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To determine the transformation that converts the graph of \( f(x) = -x + 1 \) into the graph of \( g(x) = -5x + 5 \), let's analyze the changes step by step.

### 1. **Slope Transformation:**
- The slope of \( f(x) = -x + 1 \) is \( -1 \).
- The slope of \( g(x) = -5x + 5 \) is \( -5 \).

The change in slope from \( -1 \) to \( -5 \) represents a **vertical stretch** by a factor of 5. The slopes are negative in both cases, so there is no additional reflection occurring here, as both functions already reflect across the y-axis.

### 2. **Vertical Shift:**
- \( f(x) = -x + 1 \) has a y-intercept at \( 1 \).
- \( g(x) = -5x + 5 \) has a y-intercept at \( 5 \).

The y-intercept has increased by a factor, but the primary transformation is the **vertical stretch**, which affects the steepness of the line.

Thus, the transformation that best describes the change is a **vertical stretch** by a factor of 5.

The correct answer from the options provided is:

- **Vertical stretch**

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Would you like more details or have any further questions? Here are 5 related questions to expand your understanding:

1. How does a vertical stretch affect the slope of a linear equation?
2. What is the difference between a vertical stretch and a vertical shrink?
3. What would the transformation be if the slope of \( g(x) \) was \( 5x + 5 \)?
4. Can you identify a reflection based only on the sign of the slope?
5. How does a vertical shift affect the graph of a function?

**Tip:** When comparing functions, always focus on how the slope and y-intercept change to determine the type of transformation.

What kind of transformation converts the graph of f(x) = -x + 1 into the graph of g(x) = -5x + 5?

Understand how the graph of f(x) = -x + 1 transforms into g(x) = -5x + 5. This problem covers linear function transformations, specifically vertical stretch, slope, and intercept changes. Suitable for high school students, grades 9-11.

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