https://file.aiquickdraw.com/mathbot/file/1726221959075hg6dnlzr.jpg

The graph shown in the image is a transformation of the function \( f(x) = |x| \), which is a V-shaped graph with a vertex at the origin. Let's analyze the given transformation.

### Observations:
- The original vertex of \( f(x) = |x| \) is at the origin, but in this graph, the vertex has shifted.
- The vertex of this new graph appears to be at \( (-2, -3) \), which indicates a horizontal shift by 2 units to the left and a vertical shift by 3 units down.
- The slope of the graph on both sides of the vertex is the same as \( f(x) = |x| \), meaning there is no change in slope (no stretching or compression).

### Transformation Rule:
The general transformation for an absolute value function is:
\[
g(x) = a |x - h| + k
\]
Where:
- \( h \) represents a horizontal shift (to the left if \( h \) is negative),
- \( k \) represents a vertical shift (down if \( k \) is negative),
- \( a \) affects the slope (since \( a = 1 \), there is no vertical stretch or compression).

### Equation:
Here, the graph has shifted horizontally by 2 units to the left, so \( h = -2 \), and vertically by 3 units downward, so \( k = -3 \). Therefore, the equation is:
\[
g(x) = |x + 2| - 3
\]

Would you like more details on any step, or have further questions?

### Related questions:
1. What is the effect of changing the slope \( a \) in an absolute value function?
2. How would the graph change if the transformation had a vertical shift upwards?
3. Can you identify the domain and range of the transformed function \( g(x) = |x + 2| - 3 \)?
4. How does reflecting the graph of \( f(x) = |x| \) across the x-axis affect the equation?
5. What are the key differences between horizontal and vertical transformations in graphs?

### Tip:
Always start by identifying shifts (horizontal and vertical) when graphing transformations of functions like \( |x| \).

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

This example demonstrates how to write the equation for a graph that is a transformation of the function f(x) = |x|. The solution includes analysis of horizontal and vertical shifts in absolute value functions, and how these transformations affect the equation. Suitable for high school students in Grades 9-12.

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