https://file.aiquickdraw.com/mathbot/file/1728063446296s9w6hivs.jpg

To identify the equation that matches the graph, we need to analyze the given graph and the equations provided. Let's start by examining the key features of the graph:

1. **Key Features of the Graph:**
   - The graph shows an absolute value function, \( y = a|x + b| + c \).
   - The vertex (turning point) of the graph is at the point \((-1, 3)\).
   - The slopes of the lines on either side of the vertex are different, indicating different coefficients in front of the absolute value.

2. **Given Equations:**
   The equations provided are of the form:
   \[
   -2|x + 1| + 3 = mx - k
   \]
   where the slopes \( m \) and constants \( k \) differ among the options.

3. **Steps to Match the Graph:**
   - **Identify the vertex**: The vertex of the absolute value function is at \((-1, 3)\), which aligns with the \(|x + 1|\) term.
   - **Determine the slopes**: 
     - The slope of the line to the right of the vertex can be found using two points. Let's use \((0, 1)\) and \((2, -1)\):
       \[
       \text{slope} = \frac{-1 - 1}{2 - 0} = \frac{-2}{2} = -1
       \]
     - The slope of the line to the left of the vertex is not directly needed but can be checked to verify the coefficient.

4. **Compare with Given Equations**:
   - Check each given equation by substituting points and verifying the slopes. 
   
Let's solve for one of the points:

#### Option A:
\[
-2|x + 1| + 3 = \frac{2}{5}x - \frac{11}{5}
\]
When \( x = 0 \):
\[
-2|0 + 1| + 3 = \frac{2}{5}(0) - \frac{11}{5}
\]
\[
-2(1) + 3 = -\frac{11}{5}
\]
\[
1 \neq -\frac{11}{5}
\]

Option A does not match the graph.

#### Option B:
\[
-2|x + 1| + 3 = \frac{2}{3}x - \frac{11}{3}
\]
When \( x = 0 \):
\[
-2|0 + 1| + 3 = \frac{2}{3}(0) - \frac{11}{3}
\]
\[
-2(1) + 3 = -\frac{11}{3}
\]
\[
1 \neq -\frac{11}{3}
\]

Option B does not match the graph.

#### Option C:
\[
-2|x + 1| + 3 = \frac{2}{3}x - \frac{11}{3}
\]
When \( x = 0 \):
\[
-2|0 + 1| + 3 = \frac{2}{3}(0) - \frac{11}{3}
\]
\[
-2(1) + 3 = -\frac{11}{3}
\]
\[
1 \neq -\frac{11}{3}
\]

Option C does not match the graph.

#### Option D:
\[
-2|x + 1| + 3 = \frac{2}{5}x - \frac{11}{5}
\]
When \( x = 0 \):
\[
-2|0 + 1| + 3 = \frac{2}{5}(0) - \frac{11}{5}
\]
\[
-2(1) + 3 = -\frac{11}{5}
\]
\[
1 \neq -\frac{11}{5}
\]

Option D does not match the graph.

**Conclusion:**
None of the provided equations directly match the point verification we did. Therefore, it seems like there might be a mistake in the provided options, or a different method should be used to analyze the matching.

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How can you derive the equation of an absolute value function from a graph?
2. What are the common transformations applied to absolute value functions?
3. How do you find the vertex of an absolute value function?
4. What is the significance of the slope in an absolute value function?
5. How can you check the correctness of a given equation with its graph?

**Tip:** Always verify multiple points on the graph to ensure the equation matches the graph accurately.

Choose the equation that matches the graph.

This math problem involves matching a graph of an absolute value function with its corresponding equation. Key concepts include analyzing the vertex, determining the slopes of the lines, and checking the equation form. Suitable for Grades 8-10 students learning about algebra and graphing absolute value functions.

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