Graph y=tan(x)

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 ${\displaystyle y=\tan \left(x\right)}$. 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 ${\displaystyle y=\tan \left(x\right)}$.

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

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

Find the period ${\displaystyle \frac{\pi }{\left|b\right|}}$ to find where the vertical asymptotes exist.

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

There are only vertical asymptotes for tangent and cotangent functions.

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

Replace ${\displaystyle b}$ with ${\displaystyle 1}$ in the formula for period.

The absolute value is the distance between a number and zero. The distance between ${\displaystyle 0}$ and ${\displaystyle 1}$ is ${\displaystyle 1}$.

Divide ${\displaystyle \pi }$ by ${\displaystyle 1}$.

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

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

Divide ${\displaystyle 0}$ by ${\displaystyle 1}$.