Graph y=1/2cot(pix)

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 ${\displaystyle y=\frac{\cot \left(\pi x\right)}{2}}$. 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 ${\displaystyle y=\frac{\cot \left(\pi x\right)}{2}}$.

Divide each term in ${\displaystyle \pi x=0}$ by ${\displaystyle \pi }$ and simplify.

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

Divide each term in ${\displaystyle \pi x=\pi }$ by ${\displaystyle \pi }$ and simplify.

The basic period for ${\displaystyle y=\frac{\cot \left(\pi x\right)}{2}}$ will occur at ${\displaystyle (0,1)}$, where ${\displaystyle 0}$ and ${\displaystyle 1}$ are vertical asymptotes.

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

The vertical asymptotes for ${\displaystyle y=\frac{\cot \left(\pi x\right)}{2}}$ occur at ${\displaystyle 0}$, ${\displaystyle 1}$, and every ${\displaystyle n}$, 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 ${\displaystyle \pi }$ in the formula for period.

${\displaystyle \pi }$ is approximately ${\displaystyle 3.14159265}$ which is positive so remove the absolute value

Cancel the common factor of ${\displaystyle \pi }$.

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 ${\displaystyle \pi }$.