
\def{text tab1=slib(function/tabsignes [[x,-Inf,,+Inf;f(x),,+,]])}
\def{text tab2=slib(function/tabsignes [[x,-Inf,,x_S,,+Inf;f(x),,+,0,+]])}
\def{text tab3=slib(function/tabsignes [[x,-Inf,,x_1,,x_2,,+Inf;f(x),,+,0,-,0,+]])}
\def{text tab4=slib(function/tabsignes [[x,-Inf,,+Inf;f(x),,-,]])}
\def{text tab5=slib(function/tabsignes [[x,-Inf,,x_S,,+Inf;f(x),,-,0,-]])}
\def{text tab6=slib(function/tabsignes [[x,-Inf,,x_2,,x_1,,+Inf;f(x),,-,0,+,0,-]])}

Trinôme \(f(x)=a*x^2+b*x+c) avec \(a \ne 0),
\(\Delta=b^2-4 a c),
\(f(x) = a (x - x_S)^2 + y_S) avec \(x_S=\frac{-b}{2a}) et \(y_S=f(x_S))

  \(\Delta \lt 0)
 \(\Delta = 0) \(\Delta \gt 0)
 \(a \gt 0)
 \draw{200,200}{
xrange -2,8
yrange -2,8
arrow -2,0,8,0,8, black
arrow 0,-2,0,8,8, black
dline 0,1,3,1,red
dline 3,0,3,1,red
text red,2.9,-0.1 , large, x
text red,3.4,-0.5, small, S
text red,-1,1.5, large,y
text red,-0.5,1, small, S
plot black,(x-3)^2+1
text black,1.7,4.5, large, f
}

cas 1
 \tab1
  \draw{200,200}{
xrange -2,8
yrange -2,8
arrow -2,0,8,0,8, black
text red,2.9,-0.1,large,x
text red,3.4,-0.5,small,S
plot black, (x-3)^2
text black1.7,4.5, large, f
}

cas 2
\tab2
  \draw{200,200}{
xrange -2,8
yrange -5,5
arrow -2,0,8,0,8, black
arrow 0,-5,0,5,8, black
dline 0,-2,3,-2,red
dline 3,0,3,-2,red
text red,2.9,1,large, x
text red,3.4,0.6,small, S
text red,1.1,-0.1,large, x
text red,1.6,-0.6,small, 1
text red,4.5,-0.1,large, x
text red,5,-0.6,small, 2
text red,-1,-1.5,large,y
text red,-0.5,-2,small, S
plot black, (x-3)^2-2
text black,1.7,1.5,large, f
}

cas 3
\tab3

 \(a \lt 0) \draw{200,200}{
xrange -2,8
yrange -8,2
arrow -2,0,8,0,8, black
arrow 0,-8,0,2,8, black
dline 0,-1,3,-1,red
dline 3,0,3,-1,red
text red,  2.9,1 , large, x
text red,  3.4,0.6, small, S
text red,  -1,-0.5, large,y
text red,  -0.5,-1, small, S
plot black, -(x-3)^2-1
text black,  1.7,-4.5, large, f
}
\tab4
cas 4
  \draw{200,200}{
xrange -2,8
yrange -8,2
arrow -2,0,8,0,8, black
text red,  2.9,1 , large, x
text red,  3.4,0.6, small, S
plot black, -(x-3)^2
text black,  1.7,-4.5, large, f
}

\tab5
cas 5
  \draw{200,200}{
xrange -2,8
yrange -5,5
arrow -2,0,8,0,8, black
arrow 0,-5,0,5,8, black
dline 0,2,3,2,red
dline 3,0,3,2,red
text red,  2.9,-0.1, large, x
text red,  3.4,-0.6, small, S
text red,  1.6,-0.1 , large, x
text red,  2.1,-0.6, small, 2
text red,  4,-0.1 , large, x
text red,  4.5,-0.6, small, 1
text red,  -1,2.5, large,y
text red,  -0.5,2, small, S
plot black, -(x-3)^2+2
text black,  1.7,-1.5, large, f
}

\tab6
cas 6

 Racines
Solutions de
\(f(x) = 0) Pas de racine. \(x_S = x_1 = x_2 = \frac{-b}{2 a}) \(x_1=\frac{-b-\sqrt{\Delta}}{2 a}) et \(x_2=\frac{-b+\sqrt{\Delta}}{2 a})
 Factorisation Pas de factorisation \(f(x) = a (x - x_S)^2) \(f(x) = a (x - x_1) (x - x_2))

