\documentclass[]{standalone}
\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{calc} 
\pgfplotsset{compat=1.14} 

\usepackage{numprint}
\usepackage{xcolor}

%Draw Parabolas: y=ax^2+bx+c
%a,c fixed, b varying

\begin{document}
   \begin{tikzpicture}
       \tikzstyle{texts}=[below,font={\small},blue]
       
       \pgfmathparse{1} \let\a\pgfmathresult
       \pgfmathparse{1} \let\c\pgfmathresult
       
       
       \pgfmathparse{-4.0} \let\bfirst\pgfmathresult
       \pgfmathparse{ 4.0} \let\blast\pgfmathresult
       \pgfmathparse{50} \let\N\pgfmathresult
       \pgfmathparse{2} \let\dx\pgfmathresult

       %Variation of colors - color in [0,100].
       \pgfmathparse{100/\N} \let\dn\pgfmathresult
       
       %Steps of b
       \pgfmathparse{(\blast-\bfirst)/\N} \let\db\pgfmathresult
       
       %Coordinate system
       \draw[-latex] (-4,0) -- (4,0) node [right] {$x$};
       \draw[-latex] (0,-3) -- (0,5) node [above] {$y$};
       
       %Vertex Parabola title
       \pgfmathparse{2.1} \let\x\pgfmathresult
       \coordinate (V) at ({-\x},{    -\a*\x*\x+\c  });
       \node[texts] at (V) {$y=-\a \cdot x^2+\c$};
       \coordinate (V) at ({\x},{    -\a*\x*\x+\c  });
       \node[texts] at (V) {$V \in \mathcal{P}_V$};
      
       %Figure title
       \node[texts] at (0,5.75) 
       {$y=\a \cdot x^2+b x+\c, \qquad \bfirst \leq b \leq \blast$};
       
       \foreach \n in {0,1,...,\N}
       {
           %\b and color, \color
           \pgfmathparse{\bfirst+\n*\db} \let\b\pgfmathresult
           \pgfmathparse{\n*\dn} \let\color\pgfmathresult
          
           %Discriminante
           \pgfmathparse{\b*\b-4*\a*\c} \let\d\pgfmathresult
           
           %Symmetrical value
           \pgfmathparse{-0.5*\b/\a} \let\bb\pgfmathresult
          
           %Interval to draw, symmetrical around \bb 
           \pgfmathparse{\bb-\dx} \let\xlower\pgfmathresult
           \pgfmathparse{\bb+\dx} \let\xupper\pgfmathresult
           
           \draw
              [
                 thick,smooth,
                 domain=\xlower:\xupper,
                 variable=\x,
                 %Convex combination of colors
                 color=orange!\color!cyan
              ]
              plot ({\x},{    \a*\x*\x+\b*\x+\c  })
              %node[right]
              %{\tiny{
              %   $b=$ %round to 1 decimal
              %   \nprounddigits{1}\numprint{\b}
              %}}
              ;
       }
       
       \foreach \n in {0,1,...,\N}
       {
           %\b and color, \color
           \pgfmathparse{\bfirst+\n*\db} \let\b\pgfmathresult
           \pgfmathparse{\n*\dn} \let\color\pgfmathresult
          
           %Discriminante
           \pgfmathparse{\b*\b-4*\a*\c} \let\d\pgfmathresult
           %Symmetry value
           \pgfmathparse{-0.5*\b/\a} \let\bb\pgfmathresult
           
           %Vertex
           \coordinate[gray] (V) at ($(   \bb,   -0.25*\d/\a   )$);
           \filldraw (V) circle(0.75pt);
       }
          
       %Vertex Parabola
       \draw
           [
              thick,smooth,
              domain=-2.1:2.1,
              variable=\x,
              color=blue
           ]
           plot ({\x},{    -\a*\x*\x+\c  });
       
       
   \end{tikzpicture}
\end{document}