Answer
(a) $\frac{1}{2}$
(b) the slope is also $\frac{1}{2}$
Work Step by Step
(a) We find the average rate of change from $x=a$ to $x=a+h$:
$\displaystyle \frac{f(a+h)-f(a)}{(a+h)-a}=\frac{[\frac{1}{2}(a+h)+3]-[\frac{1}{2}a+3]}{h}=\frac{\frac{1}{2}a+\frac{1}{2}h+3-\frac{1}{2}a-3}{h}=\frac{\frac{1}{2}h}{h}=\frac{1}{2}$
(b) A line has the form $f(x)=mx+b$, with $m=slope$.
For the line $f(x)=\displaystyle \frac{1}{2}x+3$, the slope is $\displaystyle \frac{1}{2}$, which equals what we got in (a).
