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Matemáticas 2 · Tema 6

Funciones cuadráticas

Mini apunte

Una función cuadrática es de la forma f(x)=ax2+bx+cf(x) = ax^2 + bx + c. Para graficarla en un intervalo dado, construimos una tabla de valores calculando la imagen (el resultado de evaluar) cada número entero en dicho intervalo. La representación gráfica es una curva llamada **parábola**:
  • Si a>0a > 0 (coeficiente principal positivo), la parábola abre **hacia arriba** (tiene un punto mínimo).
  • Si a<0a < 0 (coeficiente principal negativo), la parábola abre **hacia abajo** (tiene un punto máximo).

Ejemplo

Función: f(x)=x2−6x+5f(x) = x^2 - 6x + 5

Intervalo: f:[0,6]oRf: [0, 6] o \mathbb{R}

xf(x)
05
10
2−3
3−4
4−3
50
65
f(0)=(0)2−6(0)+5=f(0) = (0)^2 - 6(0) + 5 = 55
f(1)=(1)2−6(1)+5=1−6+5=f(1) = (1)^2 - 6(1) + 5 = 1-6+5 = 00
f(2)=(2)2−6(2)+5=4−12+5=f(2) = (2)^2 - 6(2) + 5 = 4-12+5 = −3-3
f(3)=(3)2−6(3)+5=9−18+5=f(3) = (3)^2 - 6(3) + 5 = 9-18+5 = −4-4
f(4)=(4)2−6(4)+5=16−24+5=f(4) = (4)^2 - 6(4) + 5 = 16-24+5 = −3-3
f(5)=(5)2−6(5)+5=25−30+5=f(5) = (5)^2 - 6(5) + 5 = 25-30+5 = 00
f(6)=(6)2−6(6)+5=36−36+5=f(6) = (6)^2 - 6(6) + 5 = 36-36+5 = 55
-1 1 2 3 4 5 6 7 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y

Ejercicios

Bloque 1
f(x)=x2−2x−8,[−2,4]f(x) = x^2 - 2x - 8, \quad [-2, 4]
f(x)=x2−6x+8,[0,6]f(x) = x^2 - 6x + 8, \quad [0, 6]
f(x)=x2+4x,[−5,1]f(x) = x^2 + 4x, \quad [-5, 1]
f(x)=−x2−4x,[−5,1]f(x) = -x^2 - 4x, \quad [-5, 1]
f(x)=−x2+2x+8,[−2,4]f(x) = -x^2 + 2x + 8, \quad [-2, 4]
Bloque 2
f(x)=−x2−10x−24,[−8,−2]f(x) = -x^2 - 10x - 24, \quad [-8, -2]
f(x)=x2−4x+4,[−1,5]f(x) = x^2 - 4x + 4, \quad [-1, 5]
f(x)=x2+6x+9,[−6,0]f(x) = x^2 + 6x + 9, \quad [-6, 0]
f(x)=−x2+4x−3,[−1,5]f(x) = -x^2 + 4x - 3, \quad [-1, 5]
f(x)=x2−7x+10,[1,6]f(x) = x^2 - 7x + 10, \quad [1, 6]

Resoluciones

▼

Bloque 1

f(x)=x2−2x−8f(x) = x^2 - 2x - 8 en [−2,4][-2, 4]

xf(x)
−20
−1−5
0−8
1−9
2−8
3−5
40
f(−2)=(−2)2−2(−2)−8=4+4−8=f(-2) = (-2)^2 - 2(-2) - 8 = 4+4-8 = 00
f(−1)=(−1)2−2(−1)−8=1+2−8=f(-1) = (-1)^2 - 2(-1) - 8 = 1+2-8 = −5-5
f(0)=(0)2−2(0)−8=f(0) = (0)^2 - 2(0) - 8 = −8-8
f(1)=(1)2−2(1)−8=1−2−8=f(1) = (1)^2 - 2(1) - 8 = 1-2-8 = −9-9
f(2)=(2)2−2(2)−8=4−4−8=f(2) = (2)^2 - 2(2) - 8 = 4-4-8 = −8-8
f(3)=(3)2−2(3)−8=9−6−8=f(3) = (3)^2 - 2(3) - 8 = 9-6-8 = −5-5
f(4)=(4)2−2(4)−8=16−8−8=f(4) = (4)^2 - 2(4) - 8 = 16-8-8 = 00
-3 -2 -1 1 2 3 4 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 x y

f(x)=x2−6x+8f(x) = x^2 - 6x + 8 en [0,6][0, 6]

xf(x)
08
13
20
3−1
40
53
68
f(0)=(0)2−6(0)+8=f(0) = (0)^2 - 6(0) + 8 = 88
f(1)=(1)2−6(1)+8=1−6+8=f(1) = (1)^2 - 6(1) + 8 = 1-6+8 = 33
f(2)=(2)2−6(2)+8=4−12+8=f(2) = (2)^2 - 6(2) + 8 = 4-12+8 = 00
f(3)=(3)2−6(3)+8=9−18+8=f(3) = (3)^2 - 6(3) + 8 = 9-18+8 = −1-1
f(4)=(4)2−6(4)+8=16−24+8=f(4) = (4)^2 - 6(4) + 8 = 16-24+8 = 00
f(5)=(5)2−6(5)+8=25−30+8=f(5) = (5)^2 - 6(5) + 8 = 25-30+8 = 33
f(6)=(6)2−6(6)+8=36−36+8=f(6) = (6)^2 - 6(6) + 8 = 36-36+8 = 88
-1 1 2 3 4 5 6 7 -2 -1 1 2 3 4 5 6 7 8 9 x y

f(x)=x2+4xf(x) = x^2 + 4x en [−5,1][-5, 1]

xf(x)
−55
−40
−3−3
−2−4
−1−3
00
15
f(−5)=(−5)2+4(−5)=25−20=f(-5) = (-5)^2 + 4(-5) = 25-20 = 55
f(−4)=(−4)2+4(−4)=16−16=f(-4) = (-4)^2 + 4(-4) = 16-16 = 00
f(−3)=(−3)2+4(−3)=9−12=f(-3) = (-3)^2 + 4(-3) = 9-12 = −3-3
f(−2)=(−2)2+4(−2)=4−8=f(-2) = (-2)^2 + 4(-2) = 4-8 = −4-4
f(−1)=(−1)2+4(−1)=1−4=f(-1) = (-1)^2 + 4(-1) = 1-4 = −3-3
f(0)=(0)2+4(0)=f(0) = (0)^2 + 4(0) = 00
f(1)=(1)2+4(1)=1+4=f(1) = (1)^2 + 4(1) = 1+4 = 55
-6 -5 -4 -3 -2 -1 1 2 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y

f(x)=−x2−4xf(x) = -x^2 - 4x en [−5,1][-5, 1]

xf(x)
−5−5
−40
−33
−24
−13
00
1−5
f(−5)=−(−5)2−4(−5)=−25+20=f(-5) = -(-5)^2 - 4(-5) = -25+20 = −5-5
f(−4)=−(−4)2−4(−4)=−16+16=f(-4) = -(-4)^2 - 4(-4) = -16+16 = 00
f(−3)=−(−3)2−4(−3)=−9+12=f(-3) = -(-3)^2 - 4(-3) = -9+12 = 33
f(−2)=−(−2)2−4(−2)=−4+8=f(-2) = -(-2)^2 - 4(-2) = -4+8 = 44
f(−1)=−(−1)2−4(−1)=−1+4=f(-1) = -(-1)^2 - 4(-1) = -1+4 = 33
f(0)=−(0)2−4(0)=f(0) = -(0)^2 - 4(0) = 00
f(1)=−(1)2−4(1)=−1−4=f(1) = -(1)^2 - 4(1) = -1-4 = −5-5
-6 -5 -4 -3 -2 -1 1 2 -6 -5 -4 -3 -2 -1 1 2 3 4 5 x y

f(x)=−x2+2x+8f(x) = -x^2 + 2x + 8 en [−2,4][-2, 4]

xf(x)
−20
−15
08
19
28
35
40
f(−2)=−(−2)2+2(−2)+8=−4−4+8=f(-2) = -(-2)^2 + 2(-2) + 8 = -4-4+8 = 00
f(−1)=−(−1)2+2(−1)+8=−1−2+8=f(-1) = -(-1)^2 + 2(-1) + 8 = -1-2+8 = 55
f(0)=−(0)2+2(0)+8=f(0) = -(0)^2 + 2(0) + 8 = 88
f(1)=−(1)2+2(1)+8=−1+2+8=f(1) = -(1)^2 + 2(1) + 8 = -1+2+8 = 99
f(2)=−(2)2+2(2)+8=−4+4+8=f(2) = -(2)^2 + 2(2) + 8 = -4+4+8 = 88
f(3)=−(3)2+2(3)+8=−9+6+8=f(3) = -(3)^2 + 2(3) + 8 = -9+6+8 = 55
f(4)=−(4)2+2(4)+8=−16+8+8=f(4) = -(4)^2 + 2(4) + 8 = -16+8+8 = 00
-3 -2 -1 1 2 3 4 5 -2 -1 1 2 3 4 5 6 7 8 9 10 x y

Bloque 2

f(x)=−x2−10x−24f(x) = -x^2 - 10x - 24 en [−8,−2][-8, -2]

xf(x)
−8−8
−7−3
−60
−51
−40
−3−3
−2−8
f(−8)=−(−8)2−10(−8)−24=−64+80−24=f(-8) = -(-8)^2 - 10(-8) - 24 = -64+80-24 = −8-8
f(−7)=−(−7)2−10(−7)−24=−49+70−24=f(-7) = -(-7)^2 - 10(-7) - 24 = -49+70-24 = −3-3
f(−6)=−(−6)2−10(−6)−24=−36+60−24=f(-6) = -(-6)^2 - 10(-6) - 24 = -36+60-24 = 00
f(−5)=−(−5)2−10(−5)−24=−25+50−24=f(-5) = -(-5)^2 - 10(-5) - 24 = -25+50-24 = 11
f(−4)=−(−4)2−10(−4)−24=−16+40−24=f(-4) = -(-4)^2 - 10(-4) - 24 = -16+40-24 = 00
f(−3)=−(−3)2−10(−3)−24=−9+30−24=f(-3) = -(-3)^2 - 10(-3) - 24 = -9+30-24 = −3-3
f(−2)=−(−2)2−10(−2)−24=−4+20−24=f(-2) = -(-2)^2 - 10(-2) - 24 = -4+20-24 = −8-8
-9 -8 -7 -6 -5 -4 -3 -2 -1 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 x y

f(x)=x2−4x+4f(x) = x^2 - 4x + 4 en [−1,5][-1, 5]

xf(x)
−19
04
11
20
31
44
59
f(−1)=(−1)2−4(−1)+4=1+4+4=f(-1) = (-1)^2 - 4(-1) + 4 = 1+4+4 = 99
f(0)=(0)2−4(0)+4=f(0) = (0)^2 - 4(0) + 4 = 44
f(1)=(1)2−4(1)+4=1−4+4=f(1) = (1)^2 - 4(1) + 4 = 1-4+4 = 11
f(2)=(2)2−4(2)+4=4−8+4=f(2) = (2)^2 - 4(2) + 4 = 4-8+4 = 00
f(3)=(3)2−4(3)+4=9−12+4=f(3) = (3)^2 - 4(3) + 4 = 9-12+4 = 11
f(4)=(4)2−4(4)+4=16−16+4=f(4) = (4)^2 - 4(4) + 4 = 16-16+4 = 44
f(5)=(5)2−4(5)+4=25−20+4=f(5) = (5)^2 - 4(5) + 4 = 25-20+4 = 99
-2 -1 1 2 3 4 5 6 -1 1 2 3 4 5 6 7 8 9 10 11 x y

f(x)=x2+6x+9f(x) = x^2 + 6x + 9 en [−6,0][-6, 0]

xf(x)
−69
−54
−41
−30
−21
−14
09
f(−6)=(−6)2+6(−6)+9=36−36+9=f(-6) = (-6)^2 + 6(-6) + 9 = 36-36+9 = 99
f(−5)=(−5)2+6(−5)+9=25−30+9=f(-5) = (-5)^2 + 6(-5) + 9 = 25-30+9 = 44
f(−4)=(−4)2+6(−4)+9=16−24+9=f(-4) = (-4)^2 + 6(-4) + 9 = 16-24+9 = 11
f(−3)=(−3)2+6(−3)+9=9−18+9=f(-3) = (-3)^2 + 6(-3) + 9 = 9-18+9 = 00
f(−2)=(−2)2+6(−2)+9=4−12+9=f(-2) = (-2)^2 + 6(-2) + 9 = 4-12+9 = 11
f(−1)=(−1)2+6(−1)+9=1−6+9=f(-1) = (-1)^2 + 6(-1) + 9 = 1-6+9 = 44
f(0)=(0)2+6(0)+9=f(0) = (0)^2 + 6(0) + 9 = 99
-7 -6 -5 -4 -3 -2 -1 1 -1 1 2 3 4 5 6 7 8 9 10 x y

f(x)=−x2+4x−3f(x) = -x^2 + 4x - 3 en [−1,5][-1, 5]

xf(x)
−1−8
0−3
10
21
30
4−3
5−8
f(−1)=−(−1)2+4(−1)−3=−1−4−3=f(-1) = -(-1)^2 + 4(-1) - 3 = -1-4-3 = −8-8
f(0)=−(0)2+4(0)−3=f(0) = -(0)^2 + 4(0) - 3 = −3-3
f(1)=−(1)2+4(1)−3=−1+4−3=f(1) = -(1)^2 + 4(1) - 3 = -1+4-3 = 00
f(2)=−(2)2+4(2)−3=−4+8−3=f(2) = -(2)^2 + 4(2) - 3 = -4+8-3 = 11
f(3)=−(3)2+4(3)−3=−9+12−3=f(3) = -(3)^2 + 4(3) - 3 = -9+12-3 = 00
f(4)=−(4)2+4(4)−3=−16+16−3=f(4) = -(4)^2 + 4(4) - 3 = -16+16-3 = −3-3
f(5)=−(5)2+4(5)−3=−25+20−3=f(5) = -(5)^2 + 4(5) - 3 = -25+20-3 = −8-8
-2 -1 1 2 3 4 5 6 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 x y

f(x)=x2−7x+10f(x) = x^2 - 7x + 10 en [1,6][1, 6]

xf(x)
14
20
3−2
4−2
50
64
f(1)=(1)2−7(1)+10=1−7+10=f(1) = (1)^2 - 7(1) + 10 = 1-7+10 = 44
f(2)=(2)2−7(2)+10=4−14+10=f(2) = (2)^2 - 7(2) + 10 = 4-14+10 = 00
f(3)=(3)2−7(3)+10=9−21+10=f(3) = (3)^2 - 7(3) + 10 = 9-21+10 = −2-2
f(4)=(4)2−7(4)+10=16−28+10=f(4) = (4)^2 - 7(4) + 10 = 16-28+10 = −2-2
f(5)=(5)2−7(5)+10=25−35+10=f(5) = (5)^2 - 7(5) + 10 = 25-35+10 = 00
f(6)=(6)2−7(6)+10=36−42+10=f(6) = (6)^2 - 7(6) + 10 = 36-42+10 = 44
1 2 3 4 5 6 7 -4 -3 -2 -1 1 2 3 4 5 6 x y

Ejercicios extra

▼
Bloque 1
f(x)=x2−x−2,[−2,3]f(x) = x^2 - x - 2, \quad [-2, 3]
f(x)=−x2+9,[−4,4]f(x) = -x^2 + 9, \quad [-4, 4]
f(x)=x2−4x+3,[0,4]f(x) = x^2 - 4x + 3, \quad [0, 4]
Bloque 2
f(x)=2x2−4x,[−1,3]f(x) = 2x^2 - 4x, \quad [-1, 3]
f(x)=−x2−2x+3,[−4,2]f(x) = -x^2 - 2x + 3, \quad [-4, 2]
f(x)=x2+2x+1,[−3,1]f(x) = x^2 + 2x + 1, \quad [-3, 1]
Bloque 3
f(x)=−2x2+8,[−3,3]f(x) = -2x^2 + 8, \quad [-3, 3]
f(x)=x2−9,[−4,4]f(x) = x^2 - 9, \quad [-4, 4]
f(x)=−x2+6x−8,[1,5]f(x) = -x^2 + 6x - 8, \quad [1, 5]
Bloque 4
f(x)=x2+5x+6,[−5,0]f(x) = x^2 + 5x + 6, \quad [-5, 0]
f(x)=−x2+x+6,[−3,4]f(x) = -x^2 + x + 6, \quad [-3, 4]
f(x)=x2−2x+1,[−1,3]f(x) = x^2 - 2x + 1, \quad [-1, 3]
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