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Which of the following best describes the transformation from the original function f(x)=2x+1

to the new function f(x)=5x+1
? (1 point)
Responses

a vertical stretch where the new function is farther from the x-axis but otherwise is the same
a vertical stretch where the new function is farther from the x -axis but otherwise is the same

a horizontal translation where the new function is farther to the right than its original position but otherwise is the same
a horizontal translation where the new function is farther to the right than its original position but otherwise is the same

a vertical translation where the new function is higher than its original position on the y-axis but otherwise is the same
a vertical translation where the new function is higher than its original position on the y -axis but otherwise is the same

a vertical compression where the new function is closer to the x-axis but otherwise is the same
a vertical compression where the new function is closer to the x -axis but otherwise is the same

1 answer

To analyze the transformation from the original function \( f(x) = 2x + 1 \) to the new function \( f(x) = 5x + 1 \), we can break down the changes:

  1. The slope of the original function \( f(x) = 2x + 1 \) is 2, while the slope of the new function \( f(x) = 5x + 1 \) is 5.
  2. The y-intercept in both functions is 1, meaning they intersect the y-axis at the same point.

Since the only change between the two functions is in the slope, this represents a vertical stretch of the function. The function is steeper (or farther from the x-axis) due to the increase in the slope from 2 to 5.

Therefore, the best description of the transformation from \( f(x) = 2x + 1 \) to \( f(x) = 5x + 1 \) is:

a vertical stretch where the new function is farther from the x-axis but otherwise is the same.