How are quadratic Bezier curves drawn by TikZ?

As Paul Gaborit said in his comments (a) .. controls (b) .. (c) is not a quadratic Bézier curve, but the cubic one (a) .. controls (b) and (b) .. (c).

If you want to draw a quadratic curve in TikZ you can define your own to path style. Here is an an example of how to do it using calc library.

\documentclass[tikz,border=7pt,convert={density=1400}]{standalone}
\usetikzlibrary{calc}
\tikzset{
to path={
(\tikztostart) .. controls
($#1!1/3!(\tikztostart)$) and ($#1!1/3!(\tikztotarget)$)
.. (\tikztotarget)
}
}
}
\begin{document}
\begin{tikzpicture}[nodes={scale=2,text opacity=1}]
\draw[help lines] (0,0) grid (2,4);
\filldraw[fill opacity=.07]
(0,0) coordinate[label=center:.](A)
--(1,0) coordinate[label=center:.](B)
--(2,4) coordinate[label=center:.](C);
\draw[blue] (A)..controls (B)..(C);
\draw[white,domain=0:2] plot (\x,\x^2);
\end{tikzpicture}
\end{document}


And if you prefer a code that do not use calc and use the PGF command \pgfpathquadraticcurveto you can replace the style definition with :

\makeatletter
\def\pt@get#1{%
\tikz@scan@one@point\pgfutil@firstofone#1\relax%
\pgfpoint{\the\pgf@x}{\the\pgf@y}%
}
\tikzset{