Because when you reflect a point in some line, the distance between the point and the reflection line must be the exact same as the distance between the line of refl. and the reflection point.
Every single coordinate
(x,y) on that parabola
y=(x−4)2+3 has a perpendicular distance of
∣y−(−1)∣, i.e.
∣y+1∣ to the line of reflection. Since
y>0 for all
y on this curve, then
y+1>0 and we say that
∣y+1∣=y+1, hence every single coord on this parabola has a distance of
y+1 to the line of reflection.
So when you reflect these points in this line, they will go a distance of y+1 down to the reflection line, and then ANOTHER distance of y+1 to the reflection spot.
Hence, every single coordinate
(x,y) is translated down by a distance of
2(y+1), and so we get a reflection.
Coming back to the parabola, well I'm sure you know that translating
y=f(x) down by
A is the same as writing
y=f(x)−A. Same thing here, and we get that
Y=[(x−4)2+3]−2(y+1), where
Y is the new y coordinate of the parabola. The old one,
y, is precisely just
(x−4)2+3 and so
2(y+1) is simply
2(x−4)2+8. Hence why they have the new curve as
Y=(x−4)2+3−[2(x−4)2+8].
I wouldn't worry if you can't wrap your head around this. A much simpler approach would be the following:
In order to reflect in the line y=-1, let's just shift everything by 1 unit so that our line of reflection coincides with the x-axis. Now reflect everything in the x-axis. Then shift everything back down by 1 unit. Hopefully it makes sense to you and it's much less to digest.