Question

Which of the following best describes a transformation from the original function f of x equals 2x + 1 to the new function f of x = 5x + 1
Horizontal translation where the new function is further to the right then it's original position but otherwise it is the same
A vertical stretch where the new function is further from the x-axis but otherwise is the
A vertical translation where the new function is higher than its original position on the y-axis but otherwise is the
A vertical compression or the new function is closer to the x-axis but otherwise it's the same

Answer 1

To analyze the transformation from the original function \( f(x) = 2x + 1 \) to the new function \( f(x) = 5x + 1 \), let's break down the components.

1. **Identifying the Original and New Functions:**
- Original function: \( f(x) = 2x + 1 \)
- New function: \( f(x) = 5x + 1 \)

2. **Comparing the Coefficients of \( x \):**
- The coefficient of \( x \) in the original function is 2, while in the new function it is 5. This change in the coefficient indicates a vertical stretch.

3. **Examining the Y-Intercept:**
- Both functions have the same y-intercept \( +1 \). This means there is no vertical translation.

4. **Conclusion:**
- The transformation from \( f(x) = 2x + 1 \) to \( f(x) = 5x + 1 \) is a **vertical stretch** because the slope (the coefficient of \( x \)) increases from 2 to 5, making the new function steeper.

Thus, the best description of the transformation is:

**A vertical stretch where the new function is further from the x-axis but otherwise is the same.**