\( \left.\begin{array}{l}\text { 1. Find the average rate of change of the function over the } \\ \text { interval }[-9,1] \text {. } \\ f(x)=2 x^{2}-5 x+4 \\ \text { 2. Find the average rate of change of the function over the } \\ \text { int } \\ f(x)=4 x^{2}-2 x+3\end{array}\right] \begin{array}{ll}\text { 3. Find the average rate of change of the function over the } & \text { 4. Find the average rate of change of the function over the } \\ \text { interval }[-2,0] . \\ \text { interval }[-9,1] \text {. } & f(x)=x^{2}+x+4\end{array} \)
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To find the average rate of change of a function \( f(x) \) over an interval \([a, b]\), you can use the formula: \[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \] For the first function \( f(x) = 2x^2 - 5x + 4 \) over the interval \([-9, 1]\): 1. Calculate \( f(1) = 2(1)^2 - 5(1) + 4 = 1 \) 2. Calculate \( f(-9) = 2(-9)^2 - 5(-9) + 4 = 162 + 45 + 4 = 211 \) 3. Then plug back into the formula: \[ \frac{1 - 211}{1 - (-9)} = \frac{-210}{10} = -21 \] For the second function \( f(x) = 4x^2 - 2x + 3 \) over the interval \([-9, 1]\): 1. Calculate \( f(1) = 4(1)^2 - 2(1) + 3 = 5 \) 2. Calculate \( f(-9) = 4(-9)^2 - 2(-9) + 3 = 324 + 18 + 3 = 345 \) 3. Then use the average rate of change formula: \[ \frac{5 - 345}{1 - (-9)} = \frac{-340}{10} = -34 \] Now for the function \( f(x) = x^2 + x + 4 \) over the interval \([-2, 0]\): 1. Calculate \( f(0) = 0^2 + 0 + 4 = 4 \) 2. Calculate \( f(-2) = (-2)^2 + (-2) + 4 = 4 - 2 + 4 = 6 \) 3. Then plug back into the average rate of change formula: \[ \frac{4 - 6}{0 - (-2)} = \frac{-2}{2} = -1 \] Thus, the overall results for the average rates of change are: 1. Function \( f(x) = 2x^2 - 5x + 4 \) over \([-9, 1]\): \(-21\) 2. Function \( f(x) = 4x^2 - 2x + 3 \) over \([-9, 1]\): \(-34\) 3. Function \( f(x) = x^2 + x + 4 \) over \([-2, 0]\): \(-1\)
