MHB Reflection of Parabolas: Find the Sum of Coefficients - POTW #406 Feb 27th, 2020

[anemone]

Here is this week's POTW:

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The parabola with equation $y=ax^2+bx+c$ and vertex $(h, k)$ is reflected about the line $y=k$. This results in the parabola with equation $y=dx^2+ex+f$. Find $a+b+c+d+e+f$.

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[anemone]

Congratulations to the following members for their correct solution!

1. castor28
2. MegaMoh

Solution from castor28:

Spoiler

If the generic point of the reflected parabola is $(x,z)$, we have $y+z=2k$. This gives:
$$
y+z = (ax^2+bx+c) + (dx^2+ex+f) = 2k
$$
for all $x$. Taking $x=1$ gives:
$$
a+b+c+d+e+f = 2k
$$