Question

graph two periods of the given tangent function.
y = \frac{1}{6}\tan5x
choose the correct graph of two periods of y = \frac{1}{6}\tan5x below.

Explanation:

Step1: Recall period formula for tangent

The period of the tangent function $y = A\tan(Bx)$ is given by $\frac{\pi}{|B|}$. For $y=\frac{1}{6}\tan(5x)$, $B = 5$, so the period is $\frac{\pi}{5}$.

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 $y=\frac{1}{6}\tan(5x)$, vertical asymptotes occur when $5x=(n+\frac{1}{2})\pi$, or $x=\frac{(2n + 1)\pi}{10},n\in\mathbb{Z}$. When $x = 0$, $y=\frac{1}{6}\tan(0)=0$.

Step3: Consider amplitude and shape

The coefficient $\frac{1}{6}$ in front of the tangent function compresses the graph of $y = \tan(5x)$ vertically by a factor of $\frac{1}{6}$. The basic shape of the tangent function is a set of U - shaped curves between vertical asymptotes, passing through the origin in each period.

Answer:

Without seeing the actual details of the graphs A, B and C, we can describe the correct graph characteristics: It should have vertical asymptotes at $x=\frac{\pi}{10},\frac{3\pi}{10},\frac{5\pi}{10},\frac{7\pi}{10},\cdots$ (for two periods), pass through the origin $(0,0)$ and have a vertical compression by a factor of $\frac{1}{6}$ compared to the graph of $y = \tan(5x)$. You need to match these characteristics to the given graphs to choose the correct one.