Graph y=-1/3csc(x+pi/2)

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 ${\displaystyle y=-\frac{\csc (x+\frac{\pi }{2})}{3}}$. 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 ${\displaystyle y=-\frac{\csc (x+\frac{\pi }{2})}{3}}$.

Subtract ${\displaystyle \frac{\pi }{2}}$ from both sides of the equation.

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

Move all terms not containing ${\displaystyle x}$ to the right side of the equation.

The basic period for ${\displaystyle y=-\frac{\csc (x+\frac{\pi }{2})}{3}}$ will occur at ${\displaystyle (-\frac{\pi }{2},\frac{3\pi }{2})}$, where ${\displaystyle -\frac{\pi }{2}}$ and ${\displaystyle \frac{3\pi }{2}}$ 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 ${\displaystyle y=-\frac{\csc (x+\frac{\pi }{2})}{3}}$ occur at ${\displaystyle -\frac{\pi }{2}}$, ${\displaystyle \frac{3\pi }{2}}$, and every ${\displaystyle x=-\frac{\pi }{2}+\pi n}$, where ${\displaystyle n}$ is an integer. This is half of the period.

Cosecant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: ${\displaystyle x=-\frac{\pi }{2}+\pi n}$ where ${\displaystyle n}$ is an integer

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

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 ${\displaystyle 2\pi }$ by ${\displaystyle 1}$.

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: ${\displaystyle \frac{-\frac{\pi }{2}}{1}}$

Divide ${\displaystyle -\frac{\pi }{2}}$ by ${\displaystyle 1}$.

Phase Shift: ${\displaystyle -\frac{\pi }{2}}$