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

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

reflected across the y-axis and translated up vertically

Answer 1

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

1. Reflected across the y-axis: The function \( f(-x) = |-x| \) reflects the graph of \( f(x) = |x| \) across the y-axis. However, since \( |-x| = |x| \), this reflection does not visually change the graph. Therefore, this part does not contribute to a transformation.

2. Vertical Shift: The "+6" in the expression \( f(-x) + 6 \) indicates that the entire function is translated upward by 6 units.

Thus, the overall transformations are:

• The graph is unchanged in shape due to the reflection across the y-axis (since \( |-x| = |x| \)).
• The graph is translated upward by 6 units.

Considering the answer choices, the closest option that summarizes the transformation correctly is:

Reflected across the y-axis and translated up vertically.