Question

2.9 rate of change with equations - day 4
find the average rate of change for each equation:

1. ( f(x) = x^2 - 3x + 12 ) from ( f(1) ) to ( f(2) )
2. ( y = x + 8 ) on (0, 6)
3. ( g(x) = -2x^2 - 2x + 18 ) from ( g(3) ) to ( g(4) )
4. ( y = x - 1 ) on (-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 Problem 1: $f(x)=x^2-3x+12$, from $f(1)$ to $f(2)$

Step1: Calculate $f(1)$

$f(1)=(1)^2-3(1)+12=1-3+12=10$

Step2: Calculate $f(2)$

$f(2)=(2)^2-3(2)+12=4-6+12=10$

Step3: Compute average rate

$\frac{f(2)-f(1)}{2-1}=\frac{10-10}{1}=0$

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For Problem 2: $y=x+8$, over $[0,6]$

Step1: Calculate $y(0)$

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

Step2: Calculate $y(6)$

$y(6)=6+8=14$

Step3: Compute average rate

$\frac{y(6)-y(0)}{6-0}=\frac{14-8}{6}=\frac{6}{6}=1$

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For Problem 3: $g(x)=-2x^2-2x+18$, from $g(3)$ to $g(4)$

Step1: Calculate $g(3)$

$g(3)=-2(3)^2-2(3)+18=-18-6+18=-6$

Step2: Calculate $g(4)$

$g(4)=-2(4)^2-2(4)+18=-32-8+18=-22$

Step3: Compute average rate

$\frac{g(4)-g(3)}{4-3}=\frac{-22-(-6)}{1}=-16$

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For Problem 4: $y=x-1$, over $[-4,8]$

Step1: Calculate $y(-4)$

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

Step2: Calculate $y(8)$

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

Step3: Compute average rate

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

Answer:

1. $0$
2. $1$
3. $-16$
4. $1$