3) Consider the quadratic \( f(x)=-2 x^{2}-4 x+5 \). \( \begin{array}{ll}\text { a. What is the average rate of change over } & \text { b. What is the average rate of change over } \\ \text { the interval from } x=0 \text { to } x=2 \text { ? } & \text { the interval from }-7 \leq x \leq-4 ?\end{array} \)

To determine the average rate of change of the quadratic function \( f(x) = -2x^2 - 4x + 5 \) over specified intervals, we'll use the formula: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \] where \( a \) and \( b \) are the endpoints of the interval. --- ### **a. Average Rate of Change from \( x = 0 \) to \( x = 2 \)** 1. **Calculate \( f(0) \):** \[ f(0) = -2(0)^2 - 4(0) + 5 = 0 - 0 + 5 = 5 \] 2. **Calculate \( f(2) \):** \[ f(2) = -2(2)^2 - 4(2) + 5 = -2(4) - 8 + 5 = -8 - 8 + 5 = -11 \] 3. **Compute the Average Rate of Change:** \[ \frac{f(2) - f(0)}{2 - 0} = \frac{-11 - 5}{2} = \frac{-16}{2} = -8 \] **Answer:** The average rate of change from \( x = 0 \) to \( x = 2 \) is **-8**. --- ### **b. Average Rate of Change from \( x = -7 \) to \( x = -4 \)** 1. **Calculate \( f(-7) \):** \[ f(-7) = -2(-7)^2 - 4(-7) + 5 = -2(49) + 28 + 5 = -98 + 28 + 5 = -65 \] 2. **Calculate \( f(-4) \):** \[ f(-4) = -2(-4)^2 - 4(-4) + 5 = -2(16) + 16 + 5 = -32 + 16 + 5 = -11 \] 3. **Compute the Average Rate of Change:** \[ \frac{f(-4) - f(-7)}{-4 - (-7)} = \frac{-11 - (-65)}{3} = \frac{54}{3} = 18 \] **Answer:** The average rate of change from \( x = -7 \) to \( x = -4 \) is **18**. --- **Summary:** - **a.** Average rate of change from \( x = 0 \) to \( x = 2 \) is **-8**. - **b.** Average rate of change from \( x = -7 \) to \( x = -4 \) is **18**.