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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

1 answer

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.