find the average rate of change of g(x)=-x² + 4x + 1 from x = 3 to x = 6. simplify your answer as much as…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = g(x)$ from $x = a$ to $x = b$ is $\frac{g(b)-g(a)}{b - a}$. Here, $a = 3$, $b = 6$, and $g(x)=-x^{2}+4x + 1$.

Step2: Calculate $g(6)$

Substitute $x = 6$ into $g(x)$: $g(6)=-(6)^{2}+4\times6 + 1=-36 + 24+1=-11$.

Step3: Calculate $g(3)$

Substitute $x = 3$ into $g(x)$: $g(3)=-(3)^{2}+4\times3 + 1=-9 + 12+1 = 4$.

Step4: Calculate the average rate of change

$\frac{g(6)-g(3)}{6 - 3}=\frac{-11 - 4}{3}=\frac{-15}{3}=-5$.

Answer:

$-5$