Question

find the average rate of change of $g(x) = -x^2 + 4x + 2$ from $x = 3$ to $x = 7$. simplify your answer as much as possible.

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function $g(x)$ from $x=a$ to $x=b$ is $\frac{g(b)-g(a)}{b-a}$.

Step2: Calculate $g(7)$

Substitute $x=7$ into $g(x)$:
$g(7)=-(7)^2 + 4(7)+2 = -49 + 28 + 2 = -19$

Step3: Calculate $g(3)$

Substitute $x=3$ into $g(x)$:
$g(3)=-(3)^2 + 4(3)+2 = -9 + 12 + 2 = 5$

Step4: Compute average rate of change

Substitute values into the formula:
$\frac{g(7)-g(3)}{7-3}=\frac{-19 - 5}{4}=\frac{-24}{4}=-6$

Answer:

$-6$