Problem 67 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The period of the function is \(4\pi\). The graph starts at a maximum, comes down to 0 at \(2\pi\), and returns to the maximum at \(4\pi\). It oscillates between 0 and 2.

Step 1: Understand the Function

The given function is an absolute value of cosine function which is a periodic function. The absolute function transforms the negative portion of the graph into positive. The period of the function can be calculated with \(2\pi\) divided by the absolute value of B, where y=A cos(Bx), for the given function B = 1/2.

Step 2: Calculate the Period

The period of the function is given by \(2\pi / |B|\). In this case, B = 1/2. So, the period of the function would be \(2\pi / 1/2 = 4\pi\). This means the function repeats every \(4\pi\) intervals.

Step 3: Plot the Graph

The Graph of the given function will have its maximum at |2|=2 and minimum at 0. The function being an absolute cosine function, it oscillates between 0 and 2. As it is positive cosine wave, it starts from a maximum, comes down to 0 at \(2\pi\) and goes back to maximum at \(4\pi\). Hence, one period of the function would be complete at \(4\pi\).