Graph f(x)=2cot(x-pi/4)

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 ${\displaystyle y=2\cot (x-\frac{\pi }{4})}$. 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 ${\displaystyle y=2\cot (x-\frac{\pi }{4})}$.

Add ${\displaystyle \frac{\pi }{4}}$ to both sides of the equation.

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

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

The basic period for ${\displaystyle y=2\cot (x-\frac{\pi }{4})}$ will occur at ${\displaystyle (\frac{\pi }{4},\frac{5\pi }{4})}$, where ${\displaystyle \frac{\pi }{4}}$ and ${\displaystyle \frac{5\pi }{4}}$ are vertical asymptotes.

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

The vertical asymptotes for ${\displaystyle y=2\cot (x-\frac{\pi }{4})}$ occur at ${\displaystyle \frac{\pi }{4}}$, ${\displaystyle \frac{5\pi }{4}}$, and every ${\displaystyle x=\frac{\pi }{4}+\pi n}$, where ${\displaystyle n}$ is an integer.

Cotangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: ${\displaystyle x=\frac{\pi }{4}+\pi n}$ 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 ${\displaystyle 1}$ 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 ${\displaystyle \frac{c}{b}}$.

Phase Shift: ${\displaystyle \frac{c}{b}}$

Replace the values of ${\displaystyle c}$ and ${\displaystyle b}$ in the equation for phase shift.

Phase Shift: ${\displaystyle \frac{\frac{\pi }{4}}{1}}$

Divide ${\displaystyle \frac{\pi }{4}}$ by ${\displaystyle 1}$.

Phase Shift: ${\displaystyle \frac{\pi }{4}}$