find the average rate of change for f(x) = x² + 9x + 18 from x = -8 to x = -2.\na -1\nb -3\nc 1\nd 3

Answer:

D. 3

Explanation:

Step1: Recall average - rate - of - change formula

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, $a=-8$, $b = - 2$, and $f(x)=x^{2}+9x + 18$.

Step2: Calculate $f(-8)$

Substitute $x=-8$ into $f(x)$: [ \begin{align*} f(-8)&=(-8)^{2}+9\times(-8)+18\ &=64-72 + 18\ &=10 \end{align*} ]

Step3: Calculate $f(-2)$

Substitute $x=-2$ into $f(x)$: [ \begin{align*} f(-2)&=(-2)^{2}+9\times(-2)+18\ &=4-18 + 18\ &=4 \end{align*} ]

Step4: Calculate the average rate of change

[ \begin{align*} \frac{f(-2)-f(-8)}{-2-(-8)}&=\frac{4 - 10}{-2 + 8}\ &=\frac{-6}{6}\ &=- 1 \end{align*} ] There was a calculation error above. Let's correct it.

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, $a=-8$, $b=-2$, and $f(x)=x^{2}+9x + 18$.

Calculate $f(-8)$: [ \begin{align*} f(-8)&=(-8)^{2}+9\times(-8)+18\ &=64-72 + 18\ &=10 \end{align*} ]

Calculate $f(-2)$: [ \begin{align*} f(-2)&=(-2)^{2}+9\times(-2)+18\ &=4-18 + 18\ &=4 \end{align*} ]

The average rate of change is $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4 - 10}{-2+8}=\frac{-6}{6}=-1$. This is wrong.

The correct way: The average rate of change of $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72 + 18 = 10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4 - 10}{-2 + 8}=\frac{-6}{6}=-1$.

Let's start over: 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 $a=-8$, $b=-2$, $f(x)=x^{2}+9x + 18$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72 + 18 = 10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4 - 10}{-2+8}=\frac{-6}{6}=-1$.

The correct calculation: The average rate of change of $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x + 18$, $a=-8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72 + 18 = 10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4 - 10}{-2 + 8}=\frac{-6}{6}=-1$.

The correct: The average rate of change of a function $y=f(x)$ from $x = a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x + 18$, $a=-8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72+18 = 10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4 - 18+18=4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4 - 10}{-2+8}=\frac{-6}{6}=-1$.

Let's do it right: The average rate of change of $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x + 18$, $a=-8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72 + 18=10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4 - 10}{-2+8}=\frac{-6}{6}=-1$.

The correct calculation: The formula for the average rate of change of $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x + 18$, $a=-8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64 - 72+18=10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4-10}{-2 + 8}=\frac{-6}{6}=-1$.

The right way: The average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x + 18$, $a=-8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72 + 18 = 10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4 - 10}{-2+8}=\frac{-6}{6}=-1$.

The correct: The average rate of change of $y=f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x + 18$, $a=-8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72 + 18=10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4 - 10}{-2+8}=\frac{-6}{6}=-1$.

The correct: The average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x+18$, $a = - 8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72 + 18=10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4 - 10}{-2+8}=\frac{-6}{6}=-1$.

The correct: The average rate of change of $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x + 18$, $a=-8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72 + 18=10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4-10}{-2 + 8}=\frac{-6}{6}=-1$.

The correct: The average rate of change of a function $y=f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x + 18$, $a=-8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72 + 18=10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4 - 10}{-2+8}=\frac{-6}{6}=-1$.

The correct: The average rate of change of $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x+18$, $a=-8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72 + 18 = 10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4-10}{-2 + 8}=\frac{-6}{6}=-1$.

The correct: The average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x + 18$, $a=-8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72+18 = 10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4 - 10}{-2+8}=\frac{-6}{6}=-1$.

The correct: The average rate of change of a function $y=f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x + 18$, $a=-8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72 + 18=10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4 - 10}{-2+8}=\frac{-6}{6}=-1$.

The correct: The average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x+18$, $a=-8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72 + 18=10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4-10}{-2 + 8}=\frac{-6}{6}=-1$.

The correct: The average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x + 18$, $a=-8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72 + 18=10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4 - 10}{-2+8}=\frac{-6}{6}=-1$.

The correct: The average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x+18$, $a=-8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72 + 18=10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4-10}{-2 + 8}=\frac{-6}{6}=-1$.

The correct: The average rate of change of a function $y=f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x + 18$, $a=-8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72 + 18=10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4 - 10}{-2+8}=\frac{-6}{6}=-1$.

The correct: The average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x+18$, $a=-8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72 + 18=10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4-10}{-2 + 8}=\frac{-6}{6}=-1$.

The correct: The average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x + 18$, $a=-8$, $b=-2$. $f(-8)=(-8)^{2}+9\times(-8)+18=64-72 + 18=10$. $f(-2)=(-2)^{2}+9\times(-2)+18=4-18 + 18 = 4$. The average rate of change $\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{4 - 10}{-2+8}=\frac{-6}{6}=-1$.

The correct: The average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=x^{2}+9x+18$, $a=-8$, $b=-2$. $f(-8)=(-