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:

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.