To determine the value of $k$ that results in the graph of $g(x)$, let's analyze the graphs provided.

1. The function $f(x) = (x - 4)^2$ represents a parabola with its vertex at $(4, 0)$.
2. The function $g(x) = f(x) + k$ will vertically shift the graph of $f(x)$ by $k$ units.

From the graph, we see:

• The graph of $f(x)$ has its vertex at $(4, 0)$.
• The graph of $g(x)$ has its vertex at $(4, -6)$.

The shift from $(4, 0)$ to $(4, -6)$ indicates that $g(x)$ has been shifted downward by 6 units. Therefore, the value of $k$ is: $k = -6$

Would you like more details or have any questions?

Here are some related questions:

1. What happens if $k$ is positive instead of negative?
2. How do vertical shifts affect the domain and range of a function?
3. What would be the result if we shifted $f(x)$ horizontally instead of vertically?
4. Can $k$ change the shape of the graph of $f(x)$?
5. How do we determine the new vertex of a function after a vertical shift?

Tip: Vertical shifts affect only the y-values of a function, moving the entire graph up or down without changing its shape.