The parabola with the equation $y=-x^2+4x+8$ is shifted so that it passes through the points (1,1) and (3,5). Find the equation of the new parabola. Given the points $(1,1)$ and $(3,5)$, the vertex form would be:
$1=(1-h)^2+k$  for $(1,1)$  and
$5=(3-h)^2+k$  for $(3,5)$.
With a system of equations, I obtain that $h = 1$ and $k = 1$, so the new formula of the shifted parabola would be: $y=(x-1)^2+1$
What I don't understand is the answer of my exercise sheet, which says that the new equation would be $y= -x^2+6x-4$
Can someone help me understand this?
Thanks

The original parabola is
$ y = - x^2 + 4 x + 8 $
Shifting by $(h, k)$ gives the parabola,
$ y = - (x - h)^2 + 4 (x - h) + 8 + k $
Points $(1,1)$ and $(3,5)$ are on the new parabola, so
$ 1 = - (1 - h)^2 + 4 (1 - h) + 8 + k $
$ 5 = - (3 - h)^2 + 4 (3 - h) + 8 + k $
Substracting,
$ - 4 = - (1^2 - 3^2 - 2 h (1 - 3 )) + 4 (1 - 3) $
$ - 4 = - (-8 + 4 h ) - 8 $
Therefore,
$ -8 + 4 h = - 4 $
From which
$ 4 h = 4 $
Hence, $ h = 1$.
Now substitute this into one of the new parabola equations to find $k$.
$ 1 = - (1 - 1)^2 + 4 (1 - 1) + 8 + k $
Therefore, $ k = -7 $
Now the equation of the new parabola is
$ y = - (x - 1)^2 + 4 (x - 1) + 8 + (-7) $
Expanding, this becomes
$ y = - x^2 +6 x - 4 $
