Graph f(x)=cot(1/2x-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 ${\displaystyle y=\cot (\frac{x}{2}-\pi )}$. 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 ${\displaystyle y=\cot (\frac{x}{2}-\pi )}$.

Solve for ${\displaystyle x}$.

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

Solve for ${\displaystyle x}$.

The basic period for ${\displaystyle y=\cot (\frac{x}{2}-\pi )}$ will occur at ${\displaystyle (2\pi ,4\pi )}$, where ${\displaystyle 2\pi }$ and ${\displaystyle 4\pi }$ are vertical asymptotes.

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

The vertical asymptotes for ${\displaystyle y=\cot (\frac{x}{2}-\pi )}$ occur at ${\displaystyle 2\pi }$, ${\displaystyle 4\pi }$, and every ${\displaystyle x=2\pi +2\pi n}$, where ${\displaystyle n}$ is an integer.

Cotangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: ${\displaystyle x=2\pi +2\pi n}$ 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 ${\displaystyle \frac{1}{2}}$ in the formula for period.

${\displaystyle \frac{1}{2}}$ is approximately ${\displaystyle 0.5}$ which is positive so remove the absolute value

Multiply the numerator by the reciprocal of the denominator.

Move ${\displaystyle 2}$ to the left of ${\displaystyle \pi }$.

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

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: ${\displaystyle \pi \cdot 2}$

Move ${\displaystyle 2}$ to the left of ${\displaystyle \pi }$.

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