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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 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

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

1 answer

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

  1. The slope of the original function \( f(x) = 2x + 1 \) is 2, which means for every unit increase in \( x \), \( f(x) \) increases by 2 units.
  2. In the new function \( f(x) = 5x + 1 \), the slope is 5, meaning for every unit increase in \( x \), \( f(x) \) now increases by 5 units.

Since the \( y \)-intercept (the constant term) remains the same in both functions (both are +1), and only the slope has changed, this indicates a vertical stretch.

Thus, 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.