Answer
(a) $1$
(b) $y=x+3$
Work Step by Step
(a) Given $g(x)=x^2+1$, the average rate of change $R$ is given by the formula $$R=\dfrac{g(x_2)-g(x_1)}{x_2-x_1}$$
From $x_1=-1$ to $x_2=2$, we have :
$R=\dfrac{g(2)-g(-1)}{2-(-1)}=\dfrac{(2)^2+1-[(-1)^2+1]}{3}=1$
(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 $(2, g(2))$ is on the line, find $g(2)$:
$$g(2)=(2)^2+1=5$$
Thus, the line contains the point $(2, 5)$.
Therefore, using the point-slope form, the equation of the line is:
$$\begin{align*}
y-5&=1(x-2)\\
y-5&=x-2\\
y&=x-2+5\\
y&=x+3
\end{align*}$$
