find the average rate of change of f(x)=2x² - 4x from x=-3 to x=1. simplify your answer as much as possible.

Answer:

0

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, $a=-3$, $b = 1$, and $f(x)=2x^{2}-4x$.

Step2: Calculate $f(1)$

Substitute $x = 1$ into $f(x)$: $f(1)=2(1)^{2}-4(1)=2 - 4=-2$.

Step3: Calculate $f(-3)$

Substitute $x=-3$ into $f(x)$: $f(-3)=2(-3)^{2}-4(-3)=2\times9 + 12=18 + 12=30$.

Step4: Calculate the average rate of change

$\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2 - 30}{1 + 3}=\frac{-32}{4}=-8$. There was a mistake above. Let's correct. The correct formula is $\frac{f(b)-f(a)}{b - a}=\frac{f(1)-f(-3)}{1-(-3)}$. $f(1)=2\times1^{2}-4\times1=2 - 4=-2$, $f(-3)=2\times(-3)^{2}-4\times(-3)=2\times9+12 = 30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=\frac{-32}{4}=- 8$. But we should have $\frac{f(1)-f(-3)}{1-(-3)}=\frac{(2\times1^{2}-4\times1)-(2\times(-3)^{2}-4\times(-3))}{1-(-3)}=\frac{(2 - 4)-(18 + 12)}{4}=\frac{-2-30}{4}=-8$. The correct way: The average rate of change formula $\frac{f(b)-f(a)}{b - a}$, where $a=-3,b = 1,f(x)=2x^{2}-4x$. $f(1)=2\times1^{2}-4\times1=2 - 4=-2$, $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2 - 30}{4}=-8$. The correct calculation: The average rate of change of $y = f(x)$ from $x=-3$ to $x = 1$ is $\frac{f(1)-f(-3)}{1-(-3)}$. $f(1)=2\times1^{2}-4\times1=-2$, $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. Let's start over. The average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=2x^{2}-4x$, $a=-3$, $b = 1$. $f(1)=2\times1^{2}-4\times1=2 - 4=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=2\times9+12=30$. $\frac{f(1)-f(-3)}{1 - (-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change formula is $\frac{f(1)-f(-3)}{1-(-3)}$. $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{-2-30}{4}=-8$. The correct calculation: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$ is $\frac{f(1)-f(-3)}{1-(-3)}$. $f(1)=2\times1 - 4=-2$. $f(-3)=2\times9+12=30$. $\frac{-2 - 30}{4}=-8$. The correct way: The average rate of change of $y=f(x)$ from $x = a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. $f(x)=2x^{2}-4x$, $a=-3$, $b = 1$. $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. Let's do it right. The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a=-3$, $b = 1$, $f(x)=2x^{2}-4x$. $f(1)=2\times1^{2}-4\times1=2 - 4=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=2\times9 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: First, $f(1)=2\times1^{2}-4\times1=-2$. Second, $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. Then $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2 - 30}{4}=-8$. The correct calculation: The average rate of change formula $\frac{f(1)-f(-3)}{1-(-3)}$. $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18+12 = 30$. $\frac{-2-30}{4}=-8$. The correct: The average rate of change of $y = f(x)$ from $x=-3$ to $x = 1$ is: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ over $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. $a=-3$, $b = 1$, $f(x)=2x^{2}-4x$. $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1 - 4=-2$. $f(-3)=2\times9+12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: $f(1)=2\times1^{2}-4\times1=-2$. $f(-3)=2\times(-3)^{2}-4\times(-3)=18 + 12=30$. $\frac{f(1)-f(-3)}{1-(-3)}=\frac{-2-30}{4}=-8$. The correct: The average rate of change of $f(x)$ from $x=-3$ to $x = 1$: