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We are making a plot as follows:

\documentclass{standalone}

\usepackage{pgfplots}
\usetikzlibrary{calc,shadings}
\usepackage{pgfplots}

\begin{document}
    \begin{tikzpicture}
        \def\xmin{0}
        \def\xmax{10}
        \def\ymin{0}
        \def\ymax{7}
        \draw[style=help lines, ystep=1, xstep=1] ;

        \draw (-.25,-.25) node[auto] {0};
        \draw[->] (\xmin,\ymin) -- (\xmax,\ymin) node[right] {$x$};
        \draw[->] (\xmin,\ymin) -- (\xmin,\ymax) node[above] {$f(x)$};

        \def\intersectX1{2}
        \def\intersectY1{4/5+3}
        \def\intersectX2{7}
        \def\intersectY1{9/5+3}
        \def\intersectX{4.76}
        \def\intersectY{4.26}
        \def\QPX{4}
        \def\QPY{5}
        \draw[color=red,smooth] plot [domain=0:8] (\x,{((\x-4)^2)/5+3)});

        \draw[dashed]  (2,0) node[below] {$x_1$} -- (2,3.8) node[up,left] {$f(x_1)$};
        \draw[dashed]  (7,0) node[below] {$x_1$} -- (7,4.8) node[up,right] {$f(x_2)$};

        \draw[dashed,orange]  (4,0) node[below] {$\alpha\,x_1+(1-\alpha)x_2$} -- (4,3) node[below] {$f(\alpha\,x_1+(1-\alpha)x_2)$};
        \draw[color=black,dashedd] (4,3) -- (4,4.2) node[up] {$\alpha f(x_1) + (1-\alpha)f(x_2)$};
        \draw[color=blue] (2,3.8) -- (7,4.8);
    \end{tikzpicture}
\end{document}

We want to achieve:

  1. The labels $f(x_1)$, ... needs to be smaller but more clear -- by bringing more color in them.
  2. Display that : $\alpha f(x_1) + (1-\alpha)f(x_2) \ge f (\alpha x_1 + (1-\alpha) x_2)$. Just the picture need to tell the story.
  3. Any idea to beautify this plot is very welcome.

Example

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closed as too localized by Jake, Claudio Fiandrino, Marco Daniel, diabonas, zeroth Feb 17 '13 at 16:01

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Why do you load pgfplots and don't use it? pgfplots provides a enhanced axis environment and addplot command for advanced plotting. See manual –  bloodworks Feb 16 '13 at 13:29
    
Try to avoid text over graphics? The orange color might be too light and another color might better accompany the other colors? –  N.N. Feb 16 '13 at 13:31
2  
Number 3 is too subjective for SE, please read the FAQ on don't-ask: "avoid asking subjective questions" –  Camil Staps Feb 16 '13 at 13:32
    
Maybe you could use pgfplots legends to put move some formulas further away from the plots? –  N.N. Feb 16 '13 at 14:03

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