Answer
$a.\quad-2$
$b.\quad 0$
$c.\quad5$
Work Step by Step
Average rate of change from $a$ to $b$
$=\displaystyle \frac{\Delta y}{\Delta x}=\frac{f(b)-f(a)}{b-a},\quad a\neq b$
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$a.$ from $x=-1$ to $x=1:$
$h(1)=(1)^{2}-2(1)+3=2$
$h(-1)=(-1)^{2}-2(-1)+3=6$
$\displaystyle \frac{h(1)-h(-1)}{1-(-1)}=\frac{(2)-(6)}{2}=\frac{-4}{2}=-2$
$b.$ from $x=0$ to $x=2:$
$h(2)=(2)^{2}-2(2)+3=3$
$h(0)=3$
$\displaystyle \frac{h(2)-h(0)}{2-0}=\frac{(3)-(3)}{2}=\frac{0}{2}=0$
$c.$ from $x=2$ to $x=5:$
$h(5)=(5)^{2}-2(5)+3=18$
$h(2)=(2)^{2}-2(2)+3=3$
$\displaystyle \frac{h(5)-h(2)}{5-2} =\frac{(18)-(3)}{3}=\frac{15}{3}=5$
