the period of the function is \(2\pi / 1 = 2\pi\).

Step 3: Determine the Phase Shift

The phase shift is calculated as \(-C / B\). Here, \(C = \pi / 2\) and \(B = 1\). Therefore, the phase shift is \(-\pi / 2\). It's negative, indicating a shift to the left.

Step 4: Draw the Graph

To sketch the graph of the function, start by drawing the x-axis. Place a point at the phase shift location. Then, mark the period along the x-axis from the phase shift. Draw a cosine wave that starts at a maximum, with its maximum height being the amplitude, and one wave cycle completing at the period. The function will look like a standard cosine wave shifted left by \(\pi / 2\).