Question

graph two periods of the given tangent function.
y = \frac{1}{2}\tan4x
choose the correct graph of two periods of y = \frac{1}{2}\tan4x below.

Explanation:

Step1: Recall period formula for tangent

The period of the tangent function $y = A\tan(Bx)$ is $T=\frac{\pi}{|B|}$. For $y=\frac{1}{2}\tan(4x)$, $B = 4$, so $T=\frac{\pi}{4}$.

Step2: Analyze key - points

The tangent function $y=\tan x$ has vertical asymptotes at $x=(n +\frac{1}{2})\pi,n\in\mathbb{Z}$. For $y=\frac{1}{2}\tan(4x)$, vertical asymptotes occur at $4x=(n+\frac{1}{2})\pi$, or $x=\frac{(2n + 1)\pi}{8},n\in\mathbb{Z}$. When $x = 0$, $y=\frac{1}{2}\tan(0)=0$.

Step3: Check the graphs

The graph of $y=\frac{1}{2}\tan(4x)$ has a period of $\frac{\pi}{4}$, vertical asymptotes at $x=\frac{\pi}{8},\frac{3\pi}{8},\cdots$ and passes through the origin.

Answer:

(Without seeing the actual details of the graphs A, B, C, it's not possible to give a specific letter - choice. But the correct graph should have a period of $\frac{\pi}{4}$, vertical asymptotes at $x=\frac{(2n + 1)\pi}{8},n\in\mathbb{Z}$ and pass through the origin.)