Graph y=-1/6csc(x/4)

For any ${\displaystyle y=\csc \left(x\right)}$, vertical asymptotes occur at ${\displaystyle x=n\pi }$, where ${\displaystyle n}$ is an integer. Use the basic period for ${\displaystyle y=csc\left(x\right)}$, ${\displaystyle (0,2\pi )}$, to find the vertical asymptotes for ${\displaystyle y=-\frac{\csc \left(\frac{x}{4}\right)}{6}}$. Set the inside of the cosecant function, ${\displaystyle bx+c}$, for ${\displaystyle y=a\csc (bx+c)+d}$ equal to ${\displaystyle 0}$ to find where the vertical asymptote occurs for ${\displaystyle y=-\frac{\csc \left(\frac{x}{4}\right)}{6}}$.

Set the numerator equal to zero.

Set the inside of the cosecant function ${\displaystyle \frac{x}{4}}$ equal to ${\displaystyle 2\pi }$.

Solve for ${\displaystyle x}$.

The basic period for ${\displaystyle y=-\frac{\csc \left(\frac{x}{4}\right)}{6}}$ will occur at ${\displaystyle (0,8\pi )}$, where ${\displaystyle 0}$ and ${\displaystyle 8\pi }$ are vertical asymptotes.

Find the period ${\displaystyle \frac{2\pi }{\left|b\right|}}$ to find where the vertical asymptotes exist. Vertical asymptotes occur every half period.

The vertical asymptotes for ${\displaystyle y=-\frac{\csc \left(\frac{x}{4}\right)}{6}}$ occur at ${\displaystyle 0}$, ${\displaystyle 8\pi }$, and every ${\displaystyle x=4\pi n}$, where ${\displaystyle n}$ is an integer. This is half of the period.

Cosecant only has vertical asymptotes.

The period of the function can be calculated using ${\displaystyle \frac{2\pi }{\left|b\right|}}$.

Replace ${\displaystyle b}$ with ${\displaystyle \frac{1}{4}}$ in the formula for period.

${\displaystyle \frac{1}{4}}$ is approximately ${\displaystyle 0.25}$ which is positive so remove the absolute value

Multiply the numerator by the reciprocal of the denominator.

Multiply ${\displaystyle 4}$ by ${\displaystyle 2}$.

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.

Multiply the numerator by the reciprocal of the denominator.

Multiply ${\displaystyle 0}$ by ${\displaystyle 4}$.