Answer
(a) $-5$
(b) $y=-5x$
Work Step by Step
(a) Given $h(x)=-2x^2+x$, the average rate of change $R$ is given by the formula
$$R=\frac{h(x_2)-h(x_1)}{x_2-x_1}$$
From $x_1=0$ to $x_2=3$, we have:
$$R=\frac{h(3)-h(0)}{3-0}=\frac{-2(3)^2+(3)-(-2(0)^2+(0))}{3}=-5$$
(b) The slope of the secant line is the average rate of change $m=-5$.
To find the equation of the line, we need to use the coordinates of one point on the line. Since the line contains $(0,h(0))$, find $h(0)$.
$h(0)=-2(0^2)+0=0$
Thus, the line contains the point $(0, 0)$.
Therefore, using the point-slope form, the equation of the line is:
$$\begin{align*}
y-0&=-5(x-0)\\
y&=-5x
\end{align*}$$
