Question

find the average rate of change of $g(x)=-2x^{2}-x$ from $x = 1$ to $x = 6$. simplify your answer as much as possible.

Explanation:

Step1: Recall the 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, $a = 1$, $b = 6$, and $g(x)=-2x^{2}-x$.

Step2: Calculate $g(6)$

Substitute $x = 6$ into $g(x)$:
$g(6)=-2\times6^{2}-6=-2\times36 - 6=-72-6=-78$.

Step3: Calculate $g(1)$

Substitute $x = 1$ into $g(x)$:
$g(1)=-2\times1^{2}-1=-2 - 1=-3$.

Step4: Calculate the average rate of change

Substitute $g(6)$ and $g(1)$ into the average - rate - of change formula:
$\frac{g(6)-g(1)}{6 - 1}=\frac{-78-(-3)}{5}=\frac{-78 + 3}{5}=\frac{-75}{5}=-15$.

Answer:

$-15$