the graph of f(x)=tan(bx), where b is a constant, is shown in the xy - plane. what is the value of b?

Explanation:

Step1: Recall period formula for tangent

The period of the tangent function $y = \tan(bx)$ is given by $T=\frac{\pi}{|b|}$.

Step2: Determine the period from the graph

From the graph, the vertical - asymptotes of $y = \tan(bx)$ are at $x=- 3$ and $x = 3$. The period $T$ of the function $y=\tan(bx)$ is the distance between two consecutive vertical asymptotes. So, $T=3-(-3)=6$.

Step3: Solve for $b$

Since $T = \frac{\pi}{|b|}$ and $T = 6$, we have $\frac{\pi}{|b|}=6$. Then $|b|=\frac{\pi}{6}$. Since the graph of $y = \tan(bx)$ is not reflected (the basic shape of the tangent function is maintained), $b=\frac{\pi}{6}$.

Answer:

$\frac{\pi}{6}$