Graph y = 2 cot x and list the key points on the graph

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 \( x \to \pi^- \), \( y \to -\infty \)

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### 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.

Learn how to graph the function y = 2 cot x by identifying key points such as vertical asymptotes and x-intercepts. Understand the effect of the coefficient on the graph's amplitude and how the cotangent function behaves. Suitable for students in Grades 10-12.

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