Average Rate Change B428Bf

1. **State the problem:** We need to find the average rate of change of the function $$h(x)=x^2-2x+3$$ over the intervals $[-1,1]$, $[0,2]$, and $[2,5]$. 2. **Formula for average rate of change:** The average rate of change of a function $h(x)$ from $x=a$ to $x=b$ is given by: $$\text{Average rate of change} = \frac{h(b)-h(a)}{b-a}$$ 3. **Calculate $h(x)$ at the given points:** - $h(-1) = (-1)^2 - 2(-1) + 3 = 1 + 2 + 3 = 6$ - $h(1) = 1^2 - 2(1) + 3 = 1 - 2 + 3 = 2$ - $h(0) = 0^2 - 2(0) + 3 = 0 + 0 + 3 = 3$ - $h(2) = 2^2 - 2(2) + 3 = 4 - 4 + 3 = 3$ - $h(5) = 5^2 - 2(5) + 3 = 25 - 10 + 3 = 18$ 4. **Calculate average rate of change for each interval:** - From $-1$ to $1$: $$\frac{h(1)-h(-1)}{1 - (-1)} = \frac{2 - 6}{1 + 1} = \frac{-4}{2} = -2$$ - From $0$ to $2$: $$\frac{h(2)-h(0)}{2 - 0} = \frac{3 - 3}{2} = \frac{0}{2} = 0$$ - From $2$ to $5$: $$\frac{h(5)-h(2)}{5 - 2} = \frac{18 - 3}{3} = \frac{15}{3} = 5$$ 5. **Final answers:** - Average rate of change from $-1$ to $1$ is $-2$. - Average rate of change from $0$ to $2$ is $0$. - Average rate of change from $2$ to $5$ is $5$.