To determine which table of values represents a linear function, we need to check whether the change in the $y$ values is consistent with the change in the $x$ values. For a linear function, the difference between consecutive $y$ values divided by the difference between consecutive $x$ values (i.e., the slope) should be constant.

Let's examine each table:

Table A:

x & y \\ \hline -6 & -5 \\ -2 & -2 \\ 2 & 1 \\ 5 & 4 \\ \end{array}$$ **Changes:** $$\text{slope between } (-6, -5) \text{ and } (-2, -2): \frac{-2 - (-5)}{-2 - (-6)} = \frac{3}{4}$$ $$\text{slope between } (-2, -2) \text{ and } (2, 1): \frac{1 - (-2)}{2 - (-2)} = \frac{3}{4}$$ $$\text{slope between } (2, 1) \text{ and } (5, 4): \frac{4 - 1}{5 - 2} = \frac{3}{3} = 1$$ The slopes are not consistent, so **Table A is not linear**. --- **Table B:** $$\begin{array}{c|c} x & y \\ \hline 1 & -5 \\ 2 & -3 \\ 3 & -1 \\ 6 & 4 \\ \end{array}$$ **Changes:** $$\text{slope between } (1, -5) \text{ and } (2, -3): \frac{-3 - (-5)}{2 - 1} = \frac{2}{1} = 2$$ $$\text{slope between } (2, -3) \text{ and } (3, -1): \frac{-1 - (-3)}{3 - 2} = \frac{2}{1} = 2$$ $$\text{slope between } (3, -1) \text{ and } (6, 4): \frac{4 - (-1)}{6 - 3} = \frac{5}{3}$$ The slopes are not consistent, so **Table B is not linear**. --- **Table C:** $$\begin{array}{c|c} x & y \\ \hline -5 & -7 \\ 1 & -3 \\ 4 & -1 \\ 7 & 1 \\ \end{array}$$ **Changes:** $$\text{slope between } (-5, -7) \text{ and } (1, -3): \frac{-3 - (-7)}{1 - (-5)} = \frac{4}{6} = \frac{2}{3}$$ $$\text{slope between } (1, -3) \text{ and } (4, -1): \frac{-1 - (-3)}{4 - 1} = \frac{2}{3}$$ $$\text{slope between } (4, -1) \text{ and } (7, 1): \frac{1 - (-1)}{7 - 4} = \frac{2}{3}$$ The slopes are consistent, so **Table C is linear**. --- **Table D:** $$\begin{array}{c|c} x & y \\ \hline -2 & 7 \\ 1 & 6 \\ 4 & 5 \\ 7 & 3 \\ \end{array}$$ **Changes:** $$\text{slope between } (-2, 7) \text{ and } (1, 6): \frac{6 - 7}{1 - (-2)} = \frac{-1}{3} = -\frac{1}{3}$$ $$\text{slope between } (1, 6) \text{ and } (4, 5): \frac{5 - 6}{4 - 1} = \frac{-1}{3}$$ $$\text{slope between } (4, 5) \text{ and } (7, 3): \frac{3 - 5}{7 - 4} = \frac{-2}{3}$$ The slopes are not consistent, so **Table D is not linear**. --- **Conclusion:** **Table C** represents a linear function. Would you like more details or have any questions? --- **Related Questions:** 1. How do you determine the slope of a linear function from a table of values? 2. What is the importance of consistency in slope for identifying linear functions? 3. Can a function be linear if its slope is zero? 4. How would you graph a linear function given a table of values? 5. What are real-life examples of linear functions? **Tip:** Consistent slopes between points in a table indicate that the function is linear.