To determine the correct equation of an absolute value function that has been translated horizontally to the right by 1 unit, vertically up by 2 units, and reflected on the [tex]\( x \)[/tex]-axis, follow these steps:
1. Horizontal Translation to the Right by 1 Unit:
The standard form for an absolute value function is [tex]\( f(x) = |x| \)[/tex]. To translate this function horizontally to the right by 1 unit, we replace [tex]\( x \)[/tex] with [tex]\( x-1 \)[/tex]. Therefore, the equation becomes:
[tex]\[
f(x) = |x-1|
\][/tex]
2. Vertical Translation Up by 2 Units:
To translate the function vertically upward by 2 units, we add 2 to the function:
[tex]\[
f(x) = |x-1| + 2
\][/tex]
3. Reflection on the [tex]\( x \)[/tex]-axis:
To reflect the function across the [tex]\( x \)[/tex]-axis, we multiply the entire function by -1. This changes the function to:
[tex]\[
f(x) = -|x-1| + 2
\][/tex]
Now we have the equation [tex]\( f(x) = -|x-1| + 2 \)[/tex].
Hence, the correct equation of an absolute value function that has been translated horizontally to the right by 1 unit, vertically up by 2 units, and reflected on the [tex]\( x \)[/tex]-axis is:
[tex]\[
\boxed{f(x) = -|x-1| + 2}
\][/tex]
