Question

given the function $g(x)=-x^{2}+3x + 5$, determine the average rate of change of the function over the interval $-4leq xleq6$.

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = g(x)$ over the interval $[a,b]$ is $\frac{g(b)-g(a)}{b - a}$. Here, $a=-4$ and $b = 6$.

Step2: Calculate $g(a)$

Substitute $x=-4$ into $g(x)=-x^{2}+3x + 5$.
$g(-4)=-(-4)^{2}+3\times(-4)+5=-16-12 + 5=-23$.

Step3: Calculate $g(b)$

Substitute $x = 6$ into $g(x)=-x^{2}+3x + 5$.
$g(6)=-6^{2}+3\times6+5=-36 + 18+5=-13$.

Step4: Calculate average rate of change

Use the formula $\frac{g(b)-g(a)}{b - a}=\frac{-13-(-23)}{6-(-4)}=\frac{-13 + 23}{6 + 4}=\frac{10}{10}=1$.

Answer:

$1$