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What's the form of a three-dimensional function $z = f(x, y)$ where $x$ moves in the interval $[0, X]$, $y$ in $[0, Y]$, $f(x, 0)$ is an upward parabola and $f(x, Y)$ is a downward parabola, where $X$ and $Y$ are positive integers.


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up vote 1 down vote accepted

$f(x,0) = a_2 x^2 + a_1 x + a_0$ and $f(x,Y) = b_2 x^2 + b_1 x + b_0$ where $b_2 < 0 < a_2$. For $0 < y < Y$, anything goes. Well, you might have $f(x,y) = c_2(y) x^2 + c_1(y) x + c_0(y)$ for some functions $c_0$, $c_1$ and $c_2$ with $c_2(Y) < 0 < c_2(0)$, but there are all sorts of other possibilities. I guess the simplest case might be something like $f(x,y) = x^2 (Y-2 y)$.

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Thanks for your answer!But that's not exactly what I was looking for.I meant a function z = f(x, y) that when you evaluate in y=0 and you graph the XZ axis, it's a downward parabola (like -n^2), and when you evaluate in y=Y, where Y is a fixed integer, it's a upward parabola (n^2) – federicot Mar 6 '12 at 20:14
You asked for upward at $y=0$ and downward at $y=Y$. Well, just multiply by $-1$. – Robert Israel Mar 7 '12 at 2:29

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