Find the Asymptotes y=-8tan(6x)

For any ${\displaystyle y=\tan \left(x\right)}$, vertical asymptotes occur at ${\displaystyle x=\frac{\pi }{2}+n\pi }$, where ${\displaystyle n}$ is an integer. Use the basic period for ${\displaystyle y=\tan \left(x\right)}$, ${\displaystyle (-\frac{\pi }{2},\frac{\pi }{2})}$, to find the vertical asymptotes for ${\displaystyle y=-8\tan \left(6x\right)}$. Set the inside of the tangent function, ${\displaystyle bx+c}$, for ${\displaystyle y=a\tan (bx+c)+d}$ equal to ${\displaystyle -\frac{\pi }{2}}$ to find where the vertical asymptote occurs for ${\displaystyle y=-8\tan \left(6x\right)}$.

Set the inside of the tangent function ${\displaystyle 6x}$ equal to ${\displaystyle \frac{\pi }{2}}$.

The basic period for ${\displaystyle y=-8\tan \left(6x\right)}$ will occur at ${\displaystyle (-\frac{\pi }{12},\frac{\pi }{12})}$, where ${\displaystyle -\frac{\pi }{12}}$ and ${\displaystyle \frac{\pi }{12}}$ are vertical asymptotes.

The absolute value is the distance between a number and zero. The distance between ${\displaystyle 0}$ and ${\displaystyle 6}$ is ${\displaystyle 6}$.

The vertical asymptotes for ${\displaystyle y=-8\tan \left(6x\right)}$ occur at ${\displaystyle -\frac{\pi }{12}}$, ${\displaystyle \frac{\pi }{12}}$, and every ${\displaystyle \frac{\pi n}{6}}$, where ${\displaystyle n}$ is an integer.

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: ${\displaystyle x=\frac{\pi }{12}+\frac{\pi n}{6}}$ where ${\displaystyle n}$ is an integer