Source Latex
du cours de mathématiques
\documentclass[12pt]{article}
%\usepackage{french}
\usepackage{amsfonts}\usepackage{amssymb}
\usepackage[french]{babel}
\usepackage{amsmath}
\usepackage[latin1]{inputenc}
\usepackage{a4wide}
\usepackage{graphicx}
\usepackage{epsf}
\usepackage{array}
\usepackage{color}
\usepackage{pst-plot,pst-text,pst-tree}
\usepackage{hyperref}
\hypersetup{
pdfauthor={Yoann Morel},
pdfsubject={Cours math�matiques 1�re STI2D: synth�se sur le second
degr�},
pdftitle={Trin�me du second degr� - synth�se},
pdfkeywords={Math�matiques, 1STI, premi�re, STI, STI2D,
second degr�, 2nd degr�, polyn�me
}
}
\hypersetup{
colorlinks = true,
linkcolor = red,
anchorcolor = red,
citecolor = blue,
filecolor = red,
pagecolor = red,
urlcolor = red
}
\voffset=-2.2cm
% Raccourcis diverses:
\newcommand{\nwc}{\newcommand}
\nwc{\dsp}{\displaystyle}
\nwc{\ct}{\centerline}
\nwc{\bge}{\begin{equation}}\nwc{\ene}{\end{equation}}
\nwc{\bgar}{\begin{array}}\nwc{\enar}{\end{array}}
\nwc{\bgit}{\begin{itemize}}\nwc{\enit}{\end{itemize}}
\nwc{\la}{\left\{}\nwc{\ra}{\right\}}
\nwc{\lp}{\left(}\nwc{\rp}{\right)}
\nwc{\lb}{\left[}\nwc{\rb}{\right]}
\nwc{\bgsk}{\bigskip}
\nwc{\vsp}{\vspace{0.1cm}}
\nwc{\vspd}{\vspace{0.2cm}}
\nwc{\vspt}{\vspace{0.3cm}}
\nwc{\vspq}{\vspace{0.4cm}}
\def\N{{\rm I\kern-.1567em N}} % Doppel-N
\def\D{{\rm I\kern-.1567em D}} % Doppel-N
\def\No{\N_0} % Doppel-N unten 0
\def\R{{\rm I\kern-.1567em R}} % Doppel R
\def\C{{\rm C\kern-4.7pt % Doppel C
\vrule height 7.7pt width 0.4pt depth -0.5pt \phantom {.}}}
\def\Q{\mathbb{Q}}
\def\Z{{\sf Z\kern-4.5pt Z}} % Doppel Z
\nwc{\tm}{\times}
\nwc{\V}{\overrightarrow}
\nwc{\zb}{\mbox{$0\hspace{-0.67em}\mid$}}
\nwc{\db}{\mbox{$\hspace{0.1em}|\hspace{-0.67em}\mid$}}
\nwc{\ul}[1]{\underline{#1}}
\newcounter{nex}[section]\setcounter{nex}{0}
\newenvironment{EX}{%
\stepcounter{nex}
\bgsk{\large {\bf Exercice }\arabic{nex}}\hspace{0.2cm}
}{}
\nwc{\bgex}{\begin{EX}}\nwc{\enex}{\end{EX}}
\nwc{\bgfg}{\begin{figure}}\nwc{\enfg}{\end{figure}}
\nwc{\epsx}{\epsfxsize}\nwc{\epsy}{\epsfysize}
\nwc{\bgmp}{\begin{minipage}}\nwc{\enmp}{\end{minipage}}
\textheight=25cm
\textwidth=18.5cm
\oddsidemargin=-1.4cm
\evensidemargin=0cm
\setlength{\unitlength}{1cm}
\def\euro{\mbox{\raisebox{.25ex}{{\it =}}\hspace{-.5em}{\sf C}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\thispagestyle{empty}
\vspace*{-1cm}
%\ul{Nom:}
\hspace{5cm}
{\Large Trin�me du second degr� : synth�se}
\hfill $1^{\text{�re}}\,$ STI2D
\vspace{0.8cm}
Pour un trin�me du second degr�:
\ul{$f(x)=ax^2+bx+c$}, o� $a$, $b$ et $c$ sont trois r�els, et $a\not=0$.
\vspq
Le discriminant du trin�me est \fbox{$\Delta=b^2-4ac$}.
\vspace{0.8cm}
\hspace{-1cm}
\begin{tabular}{|p{4cm}|p{4.8cm}|p{4.5cm}|p{4.5cm}|} \cline{2-4}
\multicolumn{1}{c|}{}& \vspd\ct{$\Delta>0$} & \vspd\ct{$\Delta=0$} & \vspd\ct{$\Delta<0$} \\\hline
\vspace{0.4cm}
\begin{flushleft}
Solution(s) de l'�quation $f(x)=0$
(racines de $f$)
\end{flushleft}
%\vspace{0.cm}\ \,
&2 solutions r�elles distinctes:
\[ x_1=\frac{-b-\sqrt{\Delta}}{2a}\]
\[ x_2=\frac{-b+\sqrt{\Delta}}{2a}\]
& une solution unique (double) :
\[\dsp x_0=\frac{-b}{2a}\]
& \vspace{0.5cm}pas de solution \\\hline
\rule[-1cm]{0.cm}{2cm}
\hspace{-0.2cm}Factorisation de $f(x)$
&$f(x)=a(x-x_1)(x-x_2)$
&$f(x)=a(x-x_0)^2$
&pas de factorisation \\\hline
\vspace{-2cm}
Courbe repr�sentative de $f$,
si $a>0$
\vspace{1.2cm}\ \,
&
\psset{xunit=1cm,yunit=1cm}
\begin{pspicture}(0,0)(4,4)
\psline[linewidth=0.8pt]{->}(0,1)(4.2,1)
\psline[linewidth=0.8pt]{->}(1,-0.5)(1,3.5)
\psplot{0}{4}{x -2 add x -2 add mul -0.5 add}
\psline[linewidth=0.8pt,linestyle=dotted](2,1)(2,-0.5)
\put(1.8,1.3){$\frac{-b}{2a}$}
\put(0.4,0.8){$x_1$}
\put(3.3,0.8){$x_2$}
\end{pspicture}
&
\psset{xunit=1cm,yunit=1cm}
\begin{pspicture}(0,0)(4,4)
\psline[linewidth=0.8pt]{->}(0,1)(4.2,1)
\psline[linewidth=0.8pt]{->}(1,-0.)(1,3.5)
\psplot{0.5}{3.5}{x -2 add x -2 add mul 1 add}
\put(1.8,0.5){$\frac{-b}{2a}$}
\end{pspicture}
&
\psset{xunit=1cm,yunit=1cm}
\begin{pspicture}(0,0)(4,4)
\psline[linewidth=0.8pt]{->}(0,1)(4.2,1)
\psline[linewidth=0.8pt]{->}(1,-0.)(1,3.5)
\psplot{0.5}{3.5}{x -2 add x -2 add mul 1.5 add}
\psline[linewidth=0.8pt,linestyle=dotted](2,1)(2,1.5)
\put(1.8,.5){$\frac{-b}{2a}$}
\end{pspicture}
\\\hline
\vspace{-2cm}
Courbe repr�sentative de $f$,
si $a<0$
\vspace{1cm}\ \,
&
\psset{xunit=1cm,yunit=1cm}
\begin{pspicture}(0,0)(4,4)
\psline[linewidth=0.8pt]{->}(0,1)(4.2,1)
\psline[linewidth=0.8pt]{->}(1,-0.5)(1,3.5)
\psplot{0.2}{3.8}{x -2 add x -2 add mul -1 mul 3 add}
\psline[linewidth=0.8pt,linestyle=dotted](2,1)(2,3)
\put(1.8,0.5){$\frac{-b}{2a}$}
\put(0.1,0.8){$x_1$}
\put(3.1,0.8){$x_2$}
\end{pspicture}
&
\psset{xunit=1cm,yunit=1cm}
\begin{pspicture}(0,0)(4,4)
\psline[linewidth=0.8pt]{->}(0,2.5)(4.2,2.5)
\psline[linewidth=0.8pt]{->}(1,-0.)(1,3.5)
\psplot{0.5}{3.5}{x -2 add x -2 add mul -1 mul 2.5 add}
\put(1.8,2.8){$\frac{-b}{2a}$}
\end{pspicture}
&
\psset{xunit=1cm,yunit=1cm}
\begin{pspicture}(0,0)(4,4)
\psline[linewidth=0.8pt]{->}(0,2.5)(4.2,2.5)
\psline[linewidth=0.8pt]{->}(1,-0.)(1,3.5)
\psplot{0.5}{3.5}{x -2 add x -2 add mul -1 mul 1.9 add}
\psline[linewidth=0.8pt,linestyle=dotted](2,1.9)(2,2.5)
\put(1.8,2.8){$\frac{-b}{2a}$}
\end{pspicture}
\\\hline
\vspd \vspd
Signe de $f(x)$
&
\vspace{0.2cm}\hspace*{-0.3cm}
\scalebox{0.8}{
\begin{tabular}{|c|p{0.9cm}cp{0.9cm}cp{0.9cm}|}\hline
\!$x$\!\! & $-\infty$ & $x_1$ & &$x_2$& \hfill$+\infty$ \\\hline
\!$f$\!\! & Signe de $a$ & \raisebox{-0.2cm}[0.3cm][0.5cm]{\zb} & Signe de -$a$ &\raisebox{-0.2cm}[0.3cm][0.5cm]{\zb} &
Signe de $a$ \\\hline
\end{tabular}
}\vspace{0.2cm}
&
\vspace{0.2cm}\hspace*{0.2cm}
\scalebox{0.8}{
\begin{tabular}{|c|p{0.9cm}cp{0.9cm}|}\hline
\!$x$\!\! & $-\infty$ & $x_0$ & \hfill$+\infty$ \\\hline
\!$f$\!\! & Signe de $a$ &\raisebox{-0.2cm}[0.3cm][0.5cm]{\zb} & Signe de $a$ \\\hline
\end{tabular}
}\vspace{0.2cm}
&
\vspace{0.4cm}\hspace*{0.cm}
\scalebox{0.8}{
\begin{tabular}{|c|p{0.9cm}cr|}\hline
\!$x$\!\! & $-\infty$ & &$+\infty$ \\\hline
\!$f$\!\! & &Signe de $a$& \\\hline
\end{tabular}
}\vspace{0.2cm} \\\hline
\end{tabular}
\end{document}
Télécharger le fichier source