To determine which form of the function \(g\) most quickly reveals the vertex, we should analyze each option:
Choice A: \[ g(x) = \frac{1}{2}(x - 12)(x - 4) \]
This is in factored form, and while you can find the vertex from this by first finding the x-coordinate of the vertex as the midpoint of the roots \(x = 12\) and \(x = 4\): \[ x = \frac{12 + 4}{2} = 8. \] You would then need to substitute \(x = 8\) back into the function to find the y-coordinate.
Choice B: \[ g(x) = \frac{1}{2}(x - 8)^2 - 8 \]
This is in vertex form \(g(x) = a(x - h)^2 + k\). Here, the vertex is \((h, k) = (8, -8)\). This form directly gives us the vertex without further calculations.
Choice C: \[ g(x) = \frac{1}{2}x^2 - 8x + 24 \]
This is in standard form, and you can find the vertex by completing the square or using the vertex formula \(x = -\frac{b}{2a}\): \[ x = -\frac{-8}{2 \cdot \frac{1}{2}} = \frac{8}{1} = 8. \] Then you would plug \(x = 8\) back into the function to find the y-coordinate.
From these forms, Choice B most quickly reveals the vertex because it gives it directly as \((8, -8)\) without any need of additional calculations.
So, the vertex is: \[ \boxed{(8, -8)}. \]