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We are given the function \( f(x) = |x| \) and asked to identify the transformations that result in the function \( g(x) = \frac{1}{3}|x + 5| - 6 \).

To analyze this, we will identify the transformations in terms of shifts, stretches, and reflections:

### Step-by-Step Transformations:
1. **Horizontal Shift**:  
   - In \( g(x) \), the term inside the absolute value is \( x + 5 \). This corresponds to a **horizontal shift** to the left by 5 units.  
   (If the form is \( |x - h| \), the shift is \( h \), so here it's \( -5 \)).

2. **Vertical Stretch**:  
   - The coefficient \( \frac{1}{3} \) outside the absolute value scales the function vertically. Since \( \frac{1}{3} < 1 \), this is a **vertical compression** by a factor of 3 (the graph is "squished" vertically).

3. **Vertical Shift**:  
   - The \( -6 \) at the end represents a **vertical shift** downward by 6 units.

### Summary of Transformations:
- Horizontal shift **left by 5 units**.
- Vertical **compression by a factor of 3**.
- Vertical shift **downward by 6 units**.

Would you like more details or have any further questions?

Here are five related questions:
1. How would the graph change if the coefficient outside the absolute value was negative?
2. What transformation would occur if the inside of the absolute value was \( x - 5 \) instead?
3. How would the graph change if the \( -6 \) was positive instead of negative?
4. Can you describe the effect of multiplying \( f(x) \) by 2 instead of \( \frac{1}{3} \)?
5. What happens to the graph if no horizontal shift is applied?

**Tip:** When dealing with absolute value transformations, always focus on the inside expression for horizontal changes and outside terms for vertical transformations.

List all the transformations on f(x) = |x| that result in the function g(x) = (1/3) |x + 5| - 6.

Explore the transformations that map the absolute value function f(x) = |x| to g(x) = (1/3) |x + 5| - 6. This problem involves horizontal shifts, vertical compression, and vertical shifts, and is suitable for high school students in Grades 9-12.

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