https://file.aiquickdraw.com/mathbot/file/1726677233445b2ezdpw4.jpg

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!

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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!

Which equation represents the function graphed on the coordinate plane?

Understand how to identify the correct equation for the graphed absolute value function, focusing on shifts and translations. This lesson helps students determine transformations in absolute value equations, using the formula g(x) = |x - h| + k.

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