find the average rate of change of ( f(x)=-x^{2}+2x + 2 ) from ( x = 2 ) to ( x = 6 ).\nsimplify your answer…

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function (y = f(x)) from (x = a) to (x = b) is (\frac{f(b)-f(a)}{b - a}). Here, (a = 2), (b = 6), and (f(x)=-x^{2}+2x + 2).

Step2: Calculate (f(2))

Substitute (x = 2) into (f(x)): [ \begin{align*} f(2)&=-(2)^{2}+2\times(2)+2\ &=-4 + 4+2\ &=2 \end{align*} ]

Step3: Calculate (f(6))

Substitute (x = 6) into (f(x)): [ \begin{align*} f(6)&=-(6)^{2}+2\times(6)+2\ &=-36+12 + 2\ &=-22 \end{align*} ]

Step4: Calculate the average rate of change

Substitute (f(2) = 2) and (f(6)=-22) into the formula (\frac{f(b)-f(a)}{b - a}): [ \begin{align*} \frac{f(6)-f(2)}{6 - 2}&=\frac{-22-2}{4}\ &=\frac{-24}{4}\ &=-6 \end{align*} ]

Answer:

(-6)