Answer
$a.\quad-4$
$b.\quad y=-4x+1$
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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$a.$
$f(5)=-19, \quad f(2)=-7$
Average rate of change of $f$ from 2 to 5:
$\displaystyle \frac{f(5)-f(2)}{5-2}=\frac{-19-(-7)}{3}=-\frac{12}{3}=-4$
$b.$
We have slope $m=-4$ and a point $(2,-7)$.
Point-slope equation of the secant:
$y-y_{1}=m(x-x_{1})$
$y-(-7)=-4(x-2)$
$y=-4x+8-7$
$y=-4x+1$
