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Matemáticas 3 · Tema 18

Regreso de la parábola

Ejemplo

De la ecuación general salen el vértice, el foco y la directriz: se completa el trinomio y los números del vértice cambian de signo. Si el número es negativo, la parábola abre hacia abajo.

x2−6x+20y+29=0x^2 - 6x + 20y + 29 = 0
b−6-6
mitad−3-3
cuadrado de la mitad99
x2−6x=−20y−29x2−6x+9=−20y−29+9(x−3)(x−3)=−20y−20(x−3)2=−4(5)(y+1)\begin{aligned} x^2 - 6x &= -20y - 29 \\ x^2 - 6x + 9 &= -20y - 29 + 9 \\ (x - 3)(x - 3) &= -20y - 20 \\ (x - 3)^2 &= -4(5)(y + 1) \end{aligned}
vértice(3,−1)(3, -1)
p55

Comprobación

(−7)2−6(−7)+20(−6)+29=049+42−120+29=0120−120=00=0\begin{aligned} (-7)^2 - 6(-7) + 20(-6) + 29 &= 0 \\ 49 + 42 - 120 + 29 &= 0 \\ 120 - 120 &= 0 \\ 0 &= 0 \end{aligned}
(13)2−6(13)+20(−6)+29=0169−78−120+29=0198−198=00=0\begin{aligned} (13)^2 - 6(13) + 20(-6) + 29 &= 0 \\ 169 - 78 - 120 + 29 &= 0 \\ 198 - 198 &= 0 \\ 0 &= 0 \end{aligned}

Ejercicios

Ejercicio 1
x2−4x−12y+16=0x^2 - 4x - 12y + 16 = 0
Ejercicio 2
x2−8x+16y+32=0x^2 - 8x + 16y + 32 = 0
Ejercicio 3
x2−6x−8y+17=0x^2 - 6x - 8y + 17 = 0
Ejercicio 4
x2+4x+12y−56=0x^2 + 4x + 12y - 56 = 0

Respuestas

▼
Ejercicio 1
x2−4x−12y+16=0x^2 - 4x - 12y + 16 = 0
b−4-4
mitad−2-2
cuadrado de la mitad44
x2−4x=12y−16x2−4x+4=12y−16+4(x−2)(x−2)=12y−12(x−2)2=4(3)(y−1)\begin{aligned} x^2 - 4x &= 12y - 16 \\ x^2 - 4x + 4 &= 12y - 16 + 4 \\ (x - 2)(x - 2) &= 12y - 12 \\ (x - 2)^2 &= 4(3)(y - 1) \end{aligned}
vértice(2,1)(2, 1)
p33

Comprobación

(−4)2−4(−4)−12(4)+16=016+16−48+16=048−48=00=0\begin{aligned} (-4)^2 - 4(-4) - 12(4) + 16 &= 0 \\ 16 + 16 - 48 + 16 &= 0 \\ 48 - 48 &= 0 \\ 0 &= 0 \end{aligned}
(8)2−4(8)−12(4)+16=064−32−48+16=080−80=00=0\begin{aligned} (8)^2 - 4(8) - 12(4) + 16 &= 0 \\ 64 - 32 - 48 + 16 &= 0 \\ 80 - 80 &= 0 \\ 0 &= 0 \end{aligned}
Ejercicio 2
x2−8x+16y+32=0x^2 - 8x + 16y + 32 = 0
b−8-8
mitad−4-4
cuadrado de la mitad1616
x2−8x=−16y−32x2−8x+16=−16y−32+16(x−4)(x−4)=−16y−16(x−4)2=−4(4)(y+1)\begin{aligned} x^2 - 8x &= -16y - 32 \\ x^2 - 8x + 16 &= -16y - 32 + 16 \\ (x - 4)(x - 4) &= -16y - 16 \\ (x - 4)^2 &= -4(4)(y + 1) \end{aligned}
vértice(4,−1)(4, -1)
p44

Comprobación

(−4)2−8(−4)+16(−5)+32=016+32−80+32=080−80=00=0\begin{aligned} (-4)^2 - 8(-4) + 16(-5) + 32 &= 0 \\ 16 + 32 - 80 + 32 &= 0 \\ 80 - 80 &= 0 \\ 0 &= 0 \end{aligned}
(12)2−8(12)+16(−5)+32=0144−96−80+32=0176−176=00=0\begin{aligned} (12)^2 - 8(12) + 16(-5) + 32 &= 0 \\ 144 - 96 - 80 + 32 &= 0 \\ 176 - 176 &= 0 \\ 0 &= 0 \end{aligned}
Ejercicio 3
x2−6x−8y+17=0x^2 - 6x - 8y + 17 = 0
b−6-6
mitad−3-3
cuadrado de la mitad99
x2−6x=8y−17x2−6x+9=8y−17+9(x−3)(x−3)=8y−8(x−3)2=4(2)(y−1)\begin{aligned} x^2 - 6x &= 8y - 17 \\ x^2 - 6x + 9 &= 8y - 17 + 9 \\ (x - 3)(x - 3) &= 8y - 8 \\ (x - 3)^2 &= 4(2)(y - 1) \end{aligned}
vértice(3,1)(3, 1)
p22

Comprobación

(−1)2−6(−1)−8(3)+17=01+6−24+17=024−24=00=0\begin{aligned} (-1)^2 - 6(-1) - 8(3) + 17 &= 0 \\ 1 + 6 - 24 + 17 &= 0 \\ 24 - 24 &= 0 \\ 0 &= 0 \end{aligned}
(7)2−6(7)−8(3)+17=049−42−24+17=066−66=00=0\begin{aligned} (7)^2 - 6(7) - 8(3) + 17 &= 0 \\ 49 - 42 - 24 + 17 &= 0 \\ 66 - 66 &= 0 \\ 0 &= 0 \end{aligned}
Ejercicio 4
x2+4x+12y−56=0x^2 + 4x + 12y - 56 = 0
b44
mitad22
cuadrado de la mitad44
x2+4x=−12y+56x2+4x+4=−12y+56+4(x+2)(x+2)=−12y+60(x+2)2=−4(3)(y−5)\begin{aligned} x^2 + 4x &= -12y + 56 \\ x^2 + 4x + 4 &= -12y + 56 + 4 \\ (x + 2)(x + 2) &= -12y + 60 \\ (x + 2)^2 &= -4(3)(y - 5) \end{aligned}
vértice(−2,5)(-2, 5)
p33

Comprobación

(−8)2+4(−8)+12(2)−56=064−32+24−56=088−88=00=0\begin{aligned} (-8)^2 + 4(-8) + 12(2) - 56 &= 0 \\ 64 - 32 + 24 - 56 &= 0 \\ 88 - 88 &= 0 \\ 0 &= 0 \end{aligned}
(4)2+4(4)+12(2)−56=016+16+24−56=056−56=00=0\begin{aligned} (4)^2 + 4(4) + 12(2) - 56 &= 0 \\ 16 + 16 + 24 - 56 &= 0 \\ 56 - 56 &= 0 \\ 0 &= 0 \end{aligned}

Ejercicios extra

▼
Bloque 1
x2−2x−8y+17=0x^2 - 2x - 8y + 17 = 0
x2+6x−12y+21=0x^2 + 6x - 12y + 21 = 0
x2−4x−4y−4=0x^2 - 4x - 4y - 4 = 0
x2+2x+8y−31=0x^2 + 2x + 8y - 31 = 0
x2−10x−16y+25=0x^2 - 10x - 16y + 25 = 0
Bloque 2
x2+8y−24=0x^2 + 8y - 24 = 0
x2+8x−12y+4=0x^2 + 8x - 12y + 4 = 0
x2−6x+4y−15=0x^2 - 6x + 4y - 15 = 0
x2+4x−20y+44=0x^2 + 4x - 20y + 44 = 0
x2−12x+8y+60=0x^2 - 12x + 8y + 60 = 0
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