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

Question

𝑓
(
𝑥
)
=
2
𝑥
+
1
to the new function f(x)=5x+1
𝑓
(
𝑥
)
=
5
𝑥
+
1
? (1 point)
Responses

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

Answer 1

To determine the transformation from the original function \( f(x) = 2x + 1 \) to the new function \( f(x) = 5x + 1 \), we can analyze the changes in the coefficients of \( x \).

1. Identify the key components:

   • The slope of the original function \( (m_1) \) is \( 2 \).
   • The slope of the new function \( (m_2) \) is \( 5 \).

2. Vertical and horizontal shifts:

   • The \( +1 \) in both functions indicates they have the same y-intercept, meaning they intersect the y-axis at the same point (at \( y = 1 \)).

3. Analyzing the transformation:

   • The change from a slope of \( 2 \) to a slope of \( 5 \) indicates a vertical stretch, as the new function's output values increase more rapidly compared to the original function for the same input values.

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.