what is the equation of this graph?\ny = csc(4x)\ny = sec(4x)\ny = sec(\\frac{x}{4})

Explanation:

Step1: Recall period formula

The period of $y = A\csc(Bx - C)+D$ and $y=A\sec(Bx - C)+D$ is $T=\frac{2\pi}{|B|}$.

Step2: Determine period from graph

The period of the given graph is $\frac{\pi}{2}$.

Step3: Find B - value

Set $\frac{2\pi}{|B|}=\frac{\pi}{2}$, then $|B| = 4$.

Step4: Identify function type

The graph has vertical asymptotes at $x=\frac{\pi}{4}+\frac{k\pi}{2},k\in\mathbb{Z}$, which is characteristic of the secant function $y = \sec(x)$ rather than the cosecant function.

Answer:

B. $y = \sec(4x)$