
\subsection{Signe d'un trin\^{o}me}

\begin{propriete} [Signe du trin\^{o}me $ax^2+bx+c$]
  \begin{center}\setlength{\tabcolsep}{2.5pt}
    \begin{tabular}{>{\raggedright}m{2cm}|*2{>{\centering}m{3.5cm}|}>{\centering}m{3.5cm}}
      & $\Delta{}>0$& $\Delta{}=0$& $\Delta{}<0$\tabularnewline\hline
      Signe de  $f(x) = ax^2 + bx + c$

      \vspace{1.5mm}

      $a > 0$
      & 
      \begin{tikzpicture}[scale=0.6,general]
          \tkzTabInit[lgt=1.5,espcl=1, lw=0.3mm, color, colorC=H4,
          colorV=H4, colorL=H4, colorT=A4]
          {$x$/1 , $f(x)$/1}
          {$-\infty{}$ , $x_1$ , $x_2$ , $+\infty{}$}
          \tkzTabLine{ , + , z     , $-$ , z     , + }
      \end{tikzpicture}
      & 
      \begin{tikzpicture}[scale=0.6,general]
          \tkzTabInit[lgt=1.5,espcl=1, lw=0.3mm, color, colorC=H4,
          colorV=H4, colorL=H4, colorT=A4]
          {$x$/1 , $f(x)$/1}
          {$-\infty{}$ , $x_0$ , $+\infty{}$}
          \tkzTabLine{,  + , z ,  + }
      \end{tikzpicture}
      &
      \begin{tikzpicture}[scale=0.6,general]
          \tkzTabInit[lgt=1.5,espcl=1, lw=0.3mm, color, colorC=H4,
          colorV=H4, colorL=H4, colorT=A4]
          {$x$/1 , $f(x)$/1}
          {$-\infty{}$ , $+\infty{}$}
          \tkzTabLine{,  + , }
      \end{tikzpicture}
      \tabularnewline
      & \includestandalone{\pathmanuel/figures/img-66} & 
      \includestandalone{\pathmanuel/figures/img-67} & 
      \includestandalone{\pathmanuel/figures/img-68} \tabularnewline    \hline
      Signe de $f(x) = ax^2 + bx + c$

      \vspace{1.5mm}

      $a < 0$ &   
      \begin{tikzpicture}[scale=0.6,general]
        \tkzTabInit[lgt=1.5,espcl=1, lw=0.3mm, color, colorC=H4,
        colorV=H4, colorL=H4, colorT=A4]
        {$x$/1 , $f(x)$/1}
        {$-\infty{}$ , $x_1$ , $x_2$ , $+\infty{}$}
        \tkzTabLine{ , - , z     , + , z     , - }
      \end{tikzpicture}
      & 
      \begin{tikzpicture}[scale=0.6,general]
          \tkzTabInit[lgt=1.5,espcl=1, lw=0.3mm, color, colorC=H4,
          colorV=H4, colorL=H4, colorT=A4]
          {$x$/1 , $f(x)$/1}
          {$-\infty{}$ , $x_0$ , $+\infty{}$}
          \tkzTabLine{,  - , z ,  - }
      \end{tikzpicture}
      &  
      \begin{tikzpicture}[scale=0.6,general]
          \tkzTabInit[lgt=1.5,espcl=1, lw=0.3mm, color, colorC=H4,
          colorV=H4, colorL=H4, colorT=A4]
          {$x$/1 , $f(x)$/1}
          {$-\infty{}$ , $+\infty{}$}
          \tkzTabLine{,  - , }
      \end{tikzpicture}      
      \tabularnewline
      & \includestandalone{\pathmanuel/figures/img-69} & 
      \includestandalone{\pathmanuel/figures/img-70} & 
      \includestandalone{\pathmanuel/figures/img-71} \tabularnewline
    \end{tabular}
  \end{center}
\end{propriete}

\begin{remarque}
  Pour déterminer le signe du trin\^{o}me $ax^2+bx+c$, il faut d'abord
  chercher ses racines.  Le signe du trin\^{o}me dépend :
  \begin{itemize}
  \item du signe du discriminant : la courbe $C_{f}$ coupe-t-elle
    l'axe des abscisses ?
  \item du signe de $a$ : le sommet est-il en haut ou en bas de la
    courbe ?
  \end{itemize}
\end{remarque}
