Answer
Please see below.
Work Step by Step
Please note that the absolute value for sine and cosine removes the lower half of the graph, which is below the $x$-axis, and flips it above. This results in two complete cycles, where there had been one.
So, since the period of $2\cos \frac{x}{2}$ is $\frac{2\pi }{\frac{1}{2}}=4\pi$, the period of $y=\left |2 \cos \frac{x}{2} \right |$ will become $\frac{4\pi }{2}=2\pi$.
