Problem 67 - Precalculus Essentials

Question

Graph one period of each function. $$y=\left|2 \cos \frac{x}{2}\right|$$

Step-by-Step Solution

Answer

The function should be plotted considering the period of \(4\pi\) and taking into account the absolute value transformation. Any values that would fall below the x-axis should be reflected above it.

Step 1: Find the Period of the Original Cosine Function

The formula for the period of a sinusoidal function, like cosine, is \(P = \frac{2\pi}{\left|B\right|}\), where \(B\) is the coefficient of \(x\) in the argument of the function. In this case, \(B = \frac{1}{2}\), so the period of the cosine function before applying the absolute value would be \(P = \frac{2\pi}{\left|1/2\right|} = 4\pi\).

Step 2: Sketch the Graph Considering Absolut Value

It is time to sketch the graph considering the effect of the absolute value. Normally, the cosine function alternates between positive and negative values. However, because of the absolute value, all negative values will be flipped to become positive. So, whenever \(2 \cos \frac{x}{2}\) would be negative, after applying the absolute value, |2 cos(x/2)| will be positive.

Step 3: Plot the Function

Now, use these considerations to plot the function. A good starting point might be to divide the period into equal parts. For instance, four parts in this case would give values for \(x=0\), \(x=\pi\), \(x=2\pi\), and \(x=3\pi\). Remember to reflect any part of the graph that would be below the x-axis to conform with the absolute value transformation.