https://file.aiquickdraw.com/mathbot/file/1734027988594zh23ogy0.jpg

The graph shows a transformed absolute value function. Let’s analyze it step by step to find the correct equation:

### Key Observations:
1. The basic absolute value function is \( f(x) = |x| \).
2. The graph is flipped vertically, meaning there is a negative sign in front of the absolute value (\(-|x|\)).
3. The vertex (the point where the "V" changes direction) is located at \( (1, 4) \).
   - This means the function is shifted **1 unit to the right** and **4 units up**.
4. Vertical and horizontal shifts affect the function as follows:
   - Horizontal shift: \( |x - h| \) (to the right if \( h > 0 \), left if \( h < 0 \)).
   - Vertical shift: \( +k \) (up if \( k > 0 \), down if \( k < 0 \)).

### The equation:
Considering the transformations:
- Reflect vertically: Add \(-\) in front of \( |x| \).
- Shift right by 1: Replace \( x \) with \( (x - 1) \).
- Shift up by 4: Add \( +4 \) at the end.

The equation is:
\[
f(x) = -|x - 1| + 4
\]

### Correct Answer:
**A. \( f(x) = -|x - 1| + 4 \)**

Let me know if you want further details or explanations!

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### Related Questions:
1. How do you determine horizontal shifts in absolute value functions?
2. What effect does the negative sign have on the absolute value graph?
3. How can you determine the vertex of a transformed absolute value function?
4. Why is the vertical shift added outside the absolute value expression?
5. How does the transformation change if the graph moves to the left instead of the right?

### Tip:
When working with transformations, always focus on the vertex first, as it simplifies finding horizontal and vertical shifts!

Write an equation for the graphed function by using transformations of the graphs of one of the basic functions.

Learn how to determine the equation for a graphed absolute value function. Understand transformations, including vertical reflections, shifts, and translations, with a step-by-step solution.

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