Question

find the average rate of change for each equation.

1. $f(x) = x^2 - 7x + 12$, $f(0)$ to $f(4)$
2. $y = 3x + 8$, $0, 6$
3. $g(x) = x^2 - 9x + 18$, $g(3)$ to $g(4)$
4. $y = -x + 1$, $-4,8$

Explanation:

Step1: Define average rate of change formula

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

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For 1. $f(0)$ to $f(4)$ where $f(x)=x^2-7x+12$

Step1: Calculate $f(0)$

$f(0)=0^2-7(0)+12=12$

Step2: Calculate $f(4)$

$f(4)=4^2-7(4)+12=16-28+12=0$

Step3: Compute average rate of change

$\frac{f(4)-f(0)}{4-0}=\frac{0-12}{4-0}=\frac{-12}{4}=-3$

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For 2. $[0,6]$ where $y=3x+8$

Step1: Calculate $y(0)$

$y(0)=3(0)+8=8$

Step2: Calculate $y(6)$

$y(6)=3(6)+8=18+8=26$

Step3: Compute average rate of change

$\frac{y(6)-y(0)}{6-0}=\frac{26-8}{6-0}=\frac{18}{6}=3$

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For 3. $g(3)$ to $g(4)$ where $g(x)=x^2-9x+18$

Step1: Calculate $g(3)$

$g(3)=3^2-9(3)+18=9-27+18=0$

Step2: Calculate $g(4)$

$g(4)=4^2-9(4)+18=16-36+18=-2$

Step3: Compute average rate of change

$\frac{g(4)-g(3)}{4-3}=\frac{-2-0}{4-3}=\frac{-2}{1}=-2$

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For 4. $[-4,8]$ where $y=-x+1$

Step1: Calculate $y(-4)$

$y(-4)=-(-4)+1=4+1=5$

Step2: Calculate $y(8)$

$y(8)=-(8)+1=-8+1=-7$

Step3: Compute average rate of change

$\frac{y(8)-y(-4)}{8-(-4)}=\frac{-7-5}{8+4}=\frac{-12}{12}=-1$

Answer:

1. $-3$
2. $3$
3. $-2$
4. $-1$