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
reflected across the x -axis and translated down vertically

Answer 1

To identify the transformations from \( f(x) = |x| \) to \( f(-x) + 6 = | -x | + 6 \), we can analyze the changes step by step.

1. Starting Function: \( f(x) = |x| \)
2. Reflection Across the Y-Axis: The expression \( f(-x) \) means that the function is reflected across the y-axis. Since \( | -x | = |x| \), this reflection does not change the shape or position of the graph of \( |x| \), but it is part of the transformation.
3. Translation Up Vertically: The addition of 6 to the function results in a vertical translation up by 6 units.

Therefore, the correct identification of the transformations is:

Reflected across the y-axis and translated up vertically.