Problem 42 - Algebra and Trigonometry

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: Determine the Amplitude

The amplitude of a cosine function is given by the absolute value of the coefficient of the cosine term. In this function, that value is \(-\frac{1}{2}\). So, the amplitude is given by \(|-\frac{1}{2}|\), which equals to \(\frac{1}{2}\).

Step 2: Determine the Period

The period of the cosine function can be found by the formula \(T = \frac{2\pi}{B}\), where B is the coefficient of x in the cosine function. Here, \(B=\frac{\pi}{4}\). Substituting the value of B into the formula, we get the period \(T = \frac{2\pi}{(\frac{\pi}{4})} = 8\).

Step 3: Plotting the graph

In one period of the function, the cosine wave starts from a maximal value, decreases to its minimum, and returns to the maximum. The range of the function is between \(-\frac{1}{2}\) and \(\frac{1}{2}\). The function is reflected in x-axis due to negative sign. After every 2 units, graph completes a cycle as function period is 8. These points should be enough to graph one period of the function.