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The question asks what kind of transformation converts the graph of \( f(x) = |x| + 1 \) into the graph of \( g(x) = 4|x| + 4 \).

To solve this:

1. **Transformation of the coefficient of \(|x|\)**: 
   - In \( f(x) = |x| + 1 \), the coefficient of \(|x|\) is 1.
   - In \( g(x) = 4|x| + 4 \), the coefficient of \(|x|\) is 4.
   - Multiplying the coefficient of \(|x|\) by 4 indicates a **vertical stretch** because multiplying the entire function by a factor greater than 1 stretches the graph vertically.

2. **Transformation of the constant term**:
   - In \( f(x)\), the constant is \( +1 \), while in \( g(x)\), the constant is \( +4 \). 
   - This represents a vertical shift upward by 3 units (from \(+1\) to \(+4\)).

However, the vertical stretch caused by the factor of 4 is the relevant transformation in this case.

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

Would you like more details or have any questions? Here are some related questions:

1. What happens when the coefficient of \(|x|\) is less than 1?
2. How does a horizontal shrink differ from a vertical shrink?
3. How do vertical and horizontal translations affect the graph of absolute value functions?
4. What is the general form of a transformed absolute value function?
5. How do you determine if a transformation involves a reflection?

**Tip**: A vertical stretch occurs when the function is multiplied by a factor greater than 1, making the graph appear taller.

What kind of transformation converts the graph of f(x) = |x| + 1 into the graph of g(x) = 4|x| + 4?

Discover how to transform the graph of f(x) = |x| + 1 into g(x) = 4|x| + 4 using vertical stretch and vertical shift. This problem covers key concepts of graph transformations, suitable for students in Grades 8-10.

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