What is the vertex of the absolute value function defined by f (x) = |x+5|+7?

The vertex of the function f (x) = |x - a| is in (a, 0).

f (x) + a - shift the graph of function f (x) a units up.

Therefore:

The verex of y = |x + 5| is in (-5, 0).

The vertex of y + 7 = f (x) = |x + 5| + 7 is in (-5, 0 + 7) = (-5, 7).