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To determine the transformation that converts the graph of \( f(x) = -6|x - 2| - 3 \) into the graph of \( g(x) = -3|x - 4| - 3 \), we can break it down step by step:

### Step 1: Analyze Vertical Transformation
- In \( f(x) \), the coefficient of the absolute value is \(-6\).
- In \( g(x) \), the coefficient of the absolute value is \(-3\).
  - This change from \(-6\) to \(-3\) represents a **vertical shrink** because the magnitude of the coefficient decreases (making the graph less steep).
  
### Step 2: Analyze Horizontal Transformation
- In \( f(x) \), the horizontal shift is \( x - 2 \), indicating a shift 2 units to the right.
- In \( g(x) \), the horizontal shift is \( x - 4 \), indicating a shift 4 units to the right.
  - This shows a **horizontal shift** of 2 units further to the right, but that doesn't affect the question choices.

### Step 3: Conclusion
The primary transformation between the two functions is the change in the vertical scaling from \(-6\) to \(-3\), which is a **vertical shrink**.

Thus, the correct answer is **vertical shrink**.

Would you like further details or have any additional questions?

### Related Questions:
1. What would a horizontal stretch look like in this context?
2. How does a vertical reflection affect the graph of a function?
3. Can you explain how shifting impacts the domain of absolute value functions?
4. What would happen if we changed the constant outside the absolute value expression?
5. How do horizontal shifts differ from horizontal stretches in terms of function transformation?

### Tip:
When the coefficient inside the absolute value changes, it affects horizontal transformations, while changes outside affect vertical transformations.

What kind of transformation converts the graph of f(x) = -6|x - 2| - 3 into the graph of g(x) = -3|x - 4| - 3?

Learn how to determine the transformation that converts the graph of f(x) = -6|x - 2| - 3 into g(x) = -3|x - 4| - 3. This problem covers vertical and horizontal transformations of absolute value functions. Suitable for high school students in Grades 9-12.

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