https://file.aiquickdraw.com/mathbot/file/17300453034948zffng79.jpg

Let's carefully solve this question about transformations of absolute value functions.

### Given:
1. **Reflection across the x-axis**: This flips the graph upside down, introducing a negative sign in front of the function.
2. **Vertical shift right 2 units**: This shifts the input of the function horizontally, which corresponds to modifying \(x\) as \(x - 2\).
3. **Horizontal shift down 7 units**: This is a **vertical shift** (down by 7 units), so it subtracts 7 from the function output.

### Basic Absolute Value Function:
The basic function is \(y = |x|\).

### Applying the Transformations:
1. **Reflection across the x-axis**: Add a negative sign in front, making it \(y = -|x|\).
2. **Shift right 2 units**: Modify it to \(y = -|x - 2|\).
3. **Shift down 7 units**: Subtract 7 from the function, resulting in:
   \[
   y = -|x - 2| - 7
   \]

### Final Answer:
The correct transformed equation is \(y = -|x - 2| - 7\).

### Matching with the Options:
This matches with the **third option**:  
\[
y = -|x - 2| - 7
\]

So, the correct answer is the **third choice**.

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Do you need further explanation, or have any follow-up questions? 

Here are 5 related questions for further practice:
1. What happens if a function is reflected across the y-axis instead?
2. How would a vertical shift **up** change the equation?
3. What is the difference between horizontal and vertical shifts mathematically?
4. How can we find the vertex of an absolute value function?
5. How would the equation change if we shift **left** instead of right?

**Tip:** To keep transformations organized, apply horizontal transformations inside the absolute value and vertical ones outside.

Write an absolute value function given the following transformations: -Reflection across the x-axis -Vertical shift right 2 units -Horizontal shift down 7 units.

Learn how to write an absolute value function that involves a reflection across the x-axis, a vertical shift to the right by 2 units, and a horizontal shift down by 7 units. This guide covers essential transformations and the correct approach to forming the function equation.

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