To determine the line of reflection for the given points, we start by noting the original point [tex]\( P = (4, -5) \)[/tex] and its reflected image [tex]\( P' = (-4, -5) \)[/tex].
The transformation from [tex]\( P = (4, -5) \)[/tex] to [tex]\( P' = (-4, -5) \)[/tex] changes the [tex]\( x \)[/tex]-coordinate from [tex]\( 4 \)[/tex] to [tex]\( -4 \)[/tex] while the [tex]\( y \)[/tex]-coordinate remains the same at [tex]\( -5 \)[/tex].
This type of transformation, where the [tex]\( x \)[/tex]-coordinate changes its sign and the [tex]\( y \)[/tex]-coordinate remains unchanged, is characteristic of a reflection about the [tex]\( y \)[/tex]-axis. To confirm this, recall that reflecting a point [tex]\( (x, y) \)[/tex] over the [tex]\( y \)[/tex]-axis changes it to [tex]\( (-x, y) \)[/tex].
Since the coordinates of [tex]\( P \)[/tex] transformed accordingly to [tex]\( P' \)[/tex] by this rule, the line of reflection must be the [tex]\( y \)[/tex]-axis.
Thus, the correct answer is:
- the [tex]\( y \)[/tex]-axis
So, the line of reflection is the [tex]\( y \)[/tex]-axis.
