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

Question

to f(−x)+6=|−x|+6

Responses

reflected across the x-axis and translated down vertically

reflected across the y-axis and translated down vertically

reflected across the y-axis and translated up vertically

reflected across the x-axis and translated up vertically

Answer 1

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

1. Reflected across the y-axis: The transformation from \( f(x) = |x| \) to \( f(-x) = |-x| \) represents a reflection across the y-axis. This is because substituting \(-x\) into the function reflects the graph horizontally.

2. Translated up vertically: The addition of \( +6 \) means that every point on the graph is moved up by 6 units. This transformation translates the graph vertically.

Combining these two transformations, we can see that the overall transformation from \( f(x) = |x| \) to \( f(-x) + 6 \) consists of reflecting across the y-axis and then translating vertically up by 6.

Therefore, the correct identification of the transformations is:

reflected across the y-axis and translated up vertically.