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

Question

𝑓
(
𝑥
)
=
|
𝑥
|
to f(−x)+6=|−x|+6
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 down vertically
reflected across the x -axis and translated down 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 up 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 transformation from \( f(x) = |x| \) to \( f(-x) = |-x| \) reflects the graph across the y-axis. This is because \( |−x| = |x| \), so the shape of the graph does not change with this transformation, but the input changes to negative values around the y-axis.

2. Translation upward: Adding 6 to the function, resulting in \( f(-x) + 6 = |−x| + 6 \), translates the graph vertically upward by 6 units.

Putting both transformations together: the graph is reflected across the y-axis (though this does not change its appearance since \( |−x| = |x| \)) and translated up vertically by 6 units.

Thus, the correct response would be:

reflected across the y-axis and translated up vertically.