Graph y=-1/2tan(3x)

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 ${\displaystyle y=-\frac{\tan \left(3x\right)}{2}}$. 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 ${\displaystyle y=-\frac{\tan \left(3x\right)}{2}}$.

Divide each term in ${\displaystyle 3x=-\frac{\pi }{2}}$ by ${\displaystyle 3}$ and simplify.

Set the inside of the tangent function ${\displaystyle 3x}$ equal to ${\displaystyle \frac{\pi }{2}}$.

Divide each term in ${\displaystyle 3x=\frac{\pi }{2}}$ by ${\displaystyle 3}$ and simplify.

The basic period for ${\displaystyle y=-\frac{\tan \left(3x\right)}{2}}$ will occur at ${\displaystyle (-\frac{\pi }{6},\frac{\pi }{6})}$, where ${\displaystyle -\frac{\pi }{6}}$ and ${\displaystyle \frac{\pi }{6}}$ are vertical asymptotes.

The absolute value is the distance between a number and zero. The distance between ${\displaystyle 0}$ and ${\displaystyle 3}$ is ${\displaystyle 3}$.

The vertical asymptotes for ${\displaystyle y=-\frac{\tan \left(3x\right)}{2}}$ occur at ${\displaystyle -\frac{\pi }{6}}$, ${\displaystyle \frac{\pi }{6}}$, and every ${\displaystyle \frac{\pi n}{3}}$, where ${\displaystyle n}$ is an integer.

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: ${\displaystyle x=\frac{\pi }{6}+\frac{\pi n}{3}}$ where ${\displaystyle n}$ is an integer

The period of the function can be calculated using ${\displaystyle \frac{\pi }{\left|b\right|}}$.

Replace ${\displaystyle b}$ with ${\displaystyle 3}$ in the formula for period.

The absolute value is the distance between a number and zero. The distance between ${\displaystyle 0}$ and ${\displaystyle 3}$ is ${\displaystyle 3}$.

The phase shift of the function can be calculated from ${\displaystyle \frac{c}{b}}$.

Phase Shift: ${\displaystyle \frac{c}{b}}$

Replace the values of ${\displaystyle c}$ and ${\displaystyle b}$ in the equation for phase shift.

Phase Shift: ${\displaystyle \frac{0}{3}}$

Divide ${\displaystyle 0}$ by ${\displaystyle 3}$.

Phase Shift: ${\displaystyle 0}$