Problem 42 - Precalculus Essentials

Question

Determine the amplitude and period of each function. Then graph one period of the function. $$y=-\frac{1}{2} \cos \frac{\pi}{4} x$$

Step-by-Step Solution

Answer

The amplitude of the function is \( \frac{1}{2} \) and the period is 8.

Step 1: Determining the Amplitude

The amplitude of function \(y=-\frac{1}{2} \cos \frac{\pi}{4} x\) is obtained from vertical stretching or compressing factor. In this case it is \(-\frac{1}{2}\). The amplitude is always positive, hence the amplitude is \(\frac{1}{2}\).

Step 2: Determining the Period

The period of a cosine function is \( \frac{2\pi}{|B|} \) where B is the horizontal stretch or compression factor. In our function, it is \(\frac{\pi}{4}\). So, the period is \( \frac{2\pi}{|\frac{\pi}{4}|} = 8 \).

Step 3: Graphing the function

When graphing, it's key to remember that the period will determine the width of the wave, and the amplitude its height. The Cosine wave will start at the maximum value (amplitude) since there's no phase shift, but this value is negative due to the multiplying factor. The wave will then drop down to the minimum value (negative amplitude), oscillating from top downwards, over the interval of its period, which is 8 in this case.