Answer
$a.\quad 5$
$b.\quad y=5x-2$
Work Step by Step
Average rate of change from $a$ to $b$
$=\displaystyle \frac{\Delta y}{\Delta x}=\frac{f(b)-f(a)}{b-a},\quad a\neq b$
The average rate of change of a function from $a$ to $b$ equals the slope of the secant line containing the two points $(a,f(a))$ and $(b,f(b))$ on its graph.
$m_{sec}=\displaystyle \frac{f(b)-f(a)}{b-a}$
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$a.$
$f(3)=13, \quad f(1)=3$
Average rate of change of $f$ from 1 to $3:$
$\displaystyle \frac{f(3)-f(1)}{3-1}=\frac{13-3}{3-1}=\frac{10}{2}=5$
$b.$
We have slope $m=5$ and a point $(1,3)$.
Point-slope equation of the secant:
$y-y_{1}=m(x-x_{1})$
$y-3=5(x-1)$
$y=5x-5+3$
$y=5x-2$