Subversion Repositories wimsdev

Rev

Rev 16458 | Go to most recent revision | Details | Last modification | View Log | RSS feed

Rev Author Line No. Line
16297 bpr 1 
<p>Trinôme \(f(x)=a*x^2+b*x+c) avec \(a \ne 0)&nbsp&nbsp&nbsp&nbsp&nbsp
2 
\(\Delta=b^2-4 a c)&nbsp&nbsp&nbsp&nbsp&nbsp
3 
\(f(x) = a (x - x_S)^2 + y_S) avec \(x_S=\frac{-b}{2a}) et \(y_S=f(x_S))</p>
4 
<table class="wimsborder wimscenter">
5 
<tr><td>\(\Delta \lt 0)</td>
6 
<td >\(\Delta = 0)</td><td>\(\Delta \gt 0)</td></tr>
7 
<tr><td>\(a \gt 0)</td>
8 
 <td>\draw{200,200}{
9 
xrange -2,8
10 
yrange -2,8
11 
arrow -2,0,8,0,8, black
12 
arrow 0,-2,0,8,8, black
13 
dline 0,1,3,1,red
14 
dline 3,0,3,1,red
15 
text red,  2.9,-0.1 , large, x
16 
text red,  3.4,-0.5, small, S
17 
 
18 
text red,  -1,1.5, large,y
19 
text red,  -0.5,1, small, S
20 
 
21 
plot black, (x-3)^2+1
22 
text black,  1.7,4.5, large, f
23 
}
24 
<table border=1 border="solid">
25 
     <tr><td>\(x)</td><td><table border=0 width=150><tr><td align="left">\(-\infty)</td><td align="right">\(+\infty)</td></table></td></tr>
26 
     <tr><td>\(f(x))</td><td align="center">+</td></tr>
27 
  </table>
28 
<br>cas 1
29 
 </td>
30 
 <td>
31 
\draw{200,200}{
32 
xrange -2,8
33 
yrange -2,8
34 
arrow -2,0,8,0,8, black
35 
text red,  2.9,-0.1 , large, x
36 
text red,  3.4,-0.5, small, S
37 
 
38 
 
39 
plot black, (x-3)^2
40 
text black,  1.7,4.5, large, f
41 
 
42 
}
43 
<br><br>
44 
  <table border=1 border="solid">
45 
     <tr><td>\(x)</td><td><table border=0 width=150><tr><td align="left">\(-\infty)</td><td align="center">\(x_S)</td><td align="right">\(+\infty)</td></tr></table></td></tr>
46 
     <tr><td>\(f(x))</td><td align="center"><table border=0 width=150><tr><td align="right">+</td></td><td align="center">0<td align="left">+</td></tr></table></td></tr>
47 
  </table>
48 
<br>cas 2
49 
 </td>
50 
 <td>
51 
\draw{200,200}{
52 
xrange -2,8
53 
yrange -5,5
54 
arrow -2,0,8,0,8, black
55 
arrow 0,-5,0,5,8, black
56 
dline 0,-2,3,-2,red
57 
dline 3,0,3,-2,red
58 
text red,  2.9,1 , large, x
59 
text red,  3.4,0.6, small, S
60 
 
61 
text red,  1.1,-0.1 , large, x
62 
text red,  1.6,-0.6, small, 1
63 
text red,  4.5,-0.1 , large, x
64 
text red,  5,-0.6, small, 2
65 
 
66 
 
67 
text red,  -1,-1.5, large,y
68 
text red,  -0.5,-2, small, S
69 
 
70 
plot black, (x-3)^2-2
71 
text black,  1.7,1.5, large, f
72 
 
73 
}
74 
<br><br>
75 
  <table border=1 border="solid">
76 
     <tr><td>\(x)</td><td><table border=0 width=180><tr><td align="left">\(-\infty)</td><td align="center">\(x_1)</td><td width=20%></td><td align="center">\(x_2)</td><td align="right">\(+\infty)</td></tr></table></td></tr>
77 
     <tr><td>\(f(x))</td><td align="center"><table border=0 width=180><tr><td align="right">+</td></td><td align="center" width=20%>0<td align="center">-</td><td align="center">0<td align="left">+</td></tr></table></td></tr>
78 
  </table>
79 
<br>cas 3
80 
 </td></tr>
81 
<tr>
82 
<td>\(a \lt 0)
83 
</td>
84 
 <td>
85 
\draw{200,200}{
86 
xrange -2,8
87 
yrange -8,2
88 
arrow -2,0,8,0,8, black
89 
arrow 0,-8,0,2,8, black
90 
dline 0,-1,3,-1,red
91 
dline 3,0,3,-1,red
92 
text red,  2.9,1 , large, x
93 
text red,  3.4,0.6, small, S
94 
 
95 
text red,  -1,-0.5, large,y
96 
text red,  -0.5,-1, small, S
97 
 
98 
plot black, -(x-3)^2-1
99 
text black,  1.7,-4.5, large, f
100 
 
101 
}<br><br>
102 
  <table border=1 border="solid">
103 
     <tr><td>\(x)</td><td><table border=0 width=150><tr><td align="left">\(-\infty)</td><td align="right">\(+\infty)</td></table></td></tr>
104 
     <tr><td>\(f(x))</td><td align="center">-</td></tr>
105 
  </table>
106 
<br>cas 4
107 
 </td>
108 
 <td>
109 
\draw{200,200}{
110 
xrange -2,8
111 
yrange -8,2
112 
arrow -2,0,8,0,8, black
113 
text red,  2.9,1 , large, x
114 
text red,  3.4,0.6, small, S
115 
 
116 
 
117 
plot black, -(x-3)^2
118 
text black,  1.7,-4.5, large, f
119 
 
120 
}
121 
<br><br>
122 
  <table border=1 border="solid">
123 
     <tr><td>\(x)</td><td><table border=0 width=150><tr><td align="left">\(-\infty)</td><td align="center">\(x_S)</td><td align="right">\(+\infty)</td></tr></table></td></tr>
124 
     <tr><td>\(f(x))</td><td align="center"><table border=0 width=150><tr><td align="right">-</td></td><td align="center">0<td align="left">-</td></tr></table></td></tr>
125 
  </table>
126 
<br>cas 5
127 
 </td>
128 
 <td>
129 
\draw{200,200}{
130 
xrange -2,8
131 
yrange -5,5
132 
arrow -2,0,8,0,8, black
133 
arrow 0,-5,0,5,8, black
134 
dline 0,2,3,2,red
135 
dline 3,0,3,2,red
136 
text red,  2.9,-0.1, large, x
137 
text red,  3.4,-0.6, small, S
138 
 
139 
text red,  1.6,-0.1 , large, x
140 
text red,  2.1,-0.6, small, 2
141 
text red,  4,-0.1 , large, x
142 
text red,  4.5,-0.6, small, 1
143 
 
144 
 
145 
text red,  -1,2.5, large,y
146 
text red,  -0.5,2, small, S
147 
 
148 
plot black, -(x-3)^2+2
149 
text black,  1.7,-1.5, large, f
150 
 
151 
}
152 
<br><br>
153 
  <table border=1 border="solid">
154 
     <tr><td>\(x)</td><td><table border=0 width=180><tr><td align="left">\(-\infty)</td><td align="center">\(x_2)</td><td width=20%></td><td align="center">\(x_1)</td><td align="right">\(+\infty)</td></tr></table></td></tr>
155 
     <tr><td>\(f(x))</td><td align="center"><table border=0 width=180><tr><td align="right">-</td></td><td align="center" width=20%>0<td align="center">+</td><td align="center">0<td align="left">-</td></tr></table></td></tr>
156 
  </table>
157 
<br>cas 6
158 
 </td>
159 
</tr>
160 
<tr><td>Racines<br>Solutions de<br>\(f(x) = 0)</td><td>Pas de racine.</td><td>\(x_S = x_1 = x_2 = \frac{-b}{2 a})</td><td>\(x_1=\frac{-b-\sqrt{\Delta}}{2 a}) et \(x_2=\frac{-b+\sqrt{\Delta}}{2 a})</td></tr>
161 
<tr><td>Factorisation</td><td>Pas de factorisation</td><td>\(f(x) = a (x - x_S)^2)</td><td>\(f(x) = a (x - x_1) (x - x_2))</td></tr>
162 
 
163 
</table>