which of the following correctly identifies the transformations undergone from f(x) = |x| to f(-x) + 6 = |-x| + 6

Answer 1

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

1. Starting function: \( f(x) = |x| \)

2. Transforming to \( f(-x) \):

   • The transformation \( f(-x) \) results in reflecting the graph of \( f(x) \) across the y-axis.
   • Since \( |x| = |-x| \), this reflection does not change the graph in this particular case. So \( f(-x) = |x| \).

3. Adding 6:

   • The transformation \( f(-x) + 6 \) translates the entire graph of \( f(-x) \) vertically upwards by 6 units.
   • Therefore, \( f(-x) + 6 = |x| + 6 \).

4. Final function:

   • Thus, \( f(-x) + 6 \) is equal to \( |x| + 6 \), which is indeed the same as \( |-x| + 6 \) because \( |-x| = |x| \).

Summary of Transformations:

1. Reflection across the y-axis (though in this case, it doesn't alter the graph of \( |x| \)).
2. Vertical translation upwards by 6 units.

So, the final transformation is primarily a vertical shift up by 6 from the original \( f(x) = |x| \).