Problem 39 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The amplitude of the function is 4 and the period is \(4\pi\). The function can be graphed as a reflection of the standard cosine function over the x-axis, stretching from \(0\) to \(4\pi\), with a maximum height of 4 and a minimum depth of -4.

Step 1: Find the Amplitude

The amplitude is simply the absolute value of the coefficient of the cosine function. Here, the coefficient is \(-4\), so the amplitude is \(| -4 | = 4\).

Step 2: Find the Period

The period of the function is found by dividing \(2\pi\) by the absolute value of the term with which \(x\) is being multiplied inside the cosine function, which is \(0.5\) or \(\frac{1}{2}\). Hence the period is \(2\pi/0.5 = 4\pi\).

Step 3: Graph the Function

The function is negative, so it will be a reflection of the standard cosine function over the x-axis. The amplitude tells the maximum height and depth of the function, which are \(-4\) and \(4\). The function completes one cycle over the range \(0\) to \(4\pi\). By plotting a few key points, such as the start and end points (0, -4) and (4pi, -4), and the maximum and minimum points in between, the final curve representing one period of the function can be drawn.