To determine which absolute value function, when graphed, represents the parent function [tex]\( f(x) = |x| \)[/tex] reflected over the [tex]\( x \)[/tex]-axis, let's break down what each function transformation represents.
1. Reflected over the [tex]\( x \)[/tex]-axis:
- Reflecting a function over the [tex]\( x \)[/tex]-axis involves multiplying the output of the function by [tex]\(-1\)[/tex]. For the parent function [tex]\( f(x) = |x| \)[/tex], reflecting it over the [tex]\( x \)[/tex]-axis results in [tex]\( f(x) = -|x| \)[/tex].
- Therefore, the correct function for this transformation is [tex]\( f(x) = -|x| \)[/tex].
2. Other transformations (for understanding why they're not the correct answers):
- [tex]\( f(x) = |-x| \)[/tex]:
- This represents a reflection of [tex]\( x \)[/tex] over the [tex]\( y \)[/tex]-axis before applying the absolute value, which is not the same as reflecting the entire function over the [tex]\( x \)[/tex]-axis. Mathematically, [tex]\( |-x| = |x| \)[/tex], so this is the same as the parent function [tex]\( f(x) = |x| \)[/tex].
- [tex]\( f(x) = |x| + 1 \)[/tex]:
- This represents a vertical shift of the parent function [tex]\( f(x) = |x| \)[/tex] upwards by 1 unit. It does not represent a reflection over the [tex]\( x \)[/tex]-axis.
- [tex]\( f(x) = |x-1| \)[/tex]:
- This represents a horizontal shift of the parent function [tex]\( f(x) = |x| \)[/tex] to the right by 1 unit. It does not represent a reflection over the [tex]\( x \)[/tex]-axis.
Given these analyses, the function that correctly represents the parent function [tex]\( f(x) = |x| \)[/tex] reflected over the [tex]\( x \)[/tex]-axis is:
[tex]\[ f(x) = -|x| \][/tex]
So the answer is:
[tex]\[ \boxed{1} \][/tex]
