The graph shown in the image represents a V-shaped function, which is characteristic of an absolute value function. Let’s analyze the function step by step:

1. Vertex of the Graph:

The vertex of the graph is at the point $(4, -10)$. This means the function has been shifted right by 4 units and down by 10 units.

2. Equation Format:

An absolute value function in the form of $g(x) = |x - h| + k$ has a vertex at $(h, k)$. So, the equation will look like: $g(x) = |x - 4| - 10$

This represents a shift to the right by 4 units (indicated by $x - 4$) and a shift down by 10 units (indicated by $-10$).

Conclusion:

The correct equation representing the graph is: $g(x) = |x - 4| - 10$

Thus, the first option $\boxed{g(x) = |x - 4| - 10}$ is the correct answer.

Let me know if you'd like more details or have any other questions!

Here are 5 related questions for further understanding:

1. How does the graph of $g(x) = |x - h| + k$ change when you modify $h$ and $k$?
2. What happens to the graph of an absolute value function when a coefficient is added in front of $|x - h|$?
3. Can you describe how vertical and horizontal translations work for any function?
4. What would be the effect on the graph if the equation were $g(x) = -|x - 4| - 10$?
5. How do you determine the slope of the arms of an absolute value function?

Tip: Always check the vertex and shifts when dealing with absolute value functions in graph form!