Sure, let's go through the transformations and see which one correctly transforms the point [tex]\((a, b)\)[/tex] into [tex]\((a, -b)\)[/tex].
1. A reflection over the [tex]$y$[/tex]-axis:
- Reflecting over the [tex]$y$[/tex]-axis changes the sign of the [tex]$x$[/tex]-coordinate while keeping the [tex]$y$[/tex]-coordinate the same. This would transform [tex]\((a, b)\)[/tex] into [tex]\((-a, b)\)[/tex], not [tex]\((a, -b)\)[/tex].
2. A translation of 1 unit down:
- Translating a point 1 unit down would decrease the [tex]$y$[/tex]-coordinate by 1. This would transform [tex]\((a, b)\)[/tex] into [tex]\((a, b-1)\)[/tex], not [tex]\((a, -b)\)[/tex].
3. A translation of 1 unit up:
- Translating a point 1 unit up would increase the [tex]$y$[/tex]-coordinate by 1. This would transform [tex]\((a, b)\)[/tex] into [tex]\((a, b+1)\)[/tex], not [tex]\((a, -b)\)[/tex].
4. A reflection over the [tex]$x$[/tex]-axis:
- Reflecting over the [tex]$x$[/tex]-axis changes the sign of the [tex]$y$[/tex]-coordinate while keeping the [tex]$x$[/tex]-coordinate the same. This would transform [tex]\((a, b)\)[/tex] into [tex]\((a, -b)\)[/tex], which is exactly what we are looking for.
Therefore, the correct transformation that transforms [tex]\((a, b)\)[/tex] to [tex]\((a, -b)\)[/tex] is:
A reflection over the [tex]$x$[/tex]-axis.
So the answer is:
[tex]\[
\boxed{4}
\][/tex]
