solve using a table of values (use real numbers)

To solve the function \( y = 2 \cot \left( x - \frac{3\pi}{4} \right) \) using a table of values, we'll choose specific \( x \) values, calculate the corresponding \( y \) values, and then observe the behavior of the function. 

### Steps to create the table:
1. **Select \( x \) values**: Choose \( x \) values that will give us a variety of points across one period of the cotangent function. Since the period of the cotangent function is \( \pi \), you can select \( x \) values over an interval of length \( \pi \) starting from \( \frac{3\pi}{4} \), where the horizontal shift starts.
  
2. **Calculate \( y \) values**: Substitute the selected \( x \) values into the function and compute \( y \).

### Calculation:

Let's pick \( x \) values around the interval \(\left[\frac{3\pi}{4}, \frac{7\pi}{4}\right]\) to capture one full period:

\[
\begin{array}{c|c}
x & y = 2 \cot \left( x - \frac{3\pi}{4} \right) \\
\hline
\frac{3\pi}{4} & 2 \cdot \cot \left( \frac{3\pi}{4} - \frac{3\pi}{4} \right) = 2 \cdot \cot(0) = \text{undefined} \\
\frac{\pi}{2} + \frac{3\pi}{4} = \frac{5\pi}{4} & 2 \cdot \cot \left( \frac{5\pi}{4} - \frac{3\pi}{4} \right) = 2 \cdot \cot\left(\frac{\pi}{2}\right) = 2 \cdot 0 = 0 \\
\pi + \frac{3\pi}{4} = \frac{7\pi}{4} & 2 \cdot \cot \left( \frac{7\pi}{4} - \frac{3\pi}{4} \right) = 2 \cdot \cot(\pi) = 2 \cdot 0 = \text{undefined} \\
\frac{3\pi}{2} + \frac{3\pi}{4} = \frac{9\pi}{4} & 2 \cdot \cot \left( \frac{9\pi}{4} - \frac{3\pi}{4} \right) = 2 \cdot \cot\left(\frac{\pi}{2}\right) = 2 \cdot 0 = 0 \\
\frac{\pi}{4} + \frac{3\pi}{4} = \frac{7\pi}{4} & 2 \cdot \cot \left( \frac{7\pi}{4} - \frac{3\pi}{4} \right) = 2 \cdot \cot\left(\pi\right) = 2 \cdot 0 = \text{undefined} \\
\end{array}
\]

### Table of Values:

\[
\begin{array}{c|c}
x & y = 2 \cot \left( x - \frac{3\pi}{4} \right) \\
\hline
\frac{3\pi}{4} & \text{undefined (vertical asymptote)} \\
\frac{5\pi}{4} & 0 \\
\frac{7\pi}{4} & \text{undefined (vertical asymptote)} \\
\frac{9\pi}{4} & 0 \\
\end{array}
\]

### Explanation:

- At \( x = \frac{3\pi}{4} \) and \( x = \frac{7\pi}{4} \), the function is undefined due to vertical asymptotes.
- At \( x = \frac{5\pi}{4} \) and \( x = \frac{9\pi}{4} \), the function crosses the x-axis with a value of 0.

### Observation:
The cotangent function alternates between positive and negative values around its asymptotes. The table shows that as \( x \) approaches the asymptotes, the value of \( y \) tends to \( \infty \) or \( -\infty \).

Would you like more details or have any other questions?

### Relative Questions:
1. How do vertical asymptotes affect the behavior of the graph?
2. Why is the function undefined at certain \( x \) values?
3. What happens to the function's value as \( x \) approaches the vertical asymptotes?
4. How does the period of the function relate to the interval of \( x \) chosen?
5. How would adding a vertical shift to the function affect the table of values?
6. How can you verify the values obtained in the table using a calculator?
7. What are the key characteristics of the cotangent function's graph based on the table?
8. How does scaling the function vertically by a factor of 2 affect the results?

### Tip:
When analyzing trigonometric functions, always consider the presence of asymptotes and zeros, as these greatly influence the function's behavior and graph.

Learn how to solve the function y = 2 cot (x - 3π/4) using a table of values. Explore the behavior of the cotangent function with specific x values and understand the concept of vertical asymptotes. Suitable for high school students studying trigonometry.

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