https://file.aiquickdraw.com/mathbot/file/1727665075770przxap82.jpg

From the graph you provided, we can observe that the function \( g(x) \) is a transformation of the absolute value function \( f(x) = |x| \).

### Analysis of the transformation:

1. **Vertical stretch/shrink**: 
   The slope of the lines forming the "V" shape in \( g(x) \) is 2, as the graph rises by 2 units for every 1 unit horizontally. This indicates that there is a vertical stretch by a factor of 2. Thus, \( a = 2 \).

2. **Horizontal shift**:
   The vertex of the original \( f(x) = |x| \) is at the origin (0, 0). However, in \( g(x) \), the vertex has shifted to \( (-2, 0) \). This indicates a horizontal shift to the left by 2 units. Thus, \( h = -2 \).

3. **Vertical shift**:
   The graph does not appear to have shifted vertically, as the vertex remains on the x-axis. Therefore, \( k = 0 \).

### Conclusion:
The function rule for \( g(x) \) is in the form:
\[
g(x) = a |x - h| + k
\]
Substituting \( a = 2 \), \( h = -2 \), and \( k = 0 \), we get:
\[
g(x) = 2 |x + 2|
\]

This is the function rule for \( g(x) \).

Would you like more details or have any questions?

Here are some related questions to explore:

1. How do you determine the vertex of an absolute value function from its equation?
2. What is the significance of the parameter \( a \) in an absolute value transformation?
3. How does the graph change when the value of \( h \) is positive versus negative?
4. What happens if the value of \( k \) is non-zero in the absolute value transformation?
5. How would the graph look if \( a \) were a fraction instead of an integer?

**Tip**: When analyzing transformations, always start by identifying the vertex, then check for any stretching, compressions, or shifts.

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

Learn how to determine the transformation of the absolute value function f(x) = |x| to find g(x). This problem involves identifying vertical stretches, horizontal shifts, and other key function transformations. Suitable for students in Grades 8-10.

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