graph the given function f(x) over the interval -4π, 4π. determine whether the function is periodic and, if…

graph the given function f(x) over the interval -4π, 4π. determine whether the function is periodic and, if it is, state the period. f(x)=|sin(1/4x)| + 2 all graphs are -4π, 4π by -5, 5. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the function is periodic. the period is b. the function is not periodic.

graph the given function f(x) over the interval -4π, 4π. determine whether the function is periodic and, if it is, state the period. f(x)=|sin(1/4x)| + 2 all graphs are -4π, 4π by -5, 5. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the function is periodic. the period is b. the function is not periodic.

Answer

Explanation:

Step1: Recall period of basic sine function

The period of $y = \sin(kx)$ is $T=\frac{2\pi}{k}$. For $y = \sin(\frac{1}{4}x)$, $k=\frac{1}{4}$, so $T = \frac{2\pi}{\frac{1}{4}}=8\pi$.

Step2: Consider absolute - value effect

The absolute - value function $y =|\sin(\frac{1}{4}x)|$ reflects the negative part of $y=\sin(\frac{1}{4}x)$ above the x - axis. The period of $y =|\sin(\frac{1}{4}x)|$ is half of the period of $y=\sin(\frac{1}{4}x)$. So the period of $y =|\sin(\frac{1}{4}x)|$ is $\frac{8\pi}{2}=4\pi$.

Step3: Consider vertical shift effect

The function $f(x)=|\sin(\frac{1}{4}x)| + 2$ is a vertical shift of $y =|\sin(\frac{1}{4}x)|$ by 2 units up. A vertical shift does not affect the period of the function.

Answer:

A. The function is periodic. The period is $4\pi$