[SymPy][] is a [[Python]] library for symbolic mathematics.  To give
you a feel for how it works, lets extrapolate the extremum location
for $f(x)$ given a quadratic model:

\[
  f(x) = A x^2 + B x + C
\]

and three known values:

\[
\begin{aligned}
  f(a) &= A a^2 + B a + C  \\
  f(b) &= A b^2 + B b + C  \\
  f(c) &= A c^2 + B c + C
\end{aligned}
\]

Rephrase as a matrix equation:

\[
\begin{pmatrix}
  f(a)  \\
  f(b)  \\
  f(c)
\end{pmatrix}
  =
\begin{pmatrix}
  a^2  &  a  &  1  \\
  b^2  &  b  &  1  \\
  c^2  &  c  &  1
\end{pmatrix}
  \cdot
\begin{pmatrix}
  A  \\
  B  \\
  C
\end{pmatrix}
\]

So the solutions for $A$, $B$, and $C$ are:

\[
\begin{pmatrix}
  A  \\
  B  \\
  C
\end{pmatrix}
  =
\begin{pmatrix}
  a^2  &  a  &  1  \\
  b^2  &  b  &  1  \\
  c^2  &  c  &  1
\end{pmatrix}^{-1}
  \cdot
\begin{pmatrix}
  f(a)  \\
  f(b)  \\
  f(c)
\end{pmatrix}
  =
\begin{pmatrix}
  \text{long}  \\
  \text{complicated}  \\
  \text{stuff}
\end{pmatrix}
\]

Now that we've found the model parameters, we need to find the $x$
coordinate of the extremum.

\[
  \frac{\mathrm{d}f}{\mathrm{d}x} = 2 A x + B  \;,
\]

which is zero when

\[
\begin{aligned}
  2 A x &= -B  \\
  x &= \frac{-B}{2 A}
\end{aligned}
\]

Here's the solution in SymPy:

    >>> from sympy import Symbol, Matrix, factor, expand, pprint, preview
    >>> a = Symbol('a')
    >>> b = Symbol('b')
    >>> c = Symbol('c')
    >>> fa = Symbol('fa')
    >>> fb = Symbol('fb')
    >>> fc = Symbol('fc')
    >>> M = Matrix([[a**2, a, 1], [b**2, b, 1], [c**2, c, 1]])
    >>> F = Matrix([[fa],[fb],[fc]])
    >>> ABC = M.inv() * F
    >>> A = ABC[0,0]
    >>> B = ABC[1,0]
    >>> x = -B/(2*A)
    >>> x = factor(expand(x))
    >>> pprint(x)
     2       2       2       2       2       2   
    a *fb - a *fc - b *fa + b *fc + c *fa - c *fb
    ---------------------------------------------
     2*(a*fb - a*fc - b*fa + b*fc + c*fa - c*fb) 
    >>> preview(x, viewer='pqiv')

Where `pqiv` is the executable for [pqiv][], my preferred image
viewer.  With a bit of additional factoring, that is:

\[
  x = \frac{a^2 [f(b)-f(c)] + b^2 [f(c) - f(a)] + c^2 [f(a) - f(b)]}
           {2\cdot\{a [f(b) - f(c)] + b [f(c) - f(a)] + c [f(a) - f(b)]\}}
\]

[SymPy]: http://sympy.org/
[pqiv]: http://www.pberndt.com/Programme/Linux/pqiv/

[[!tag tags/python]]
[[!tag tags/teaching]]
[[!tag tags/tools]]