Lecture Parabole 2
Author: Joël Gauvain
Date: 20 Mars 2013

-3/25(x+2)(x-8)

Nouvelle courbe

?

?

?

f(x)=a\,\left(x-x_1)(x-x_2)

\text{La courbe tracée représente la fonction }

\text{où }a , \  x_1 \text{ et }x_2 \text{ sont trois réels à déterminer.}

Valider

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

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

\text{pour tout réel } x \;:\

f(x)=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

a\left(x-x_1\right)\,\left(x-x_2\right)=a\left(x^2-\left(x_1+x_2\right)\,x+x_1x_2\right)

\text{donc } f(x)=a\,x^2-a\left(x_1+x_2\right)\,x+a\,x_1\,x_2

\text{ainsi }\ovalbox{\,\text{b}=-a\left(x_1+x_2\right)}\text{ et }\ovalbox{\,\text{c}=a\,x_1\,x_2}

\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}(x\,;\,y)\text{ est un autre point de la parabole,}

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

\text{ainsi }\ovalbox{\text{a}=\frac{y-\beta}{(x-\alpha)^2}}

Cacher les vecteurs

\text{Vous devez utiliser des valeurs exactes !}

\text{Vous devez utiliser des valeurs exactes !}

\text{Dernier exercice}

f(x) =

f(x)=

rc: 3
rb: 2
yA = 3
xA = 3
A: (xA, yA)
D: x = xA
vy = -3
vx = 5
k_5(x) = (vy / vx^(2) * x^(2)) + (rb * x) + rc
B: A + (vx, vy)
B' = Mirror(B, D)
f_5 = Polynomial({B', A, B})
ra: 2
a1 = SurdText(vy / vx^(2))
aentexte = SurdText(vy / vx^(2))
h(x) = ((-3 / 25 * (x + 2)) * (x - 8))
texte48 = LaTeX(h, true, true)
posh = IndexOf("=", texte48)
texteh = Take(texte48, posh + 1)
poscarré = IndexOf("^{2}", texteh)
sanscarre: ¬Defined[poscarré]
ralpha: 3
ralpha2: (-ralpha)
rbeta: 4
a2 = SurdText(ralpha2, {3.141592653589793})
a3 = SurdText(rbeta, {3.141592653589793})
signa2 = If(ralpha  <  0, "+", "")
a1b = If(a1 ≟ "1", "", aentexte)
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)
texte2: "f(x)=" + (LaTeX[a11]) + "x^2" + (LaTeX[signa3]) + "" + (LaTeX[a3]) + ""
texte3: "f(x)=" + (LaTeX[a11]) + "x^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"
juste1: h(11) ≟ f_5(11) ∧ h(13) ≟ f_5(13) ∧ h(-13) ≟ f_5(-13) ∧ sanscarre
tester1: true
gain1 = 0
score1 = 3 (slider from -5 to 5, step 0.1)
a = Segment(C, G)
ChampTexte4 = Textfield(tb)
ChampTexte5 = Textfield(tc)
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]) + ""
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)
def61: Defined[rb] ∧ Defined[rc] ∧ rb ≟ 1 ∧ rc ≟ 0
def62: Defined[rb] ∧ Defined[rc] ∧ rb ≟ -1 ∧ rc ≟ 0
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)
def02: Defined[rb] ∧ Defined[rc] ∧ rb ≟ 0 ∧ (rc ≠ 0)
estfrac: ¬(Mod[vy, vx^(2)] ≟ 0)
texte28: "\text{a}=" + (LaTeX[a1]) + ""
juste2: f_5(1) ≟ k_5(1) ∧ f_5(3) ≟ k_5(3) ∧ f_5(-1) ≟ k_5(-1)
tester2: false
gain2 = 0
demar: false
score2 = 0
numexo = 5
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
texte43: "\text{Formes développées trouvées : }" + (LaTeX[score2]) + "\text{ sur }" + (LaTeX[numexo]) + ""
texte44: "\text{Ici on a : }\fgcolor{ff0000}{\text{a}}=\frac{\fgcolor{004000}{" + (LaTeX[vy]) + "}}{\fgcolor{ff00ff}{" + (LaTeX[vx]) + "}^2}"
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
troplong = false
troplong2 = false
grade: (score1 + score2) / 2
notemax: 10
ChampTexte1 = Textfield(tff)
factf = If(Defined[h], texteh, "?")
texte1: "f(x)=" + (LaTeX[factf]) + "\phantom{\frac{2}{5}}"
juste0: h(11) ≟ f_5(11) ∧ h(13) ≟ f_5(13) ∧ h(-13) ≟ f_5(-13)
texte24: "f(x)=" + (LaTeX[a11]) + "x^2+x" + (LaTeX[signc]) + "" + (LaTeX[c1]) + "\phantom{\frac{2}{2}}"
texte42: "\text{Formes factorisées trouvées  : }" + (LaTeX[score1]) + "\text{ sur }" + (LaTeX[numexo]) + ""
texte26: "f(x)=" + (LaTeX[a11]) + "x^2" + (LaTeX[signc]) + "" + (LaTeX[c1]) + "\phantom{\frac{2}{2}}"
texte25: "f(x)=" + (LaTeX[a11]) + "x^2-x" + (LaTeX[signc]) + "" + (LaTeX[c1]) + "\phantom{\frac{2}{2}}"
texte22: "f(x)=" + (LaTeX[a11]) + "x^2-x\phantom{\frac{2}{2}}"
texte21: "f(x)=" + (LaTeX[a11]) + "x^2+x\phantom{\frac{2}{2}}"
texte27: "a=" + (LaTeX[aentexte]) + "\phantom{\frac{2}{2}}"