Which of the following correctly identifies the transformations undergone from f(x)=|x|

Question

to f(−x)+6=|−x|+6
?(1 point)
Responses

reflected across the y-axis and translated down vertically
reflected across the y -axis and translated down vertically

reflected across the x-axis and translated down vertically
reflected across the x -axis and translated down vertically

reflected across the x-axis and translated up vertically
reflected across the x -axis and translated up vertically

reflected across the y-axis and translated up vertically

Answer 1

To analyze the transformations from the function \( f(x) = |x| \) to the function \( f(-x) + 6 = | -x | + 6 \):

1. Substituting \(-x\): \( f(-x) = |-x| = |x| \)

   • This transformation reflects the function across the y-axis.

2. Adding 6: When we add 6, we translate the entire graph upward by 6 units.

Therefore, the full transformation from \( f(x) = |x| \) to \( f(-x) + 6 = | -x | + 6 \) involves:

1. Reflection across the y-axis (due to the \(-x\) input),
2. Translation upward vertically by 6 units (due to the \(+6\)).

So, the correct response is:

reflected across the y-axis and translated up vertically.