https://file.aiquickdraw.com/mathbot/file/172908207886982doxgv8.jpg

The graph in the image shows a V-shaped function, which is typical of an absolute value function. The key is to identify the vertex of the graph and any shifts.

1. **Vertex**: The vertex is at \((-1, -2)\), meaning the graph has been shifted 1 unit left and 2 units down from the origin.

2. **General form of an absolute value function**:
   \[
   y = a |x - h| + k
   \]
   where \((h, k)\) is the vertex.

   In this case, the vertex is \((-1, -2)\), so \(h = -1\) and \(k = -2\). This means the equation should look like:
   \[
   y + 2 = |x + 1|
   \]
   to account for the downward shift of 2 and the leftward shift of 1.

Thus, the correct answer from the options provided is:

\[
y + 2 = |x + 1|
\]

Would you like further explanation on this? Here are some related questions to deepen understanding:

1. What does the "absolute value" part of the equation represent graphically?
2. How would the graph change if the equation were \(y - 2 = |x - 1|\)?
3. What is the slope of the two lines forming the "V" shape in an absolute value function?
4. How would you shift an absolute value function up or down?
5. What role does the coefficient of the absolute value term (if any) play in stretching or compressing the graph?

**Tip:** The vertex form of an absolute value function is key to understanding transformations like shifts and reflections.

Identify the equation for the absolute value graph displayed, with the options given as: y + 2 = |x - 1|, y - 2 = |x - 1|, y + 2 = |x + 1|, and y - 2 = |x + 1|.

Learn how to identify the equation of an absolute value graph, focusing on shifts and transformations. This problem demonstrates how to use the vertex form of an absolute value function to match equations with the graph, ideal for students in Grades 8-10.

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