To graph the equation \( y = \frac{2}{3} \cot\left(\frac{1}{2}x\right) \), we need to analyze the behavior of the function. Here's a step-by-step breakdown:

### 1. **General Form**: 
   - The equation \( y = A \cot(Bx) \) is a transformation of the basic cotangent function \( y = \cot(x) \), where:
     - \( A = \frac{2}{3} \) is the amplitude factor (affecting the vertical stretch or compression),
     - \( B = \frac{1}{2} \) affects the period of the cotangent function.

### 2. **Period**:
   - The period of the cotangent function is normally \( \pi \). However, in this case, the period is scaled by \( B = \frac{1}{2} \), so the new period becomes:
     \[
     \text{Period} = \frac{\pi}{B} = \frac{\pi}{\frac{1}{2}} = 2\pi
     \]
   - This means the graph will repeat every \( 2\pi \) units.

### 3. **Vertical Stretch/Compression**:
   - The factor \( \frac{2}{3} \) vertically compresses the graph by a factor of \( \frac{2}{3} \). Therefore, the typical values of cotangent are scaled down by this factor.

### 4. **Asymptotes**:
   - Cotangent has vertical asymptotes where \( \cot(x) = 0 \), which occur at \( x = 0, \pi, 2\pi, \dots \). For the given transformation, the asymptotes occur at multiples of the period \( 2\pi \), starting from \( x = 0 \).

### 5. **Key Points**:
   - For a regular cotangent function, \( \cot(x) \) has key points at \( \left(0, \infty \right) \), \( \left(\frac{\pi}{2}, 0 \right) \), and asymptotes at \( x = 0, \pi, 2\pi, \dots \).
   - With the transformations applied:
     - The zeros occur at \( x = \pi, 2\pi, 3\pi, \dots \).
     - The vertical asymptotes occur at \( x = 0, 2\pi, 4\pi, \dots \).
     - The graph passes through \( \left(\frac{\pi}{2}, 0\right) \).

Would you like me to graph this function or provide more details on how to draw it?

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### Related Questions:
1. How does the transformation affect the domain and range of the cotangent function?
2. How would the graph look if the vertical stretch factor were larger or smaller?
3. How would the graph change if the period were doubled or halved?
4. What are the differences between the graphs of \( y = \cot(x) \) and \( y = \frac{2}{3} \cot\left(\frac{1}{2}x\right) \)?
5. How do vertical asymptotes influence the behavior of the graph?

**Tip**: Understanding the effect of transformation parameters (like amplitude and period) helps when interpreting trigonometric graphs.

Learn how to graph the trigonometric function y = 2/3 cot(1/2 x). This solution explains the transformation of the cotangent function, including vertical compression, changes in period, and identifying key features like zeros and asymptotes. Suitable for Grades 10-12 students.

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