1. **Problem Statement:** We are given a graph of a function with points approximately (-4, -4) and (2, 6), indicating a positive slope. We need to determine which of the given relationships (A, B, C, or D) represents a function with a lesser rate of change (slope) than the graphed function. 2. **Step 1: Find the slope of the graphed function.** The slope formula is: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Using points (-4, -4) and (2, 6): $$m = \frac{6 - (-4)}{2 - (-4)} = \frac{6 + 4}{2 + 4} = \frac{10}{6} = \frac{5}{3} \approx 1.67$$ 3. **Step 2: Calculate the slope (rate of change) for each option.** - **Option A:** Points (-2, -14) and (2, 6) $$m_A = \frac{6 - (-14)}{2 - (-2)} = \frac{20}{4} = 5$$ - **Option B:** Equation $y = 2x - 5$ has slope $m_B = 2$ - **Option C:** Points (-1, 2) and (3, -14) $$m_C = \frac{-14 - 2}{3 - (-1)} = \frac{-16}{4} = -4$$ - **Option D:** Equation $y = \frac{5}{2}x + 5$ has slope $m_D = \frac{5}{2} = 2.5$ 4. **Step 3: Compare slopes to the graphed function's slope $\frac{5}{3} \approx 1.67$.** - $m_A = 5$ (greater than 1.67) - $m_B = 2$ (greater than 1.67) - $m_C = -4$ (less than 1.67, but negative slope means decreasing function) - $m_D = 2.5$ (greater than 1.67) 5. **Step 4: Interpretation** The question asks for a function with a lesser rate of change than the graphed function. Since the graphed function has a positive slope of about 1.67, a lesser positive slope would be between 0 and 1.67. - Option C has a negative slope, which is less than 1.67 but represents a decreasing function, not a lesser positive rate of change. - None of the options have a positive slope less than 1.67. 6. **Conclusion:** If we consider absolute value of rate of change, option C has a lesser rate of change in terms of magnitude but negative slope. If the question implies positive slope lesser than 1.67, none fit. Since the question likely means lesser positive rate of change, none of the options have a positive slope less than 1.67. **Final answer:** Option C represents a function with a lesser rate of change (negative slope) than the graphed function.