Graph

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 ${\displaystyle y=3\cot \left(4x\right)}$. 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 ${\displaystyle y=3\cot \left(4x\right)}$.

Divide each term in ${\displaystyle 4x=0}$ by ${\displaystyle 4}$ and simplify.

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

Divide each term in ${\displaystyle 4x=\pi }$ by ${\displaystyle 4}$ and simplify.

The basic period for ${\displaystyle y=3\cot \left(4x\right)}$ will occur at ${\displaystyle (0,\frac{\pi }{4})}$, where ${\displaystyle 0}$ and ${\displaystyle \frac{\pi }{4}}$ are vertical asymptotes.

The absolute value is the distance between a number and zero. The distance between ${\displaystyle 0}$ and ${\displaystyle 4}$ is ${\displaystyle 4}$.

The vertical asymptotes for ${\displaystyle y=3\cot \left(4x\right)}$ occur at ${\displaystyle 0}$, ${\displaystyle \frac{\pi }{4}}$, and every ${\displaystyle \frac{\pi n}{4}}$, 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 ${\displaystyle 4}$ in the formula for period.

The absolute value is the distance between a number and zero. The distance between ${\displaystyle 0}$ and ${\displaystyle 4}$ is ${\displaystyle 4}$.

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 ${\displaystyle 4}$.