We find the vertex by setting what is between the
absolute value bars equal to 0.

y = |x + 2| + 1

x + 2 = 0
    x = -2

Then substitute that into 
y = |x + 2| + 1
y = |-2 + 2| + 1
y = 0 + 1 
y = 1

So the vertex is (-2,1).  Get a couple more points, 
say (0,3) and (-4,3).  Since we are told to graph it
only on the interval -5 < x < 1, we draw little open 
circles at the endpoints to show that the graph does 
not include those endpoints. 

 

The domain is {x | -5 < x < 1} and the range is {y | 1 < y < 4}.

Edwin