canonique3ahtml
Author: Joël Gauvain
Date: 20 Mars 2013

Nouvelle courbe

2

0

1

a = .

α = .

β = .

f(x)=\text{a}(x-\alpha)^2+\beta

\text{Entrez ci-dessous les valeurs de  a } ,\,\,\alpha\text{ et }\beta\,.

\text{a}=

\alpha=

\beta=

\text{La courbe tracée représente la fonction }

\text{où a , }\alpha \text{ et }\beta\text{ sont trois réels à déterminer.}

Valider

\text{Soit a , b, et c les trois réels tels que}

f\text{ définie pour tout réel }x \text{ par :}

\text{pour tout réel } x \;:\

f(x)=\text{a}\,x^2+\text{b}\,x+\text{c}

\text{Entrez ci-dessous les valeurs de b et c.}

2

3

b = .

c =  .

\text{a est non nul}

\text{La courbe rouge correspond }

\text{à votre réponse. Corrigez !}

\text{b}=

\text{c}=

Valider

\text{La courbe rouge correspond}

\text{à votre réponse. Corrigez !}

Commencer la série d'exercices

\text{a}(x-\alpha)^2+\beta=\text{a}\left(x^2-2\alpha x+\alpha^2\right)+\beta

\text{donc } f(x)=\text{a}x^2-2\text{a}\alpha x+\text{a}\alpha^2+\beta

\text{ainsi }\ovalbox{\,\text{b}=-2\text{a}\alpha\,}\text{ et }\ovalbox{\,\text{c}=\text{a}\alpha^2+\beta}

\text{Remarque : }\ovalbox{\text{c}=f(0)}

Cacher l'aide

\text{Le sommet S de la parabole a pour  }

\text{coordonnées }(\alpha\,;\,\beta\,).

\text{Si M}(\alpha+1\,;\,y)\text{ est un point de la parabole,}

y=\text{a}(\alpha+1-\alpha)^2+\beta\text{  donc  }y=\text{a}+\beta

\text{ainsi }\ovalbox{\text{a}=y-\beta}

Cacher les vecteurs

\text{Dernier exercice}

ra: 2
a1 = SurdText(ra, {3.141592653589793})
a1b = If(a1 ≟ "1", "", a1)
ralpha: 0
ralpha2: (-ralpha)
xA = 0
yA = 1
A: (xA, yA)
vx = 1
vy = 2
B: A + (vx, vy)
D: x = xA
B' = Mirror(B, D)
f_5 = Polynomial({B', A, B})
rbeta: 1
ChampTexte1 = Textfield(ta)
ChampTexte2 = Textfield(talpha)
ChampTexte3 = Textfield(tbeta)
a2 = SurdText(ralpha2, {3.141592653589793})
a3 = SurdText(rbeta, {3.141592653589793})
signa2 = If(ralpha  <  0, "+", "")
a11 = If(ra ≟ -1, "-", a1b)
signa3 = If(rbeta > 0, "+", "")
def1: Defined[ra] ∧ Defined[ralpha] ∧ Defined[rbeta] ∧ (ralpha ≠ 0) ∧ (rbeta ≠ 0) ∧ (ra ≠ 0)
def2: Defined[a1] ∧ Defined[a2] ∧ Defined[a3] ∧ ralpha ≟ 0 ∧ (rbeta ≠ 0) ∧ (ra ≠ 0)
def3: Defined[a1] ∧ Defined[a2] ∧ Defined[a3] ∧ ralpha ≟ 0 ∧ rbeta ≟ 0 ∧ (ra ≠ 0)
def4: Defined[a1] ∧ Defined[a2] ∧ Defined[a3] ∧ (ralpha ≠ 0) ∧ rbeta ≟ 0 ∧ (ra ≠ 0)
texte1: "f(x)=" + (LaTeX[a11]) + "\left(x" + (LaTeX[signa2]) + "" + (LaTeX[a2]) + "\right)^2" + (LaTeX[signa3]) + "" + (LaTeX[a3]) + ""
texte2: "f(x)=" + (LaTeX[a11]) + "x^2" + (LaTeX[signa3]) + "" + (LaTeX[a3]) + ""
texte3: "f(x)=" + (LaTeX[a11]) + "x^2"
juste1: xA ≟ ralpha ∧ yA ≟ rbeta ∧ ra ≟ vy / vx^(2)
C = (9, 30)
E = (50, 30)
F = (50, -30)
G = (9, -30)
poly1, cs, e, f_1, g_1 = Polygon(C, E, F, G)
texte4: "f(x)=" + (LaTeX[a11]) + "\left(x" + (LaTeX[signa2]) + "" + (LaTeX[a2]) + "\right)^2"
tester1: true
gain1 = 1
score1 = 1 (slider from -5 to 5, step 0.1)
a = Segment(C, G)
ChampTexte4 = Textfield(tb)
ChampTexte5 = Textfield(tc)
rb: 2
rc: 3
b1 = SurdText(rb, {3.141592653589793})
c1 = SurdText(rc, {3.141592653589793})
signb = If(rb > 0, "+", "")
signc = If(rc > 0, "+", "")
textef: "f(x)=" + (LaTeX[a11]) + "x^2" + (LaTeX[signb]) + "" + (LaTeX[b1]) + "x" + (LaTeX[signc]) + "" + (LaTeX[c1]) + "\phantom{\frac{2}{2}}"
g_5(x) = (ra * (x - ralpha)^(2)) + rbeta
juste2: rb ≟ ((-2 * ra) * ralpha) ∧ rc ≟ (ra * ralpha^(2)) + rbeta
texte20: "f(x)=" + (LaTeX[a11]) + "x^2" + (LaTeX[signb]) + "" + (LaTeX[b1]) + "x"
def5: Defined[rb] ∧ Defined[rc] ∧ (rb ≠ 0) ∧ (rc ≠ 0) ∧ (abs(rb) ≠ 1)
def6: Defined[rb] ∧ Defined[rc] ∧ (rb ≠ 0) ∧ rc ≟ 0 ∧ (abs(rb) ≠ 1)
texte21: "f(x)=" + (LaTeX[a11]) + "x^2+x"
def61: Defined[rb] ∧ Defined[rc] ∧ rb ≟ 1 ∧ rc ≟ 0
def62: Defined[rb] ∧ Defined[rc] ∧ rb ≟ -1 ∧ rc ≟ 0
texte22: "f(x)=" + (LaTeX[a11]) + "x^2-x"
texte23: "f(x)=" + (LaTeX[a11]) + "x^2"
def00: Defined[rb] ∧ Defined[rc] ∧ rb ≟ 0 ∧ rc ≟ 0
def12: Defined[rb] ∧ Defined[rc] ∧ rb ≟ 1 ∧ (rc ≠ 0)
def22: Defined[rb] ∧ Defined[rc] ∧ rb ≟ -1 ∧ (rc ≠ 0)
texte25: "f(x)=" + (LaTeX[a11]) + "x^2-x" + (LaTeX[signc]) + "" + (LaTeX[c1]) + ""
def02: Defined[rb] ∧ Defined[rc] ∧ rb ≟ 0 ∧ (rc ≠ 0)
texte26: "f(x)=" + (LaTeX[a11]) + "x^2" + (LaTeX[signc]) + "" + (LaTeX[c1]) + ""
texte27: "\text{a}=" + (LaTeX[a11]) + ""
estfrac: ¬(Mod[vy, vx^(2)] ≟ 0)
texte28: "\text{a}=" + (LaTeX[a1]) + ""
tester2: false
h(x) = (ra * x^(2)) + (rb * x) + rc
gain2 = 1
demar: false
score2 = 0
numexo = 2
H = (-4, -2)
I = (8.5, -2)
J = (8.5, -8)
K = (-4, -8)
poly2, h_1, i, j, k = Polygon(H, I, J, K)
aide2: false
aide1: false
texte42: "\text{Formes canoniques trouvées : }" + (LaTeX[score1]) + "\text{ sur }" + (LaTeX[numexo]) + ""
texte43: "\text{Formes développées trouvées : }" + (LaTeX[score2]) + "\text{ sur }" + (LaTeX[numexo]) + ""
A': A + (vx, 0)
u = Vector(A, A')
v = Vector(A', B)
L: A + (1, 0)
M: L + (0, vy / vx^(2))
w = Vector(L, M)
voirvec: false
N = Corner(texte1, 1)
O = Corner(texte1, 4)
P = Midpoint(O, N)
Q = Corner(texte1, 2)
R = Corner(texte1, 3)
S = Midpoint(R, Q)
T = Corner(textef, 1)
U = Corner(textef, 2)
V = Corner(textef, 3)
W = Corner(textef, 4)
Z = Midpoint(W, T)
A_1 = Midpoint(V, U)
b = Segment(L, M)
texte44: "\text{Ici on a : }\fgcolor{ff0000}{\text{a}}=\fgcolor{ff00000}{" + (LaTeX[vy]) + "}"
grade: (score1 + score2) / 2
notemax: 10
texte24: "f(x)=" + (LaTeX[a11]) + "x^2+x" + (LaTeX[signc]) + "" + (LaTeX[c1]) + "\phantom{\frac{2}{2}}"