Question

graph the function over a one - period interval.
y = tan 6x
which graph below shows one period of the function?

Explanation:

Step1: Recall period formula for tangent

The period of the tangent function $y = A\tan(Bx - C)+D$ is $\frac{\pi}{|B|}$. For $y=\tan(6x)$, $B = 6$, so the period is $\frac{\pi}{6}$.

Step2: Analyze asymptotes

The tangent function $y = \tan x$ has vertical asymptotes at $x=(n+\frac{1}{2})\pi$, $n\in\mathbb{Z}$. For $y=\tan(6x)$, the vertical asymptotes are at $6x=(n +\frac{1}{2})\pi$, or $x=\frac{(2n + 1)\pi}{12}$, $n\in\mathbb{Z}$. In one - period interval, when $n = 0$, $x=\frac{\pi}{12}$ and when $n=- 1$, $x=-\frac{\pi}{12}$. The function passes through the origin $(0,0)$.

Step3: Match with graphs

The graph of $y = \tan(6x)$ in the interval $(-\frac{\pi}{12},\frac{\pi}{12})$ has a shape similar to the basic tangent function, passing through the origin and having vertical asymptotes at $x =-\frac{\pi}{12}$ and $x=\frac{\pi}{12}$.

Answer:

C.