Graph y=1/4cot(x+pi)

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

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

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

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

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

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

The vertical asymptotes for ${\displaystyle y=\frac{\cot (x+\pi )}{4}}$ occur at ${\displaystyle -\pi }$, ${\displaystyle 0}$, and every ${\displaystyle x=-\pi +\pi n}$, where ${\displaystyle n}$ is an integer.

Cotangent only has vertical asymptotes.

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