Answer
(a) $-4$
(b) $y=-4x+1$
Work Step by Step
(a) Given $f(x)=-4x+1$, the average rate of change is given by the formula $$R=\dfrac{f(x_2)-f(x_1)}{x_2-x_1}$$
From $x_1=2$ to $x_2=5$, we have:
$R=\dfrac{f(5)-f(2)}{5-2}=\dfrac{-4(5)+1-(-4(2)+1)}{3}=-4$
(b) Slope for the line is the average rate of change $m=-4$.
To find the equation of the line, we need to find the coordinates of one point o the line.
Since the line contains the point $(2, f(2))$, solve for $f(2)$:
$f(2)=-4(2)+1=-7$
Thus, the line contains the point $(2, -7)$.
Therefore, the equation of the line sing the point-slope form is:
$$\begin{align*}
y-(-7)&=-4(x-2)\\
y+7&=-4x+8\\
y&=-4x+8-7\\
y&=-4x+1
\end{align*}$$
