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Matemáticas 3 · Tema 10

Gráfica de una ecuación cuadrática

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Ejemplo

f(x)=x2−8x+13f(x) = x^2 - 8x + 13
f(x)=x2−8x+16−16+13f(x)=(x−4)(x−4)−3f(x)=(x−4)2−3\begin{aligned} f(x) &= x^2 - 8x + 16 - 16 + 13 \\ f(x) &= (x - 4)(x - 4) - 3 \\ f(x) &= (x - 4)^2 - 3 \end{aligned}
b−8-8
mitad−4-4
cuadrado de la mitad1616
V=(4, −3)V = (4,\, -3)
xxf(x)f(x)(x)2−8(x)+13(x)^2 - 8(x) + 13
221\mathbf{1}(2)2−8(2)+13=4−16+13=1(2)^2 - 8(2) + 13 = 4 - 16 + 13 = \mathbf{1}
33−2\mathbf{-2}(3)2−8(3)+13=9−24+13=−2(3)^2 - 8(3) + 13 = 9 - 24 + 13 = \mathbf{-2}
44−3\mathbf{-3}(4)2−8(4)+13=16−32+13=−3(4)^2 - 8(4) + 13 = 16 - 32 + 13 = \mathbf{-3}
55−2\mathbf{-2}(5)2−8(5)+13=25−40+13=−2(5)^2 - 8(5) + 13 = 25 - 40 + 13 = \mathbf{-2}
661\mathbf{1}(6)2−8(6)+13=36−48+13=1(6)^2 - 8(6) + 13 = 36 - 48 + 13 = \mathbf{1}

Ejercicios

Ejercicio 1
f(x)=x2−6x+18f(x) = x^2 - 6x + 18
Ejercicio 2
f(x)=x2+10x+16f(x) = x^2 + 10x + 16
Ejercicio 3
f(x)=x2+8x−9f(x) = x^2 + 8x - 9
Ejercicio 4
f(x)=x2+2x−12f(x) = x^2 + 2x - 12

Respuestas

▼
Ejercicio 1
f(x)=x2−6x+18f(x) = x^2 - 6x + 18
f(x)=x2−6x+9−9+18f(x)=(x−3)(x−3)+9f(x)=(x−3)2+9\begin{aligned} f(x) &= x^2 - 6x + 9 - 9 + 18 \\ f(x) &= (x - 3)(x - 3) + 9 \\ f(x) &= (x - 3)^2 + 9 \end{aligned}
b−6-6
mitad−3-3
cuadrado de la mitad99
V=(3, 9)V = (3,\, 9)
xxf(x)f(x)(x)2−6(x)+18(x)^2 - 6(x) + 18
1113\mathbf{13}(1)2−6(1)+18=1−6+18=13(1)^2 - 6(1) + 18 = 1 - 6 + 18 = \mathbf{13}
2210\mathbf{10}(2)2−6(2)+18=4−12+18=10(2)^2 - 6(2) + 18 = 4 - 12 + 18 = \mathbf{10}
339\mathbf{9}(3)2−6(3)+18=9−18+18=9(3)^2 - 6(3) + 18 = 9 - 18 + 18 = \mathbf{9}
4410\mathbf{10}(4)2−6(4)+18=16−24+18=10(4)^2 - 6(4) + 18 = 16 - 24 + 18 = \mathbf{10}
5513\mathbf{13}(5)2−6(5)+18=25−30+18=13(5)^2 - 6(5) + 18 = 25 - 30 + 18 = \mathbf{13}
Ejercicio 2
f(x)=x2+10x+16f(x) = x^2 + 10x + 16
f(x)=x2+10x+25−25+16f(x)=(x+5)(x+5)−9f(x)=(x+5)2−9\begin{aligned} f(x) &= x^2 + 10x + 25 - 25 + 16 \\ f(x) &= (x + 5)(x + 5) - 9 \\ f(x) &= (x + 5)^2 - 9 \end{aligned}
b1010
mitad55
cuadrado de la mitad2525
V=(−5, −9)V = (-5,\, -9)
xxf(x)f(x)(x)2+10(x)+16(x)^2 + 10(x) + 16
−3-3−5\mathbf{-5}(−3)2+10(−3)+16=9−30+16=−5(-3)^2 + 10(-3) + 16 = 9 - 30 + 16 = \mathbf{-5}
−4-4−8\mathbf{-8}(−4)2+10(−4)+16=16−40+16=−8(-4)^2 + 10(-4) + 16 = 16 - 40 + 16 = \mathbf{-8}
−5-5−9\mathbf{-9}(−5)2+10(−5)+16=25−50+16=−9(-5)^2 + 10(-5) + 16 = 25 - 50 + 16 = \mathbf{-9}
−6-6−8\mathbf{-8}(−6)2+10(−6)+16=36−60+16=−8(-6)^2 + 10(-6) + 16 = 36 - 60 + 16 = \mathbf{-8}
−7-7−5\mathbf{-5}(−7)2+10(−7)+16=49−70+16=−5(-7)^2 + 10(-7) + 16 = 49 - 70 + 16 = \mathbf{-5}
Ejercicio 3
f(x)=x2+8x−9f(x) = x^2 + 8x - 9
f(x)=x2+8x+16−16−9f(x)=(x+4)(x+4)−25f(x)=(x+4)2−25\begin{aligned} f(x) &= x^2 + 8x + 16 - 16 - 9 \\ f(x) &= (x + 4)(x + 4) - 25 \\ f(x) &= (x + 4)^2 - 25 \end{aligned}
b88
mitad44
cuadrado de la mitad1616
V=(−4, −25)V = (-4,\, -25)
xxf(x)f(x)(x)2+8(x)−9(x)^2 + 8(x) - 9
−2-2−21\mathbf{-21}(−2)2+8(−2)−9=4−16−9=−21(-2)^2 + 8(-2) - 9 = 4 - 16 - 9 = \mathbf{-21}
−3-3−24\mathbf{-24}(−3)2+8(−3)−9=9−24−9=−24(-3)^2 + 8(-3) - 9 = 9 - 24 - 9 = \mathbf{-24}
−4-4−25\mathbf{-25}(−4)2+8(−4)−9=16−32−9=−25(-4)^2 + 8(-4) - 9 = 16 - 32 - 9 = \mathbf{-25}
−5-5−24\mathbf{-24}(−5)2+8(−5)−9=25−40−9=−24(-5)^2 + 8(-5) - 9 = 25 - 40 - 9 = \mathbf{-24}
−6-6−21\mathbf{-21}(−6)2+8(−6)−9=36−48−9=−21(-6)^2 + 8(-6) - 9 = 36 - 48 - 9 = \mathbf{-21}
Ejercicio 4
f(x)=x2+2x−12f(x) = x^2 + 2x - 12
f(x)=x2+2x+1−1−12f(x)=(x+1)(x+1)−13f(x)=(x+1)2−13\begin{aligned} f(x) &= x^2 + 2x + 1 - 1 - 12 \\ f(x) &= (x + 1)(x + 1) - 13 \\ f(x) &= (x + 1)^2 - 13 \end{aligned}
b22
mitad11
cuadrado de la mitad11
V=(−1, −13)V = (-1,\, -13)
xxf(x)f(x)(x)2+2(x)−12(x)^2 + 2(x) - 12
−3-3−9\mathbf{-9}(−3)2+2(−3)−12=9−6−12=−9(-3)^2 + 2(-3) - 12 = 9 - 6 - 12 = \mathbf{-9}
−2-2−12\mathbf{-12}(−2)2+2(−2)−12=4−4−12=−12(-2)^2 + 2(-2) - 12 = 4 - 4 - 12 = \mathbf{-12}
−1-1−13\mathbf{-13}(−1)2+2(−1)−12=1−2−12=−13(-1)^2 + 2(-1) - 12 = 1 - 2 - 12 = \mathbf{-13}
00−12\mathbf{-12}(0)2+2(0)−12=0+0−12=−12(0)^2 + 2(0) - 12 = 0 + 0 - 12 = \mathbf{-12}
11−9\mathbf{-9}(1)2+2(1)−12=1+2−12=−9(1)^2 + 2(1) - 12 = 1 + 2 - 12 = \mathbf{-9}

Ejercicios extra

▼
Bloque 1
f(x)=x2−4x+7f(x) = x^2 - 4x + 7
f(x)=x2+6x+5f(x) = x^2 + 6x + 5
f(x)=x2−2x−8f(x) = x^2 - 2x - 8
f(x)=x2+4x−5f(x) = x^2 + 4x - 5
f(x)=x2−10x+21f(x) = x^2 - 10x + 21
Bloque 2
f(x)=x2+8x+12f(x) = x^2 + 8x + 12
f(x)=x2−6x−7f(x) = x^2 - 6x - 7
f(x)=x2+12x+32f(x) = x^2 + 12x + 32
f(x)=x2−8x+20f(x) = x^2 - 8x + 20
f(x)=x2+2x−3f(x) = x^2 + 2x - 3
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