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The problem is dealing with transformations of the absolute value function, where the basic function is given as \( f(x) = |x| \). The transformation form provided is \( y = \pm A |x - C| + D \).

### Analysis of the Transformed Function \( g(x) \)

From the tables:
- **Basic Function \( f(x) = |x| \)**:
  - Values: 
    - \( f(-3) = 3 \)
    - \( f(-1) = 1 \)
    - \( f(0) = 0 \)
    - \( f(1) = 1 \)
    - \( f(3) = 3 \)

- **Transformed Function \( g(x) \)**:
  - Values:
    - \( g(-1) = -3 \)
    - \( g(1) = -1 \)
    - \( g(2) = 0 \)
    - \( g(3) = -1 \)
    - \( g(5) = -3 \)

### Identifying the Transformation
To find the transformation applied:
- \( f(x) = |x| \) was transformed into \( g(x) \).
- By examining the transformed points, it appears that the graph of the function is vertically flipped (indicating a negative sign in front), translated downwards, and shifted horizontally.

### Possible Transformation
Comparing the values:
- A flip and a vertical shift down of \( D = 0 \).
- Horizontal shift by \( C \) and a vertical stretch/shrink factor \( A \).

Considering the patterns, the transformation seems to be:
\[ g(x) = -|x - 2| \]
This matches the points:
- Shifts right by 2 ( \( C = 2 \) )
- Flipped vertically and centered around \( x = 2 \).

Would you like more details on how the transformation was determined, or do you have specific questions?

### Further Questions
1. How do vertical flips affect absolute value functions?
2. What is the effect of horizontal translations on the graph?
3. How do you determine the transformation factor \( A \)?
4. What role does the constant \( D \) play in transformations?
5. Can you derive the transformation equation using other points?

**Tip**: When working with transformations, it’s helpful to plot key points and observe changes to the basic function visually.

The problem involves transforming the absolute value function f(x) = |x| to g(x) and analyzing the transformed output table. The task is to identify the transformations applied to the function.

Explore how to transform the absolute value function f(x) = |x| into g(x) through horizontal shifts and vertical flips. This step-by-step analysis covers core concepts such as vertical flips, translations, and stretching of functions. Suitable for high school students (Grades 9-12).

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