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

reflected across the x-axis and translated down vertically

Answer 1

To analyze the transformation from \( f(x) = |x| \) to \( f(-x) + 6 = | -x | + 6 \), let's break down the steps:

1. Reflection across the y-axis: The function \( f(-x) = | -x | \) reflects the original function \( f(x) = |x| \) across the y-axis. This is because replacing \( x \) with \( -x \) is a reflection across the y-axis.

2. Vertical Translation: Next, adding 6 in \( f(-x) + 6 \) translates the graph of \( | -x | \) upwards by 6 units.

Putting this together, the transformation that occurs is a reflection across the y-axis followed by a vertical translation upwards.

Therefore, the correct response is:

reflected across the y-axis and translated up vertically.