This will be the graph of #y = |x|# that has been reflected over that opens downward and has had a vertical transformation of #4# units.
The domain, like #y= |x|#, will be #x in RR#. The range of any absolute value function depends on the maximum/minimum of that function.
The graph of #y = |x|# would open upward, so it would have a minimum, and the range would be #y ≥ C#, where #C# is the minimum.
However, our function opens downwards, so we will have a maximum. The vertex, or maximum point of the function will occur at #(p, q)#, in #y = a|x - p| + q#. Hence, our vertex is at #(0, -4)#. Our true "maximum" will occur at #q#, or the y-coordinate. So, the maximum is #y = -4#.
We know the maximum, and that the function opens down. Hence, the range will be #y ≤ -4#.
Hopefully this helps!
