For an average rate of change all that matters is the endpoints. If -1≤x≤0, then the endpoints are -1 and 0. So on your graph draw a vertical line from x=-1 on the x-axis up to where it meets the curve. It should meet the curve at y=f(x)=2, so that's 2 on the vertical axis. This corresponds to f(-1)=2.
x=0 happens to be the vertical y-axis so read off where the curve crosses the y-axis. This is f(0). This should be at y=f(x)=-2. This corresponds to f(0)=-2.
Join these endpoints with a straight line, that is, join (-1,2) to (0,-2). The slope of this line is the average rate of change and we just work out (f(0)-f(-1))/(0-(-1))=(-2-2)/1=-4, a negative rate of change, which means that the curve has dropped from its starting point at (-1,2) to its endpoint at (0,-2). You can count the number of units it's dropped to confirm the calculation. And the curve has dropped 4 units vertically but over a unit interval (-1 to 0 measures one unit). Hence the average rate is -4, as the calculation shows.