Answer
Concave up
Work Step by Step
Between $x=-2$ and $x=0$, the average rate of change is:
$$
\frac{\Delta f(x)}{\Delta x}=\frac{f(0)-f(-2)}{0-(-2)}=\frac{(0-1)^2+2-\left((-2-1)^2+2\right)}{2}=-4 .
$$
Between $x=0$ and $x=2$, the average rate of change is:
$$
\frac{\Delta f(x)}{\Delta x}=\frac{f(2)-f(0)}{2-0}=\frac{(2-1)^2+2-\left((0-1)^2+2\right)}{2}=0 .
$$
Between $x=2$ and $x=4$, the average rate of change is:
$$
\frac{\Delta f(x)}{\Delta x}=\frac{f(4)-f(2)}{4-2}=\frac{(4-1)^2+2-\left((2-1)^2+2\right)}{2}=4
$$
Since rates of change are increasing, the graph of $f(x)$ is concave up.
