https://file.aiquickdraw.com/mathbot/file/1726443386603dxua1tri.jpg

The graph shown represents a transformation of the basic absolute value function \( f(x) = |x| \). To determine the equation for \( g(x) \), we need to identify the transformations applied to \( f(x) \).

### Observations:
1. **Reflection**: The graph opens downward, suggesting that the function has been reflected over the x-axis. This means a negative sign is introduced.
2. **Vertical Stretch**: The "V" shape of the graph is more pronounced, indicating a vertical stretch. The slope appears to be steeper than the standard \( f(x) = |x| \).
3. **Translation**: The graph is centered at the origin, so no horizontal or vertical shifts are evident.

### Step-by-Step Analysis:
1. The reflection introduces a negative sign, so we start with \( -|x| \).
2. The slope on each side of the vertex is 2, indicating a vertical stretch by a factor of 2.

Thus, the transformation is described by the function:
\[
g(x) = -2|x|
\]

This is the formula for the transformed graph.

Would you like further clarification or more details? Here are 5 related questions:

1. What are other types of transformations (besides reflection and stretch) that can be applied to functions?
2. How would you modify the function if the graph had a horizontal shift?
3. What happens if there is both a vertical stretch and a horizontal compression?
4. How does a horizontal reflection affect the graph of a function like \( f(x) = |x| \)?
5. What are the key differences between vertical and horizontal transformations?

**Tip**: Always check the vertex of an absolute value function graph, as it gives clues about translations and reflections.

The graph above is a transformation of the function f(x) = |x|. Write a formula for the function graphed above.

Learn how to transform the absolute value function f(x) = |x| to g(x) = -2|x| by applying a reflection and vertical stretch. This problem explores transformations like reflection and vertical stretch, suitable for students in Grades 9-12.

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