Given $(a,b)$, $(c,d)$, and $(e,f)$ (assume non-collinear and $a\neq c$, $c\neq e$, and $a\neq e$), is there a generic way to find a parabolic function between the three?
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Yes. Write $y = Ax^2 + Bx + C$. Substitute in the three points; if the values $a$, $c$ and $e$ are distinct, you get a nondegenerate system of three linear equations in three unknowns. Solve for $A$, $B$ and $C$ and you are there.