Here is the graph of $f(x) = 3 \cot(2x)$ over one period, $\left( 0, \frac{\pi}{2} \right)$, with the critical points labeled. You can observe the vertical asymptotes at $x = 0$ and $x = \frac{\pi}{2}$, as well as the key points:

• Maximum: $\left( \frac{\pi}{8}, 3 \right)$,
• Zero: $\left( \frac{\pi}{4}, 0 \right)$,
• Minimum: $\left( \frac{3\pi}{8}, -3 \right)$.

These points highlight the behavior of the cotangent function as it approaches infinity near the asymptotes and crosses the x-axis at $\frac{\pi}{4}$.

Would you like any more details or further questions? Here are five related questions:

1. How do the transformations (such as amplitude and period changes) affect other trigonometric functions?
2. What is the general form of a cotangent function and how do we find phase shifts?
3. What happens to the graph if we modify the function to $f(x) = 3 \cot(2x) + 1$?
4. How do you determine the asymptotes for other trigonometric functions?
5. Can you explain the relationship between cotangent and tangent graphs?

Tip: Vertical asymptotes in cotangent functions occur where the sine in $\frac{\cos(x)}{\sin(x)}$ equals zero, making the function undefined.