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

reflected across the x-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 \( f(-x) + 6 = |−x| + 6 \):

1. Reflection 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 change the graph. Therefore, this part does not contribute a visible transformation.

2. Translation upward: The term "+6" translates the graph of the function vertically upward by 6 units.

So, the transformation from \( f(x) = |x| \) to \( f(-x) + 6 = |−x| + 6 \) involves only the vertical translation up by 6, and the reflection across the y-axis does not affect the shape of the graph.

Thus, the correct identification of the transformations is:

Reflected across the y-axis and translated up vertically.