https://file.aiquickdraw.com/mathbot/file/17255918850650bponqbi.jpg

The graph in the image represents a transformation of the function \( f(x) = |x| \), which is the absolute value function.

To determine the function rule for \( g(x) \), let's analyze the transformation:

1. The vertex of the parent function \( f(x) = |x| \) is at the origin \( (0, 0) \). In the graph of \( g(x) \), the vertex has shifted to \( (-3, 2) \).
   
2. The slope of the lines on both sides of the vertex appears unchanged, meaning that the function has not been vertically stretched or compressed.

This suggests the following transformations:
- Horizontal shift to the left by 3 units.
- Vertical shift up by 2 units.

Thus, the function rule for \( g(x) \) is:

\[
g(x) = |x + 3| + 2
\]

This is the transformed absolute value function where:
- \( +3 \) inside the absolute value shifts the graph to the left.
- \( +2 \) outside shifts the graph upwards.

Would you like more details on the transformation steps? 

Here are 5 related questions:
1. How do vertical and horizontal shifts affect the graph of a function?
2. What would happen if we added a coefficient in front of the absolute value, such as \( 2|x + 3| + 2 \)?
3. How can we use transformations to reflect a graph across the x-axis or y-axis?
4. What is the general form of a transformed absolute value function?
5. Can transformations like stretching or compressing occur together with shifts?

**Tip:** When analyzing graph transformations, always start by identifying the vertex and check for any changes in slope or direction.

The graph shows g(x), which is a transformation of f(x) = |x|. Write the function rule for g(x).

Learn how to find the transformation of the absolute value function f(x) = |x| into g(x) = |x + 3| + 2. This solution covers vertical and horizontal shifts in graph transformations, perfect for students in Grades 8-10.

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