Answer
(a) $-1$
(b) $y=-x$
Work Step by Step
(a) Given $g(x)=x^2-2$, the average rate of change $R$ is given by the formula
$$R=\dfrac{g(x_2)-g(x_1)}{x_2-x_1}$$
From $x_1=-2$ to $x_2=1$, we have:
$R=\dfrac{g(1)-g(-2)}{1-(-2)}=\dfrac{(1)^2-2-[(-2)^2-2]}{3}=-1$
(b) The slope for the secant line is the average rate of change $m=-1$.
To find the equation of the line, we need to find the coordinates of one point.
Since the line contains the point $(1, g(1))$, find the value of $g(1)$:
$g(1)=(1)^2-2=-1$
Thus, the line contains the point $(1,-1)$.
Therefore, using the point-slope form of a line's equation yields:
$$\begin{align*}
y-(-1)&=-1(x-1)\\
y+1=-x+1\\
y&=-x+1-1\\
y&=-x
\end{align*}$$
