To graph $y = 2 \cot x$, it's important to understand the properties of the cotangent function and how the coefficient affects its behavior.

Steps to Graph $y = 2 \cot x$:

1. Basic Shape of $\cot x$: The cotangent function, $\cot x = \frac{\cos x}{\sin x}$, has vertical asymptotes at points where $\sin x = 0$, which occur at multiples of $\pi$ (i.e., $x = 0, \pi, 2\pi, -\pi, -2\pi$, etc.).

   The cotangent function is undefined at these points, and the graph of $\cot x$ approaches these vertical asymptotes.

2. Period: The period of $\cot x$ is $\pi$, so the function repeats its behavior every $\pi$ units. This means that the graph of $y = 2 \cot x$ will also repeat every $\pi$.

3. Amplitude: The coefficient $2$ stretches the cotangent graph vertically by a factor of 2. This makes the graph steeper compared to $y = \cot x$.

4. Key Points: Let's list the key points for one period (from $0$ to $\pi$):

   • At $x = \frac{\pi}{2}$, the cotangent function crosses the x-axis. So, $y( \frac{\pi}{2} ) = 2 \times \cot( \frac{\pi}{2} ) = 0$.
   • As $x \to 0^+$, $\cot x \to \infty$, and since it's multiplied by 2, the value of $y \to \infty$.
   • As $x \to \pi^-$, $\cot x \to -\infty$, so $y \to -\infty$.

5. Graph Behavior:

   • The function decreases as $x$ moves from 0 to $\pi$, starting from $+\infty$, crossing the x-axis at $x = \frac{\pi}{2}$, and approaching $-\infty$ as $x \to \pi^-$.

Summary of Key Points:

• Vertical asymptotes: $x = 0, \pi, 2\pi, -\pi, -2\pi, \dots$
• X-intercepts: $x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \dots$
• Behavior near asymptotes:
  • As $x \to 0^+$, $y \to +\infty$
  • As $x \to \pi^-$, $y \to -\infty$

Would you like me to graph this function for you?

Follow-up Questions:

1. What is the effect of changing the coefficient in front of $\cot x$?
2. How does the period of $\cot x$ compare to that of $\sin x$?
3. What are the key points for $y = \cot x$ without the 2 as a multiplier?
4. Can you explain the differences between $\tan x$ and $\cot x$?
5. How do vertical asymptotes influence the graph of trigonometric functions?

Tip:

Always remember that vertical asymptotes occur where the denominator of a fraction (like $\cot x = \frac{\cos x}{\sin x}$) is zero.