https://file.aiquickdraw.com/mathbot/file/1726280878805k985oedg.jpg

From the graph in the image, it appears that you're dealing with a piecewise linear function, \( g(x) \), that has a sharp corner or "V" shape at the point \( (-2, 2) \). This suggests that \( g(x) \) is likely an absolute value function.

### Step-by-Step Analysis:
1. The vertex of the V-shape is at \( (-2, 2) \).
2. The function has two distinct linear behaviors on either side of the vertex:
   - For \( x < -2 \), the graph appears to have a negative slope.
   - For \( x > -2 \), the graph has a positive slope.

### Equation Form:
The graph suggests a shifted and scaled absolute value function, which typically has the form:
\[
g(x) = a |x + 2| + 2
\]
where:
- \( a \) is the slope of the lines on either side of \( x = -2 \),
- \( x + 2 \) shifts the function horizontally to have the vertex at \( x = -2 \),
- The \( +2 \) shifts the function vertically to the point \( (-2, 2) \).

### Finding \( a \) (the slope):
- The left side of the graph seems to have a slope of -1, and the right side has a slope of 1.
- So, \( a = 1 \).

Thus, the equation of the function \( g(x) \) is:
\[
g(x) = |x + 2| + 2
\]

Would you like a more detailed explanation or have any questions? Here are five related questions you might explore:

1. How can you determine the slope from a graph like this?
2. What happens to the graph if we change the vertex to a different point?
3. How do absolute value functions behave compared to linear functions?
4. What if \( g(x) \) had a different slope on each side of the vertex?
5. Can you describe how shifting and scaling affect the graph of an absolute value function?

**Tip:** When analyzing a graph, always look for key points like the vertex and slopes to determine the function's structure.

Find the equation of the piecewise function g(x) based on the graph.

This example demonstrates how to find the equation of an absolute value function based on its graph. The problem covers the identification of the vertex and the slopes of the linear parts of the function.

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