Question

Calcule as raízes e o vertice, caso existam, das seguintes funções abaixo:

a) y = x² - 1

b) y = x² + 3x + 2

c) y = x² + x - 2

d) y = x² - 6x + 9

e) y = x² - 4x + 3

f) y = x² + 4x + 3

g) y = x² - x - 2

h) y = x² - 2x - 3

Answer 1

Boa noite Clarissa

a) f(x) = x² - 1

a = 1, b = 0, c = -1
raízes x1 = 1, x2 = -1
vértice Vx = -b/2a = 0 , Vy = f(0) = - 1 

b) f(x) = x² + 3x + 2

a = 1, b = 3 , c = 2
(x + 2)*(x + 1) = 0
x1 = -2, x2 = -1
vértice Vx = -b/2a = -3/2 , Vy =  f(-3/2) = 9/4 - 9/2 + 2 = -9/4 + 8/4 = -1/4 

c) f(x)  = x² + x - 2

a = 1 , b = 1, c = -2
(x - 1)*(x + 2) = 0
x1 = 1, x2 = -2
vérice Vx = -b/2a = -1/2, Vy = f(-1/2) = 1/4 - 2/4 - 8/4 = -9/4

d) f(x) = x² - 6x + 9

a = 1, b = -6 , c = 9

(x - 3)² = 0
x1 = x2 = 3
vértice Vx = -b/2a = 6/2 = 3, Vy = f(3) = 0

e) f(x) = x² - 4x + 3

a = 1, b = -4, c = 3
(x - 3)*(x - 1) = 0
x1 = 3, x2 = 1
vértice vx = -b/2a = 4/2 = 2, Vy = f(2) = 4 - 8 + 3 = -1

f) f(x) = x² + 4x + 3

a = 1, b = 4, c = 3
(x + 3)*(x + 1) = 0
x1 = -3, x2 = -1
vértice Vx = -b/2a = -4/2 = -2, Vy = f(-2) = 4 - 8 + 3 = -1

g) f(x) = x² - x - 2

a = 1, b = -1 ,c = -2
(x - 2)*(x + 1) = 0
x1 = 2, x2 = -1
vértice Vx = -b/2a = 1/2 , Vy = f(1/2) = 1/4 - 2/4 - 8/4 = -9/4 

h) f(x) = x² - 2x - 3

a = 1, b = -2 , c = -3

(x - 3)*(x + 1) = 0
x1 = 3, x1 = -1
vértice Vx = -b/2a = 2/2 = 1, Vy = f(1) = 1 - 2 - 3 = -4 

Answer 2

a) y = x² - 1
x² - 1= 0
a= 1 ; b= 0 c= -1
∆ = b² - 4.a.c
∆ = (0)² - 4.(1).(-1)
∆ = 4
x = -b ± √∆) / 2.a
x = - (0)± √4/ 2.(1)
x = ± 2/2

x'= 2/2
x'= 1

x"= -2/2
x"= -1
Raízes { 1 , -1 }

As coordenadas do vértice:
Vx= - b/2.a
Vx= - (0)/ 2.(1)
Vx= 0/2
Vx = 0

Vy= - Δ/4.a
Vy= - (4)/4.(1)
Vy= -4/4
Vy= -1
Vértices= ( 0 , -1 )


b) y = x² + 3x + 2
x² + 3x + 2
a = 1 ; b = 3 ; c = 2
∆ = b² - 4.a.c
∆ = (3)² - 4.(1).(2)
∆= 9 -8
∆= 1
x = -b ± √∆) / 2.a
x = - (3)± √1/ 2.(1)
x = -3± 1/2
x' = -3 +1/2 → -2/2
x'= -1
x" = -3 -1/2 → -4/2
x" = -2
Raízes { -1 , -2 }

As coordenadas do vértice:
Vx= - b/2.a
Vx= - (3)/ 2.(1)
Vx= -3/2

Vy= - Δ/4.a
Vy= - (1)/4.(1)
Vy= -1/4
Vértices = (-3/2 , -1/4)


c) y = x² + x - 2
x² +x - 2 = 0
a = 1 ; b = 1 ; c = -2
∆ = b² - 4.a.c
∆ = (-1)² - 4.(1).(-2)
∆= 1 +8
∆= 9
x = -b ± √∆) / 2.a
x = - (1)± √9/ 2.(1)
x = -1± 3/2

x' = -1+3/2 →2/2
x'= 1
x" = -1-3/2 → -4/2
x" = -2
Raízes { 1, -2 }

As coordenadas do vértice:
Vx= - b/2.a
Vx= - (1)/ 2.(1)
Vx= -1/2

Vy= - Δ/4.a
Vy= - (9)/4.(1)
Vy= -9/4
Vértices= (-1/2 , -9/4)

d) y = x² - 6x + 9
x²-6x+9 = 0
a= 1 ; b= -6 ; c= 9
∆ = b² - 4.a.c
∆ = (-6)² - 4.(1).(9)
∆ = 36 -36
∆ = 0
x = -b ± √∆) / 2.a
x = - (-6)± √0/ 2.(1)
x = 6 ± 0/2
x' = x" = 3

As coordenadas do vértice:
Vx = - b/2a
Vx = - (-6)/ 2.(1)
Vx = 6/2
Vx = 3

Vy = - Δ/4.a
Vy = 0/4.(1)
Vy= 0
Vértices=(3, 0)

e) y = x² - 4x + 3
x² - 4x + 3=0
a = 1 ; b = -4 ; c = 3
∆ = b² - 4.a.c
∆ = (-4)² - 4.(1).(3)
∆ = 16 -12
∆ = 4
x = -b ± √∆) / 2.a
x = - (-4)± √4/ 2.(1)
x = 4 ± 2/2

x' = 4 +2/2 → 6/2
x'= 3

x" = 4 -2/2 → 2/2
x" = 1
Raízes : { 3 , 1}

As coordenadas do vértice:
Vx= - b/2a
Vx= - (-4)/ 2.(1)
Vx= 4/2
Vx = 2

Vy= - Δ/4.a
Vy= - (4)/4.(1)
Vy= -4/4
Vy= -1
Vértices= (2 , -1)

f) y = x² + 4x + 3
x²+4x+3=0
a =1 ; b = 4 ; c = 3
∆ = b² - 4.a.c
∆ = (4)² - 4.(1).(3)
∆ = 16 -12
∆ = 4
x = -b ± √∆) / 2.a
x = - (4)± √4/ 2.(1)
x = -4 ± 2/2

x' = -4 +2/2 →-2/2
x'= -1

x" = -4 -2/2 →-6/2
x" = -3
Raízes : { -1 , -3}

As coordenadas do vértice:
Vx= - b/2.a
Vx= - (4)/ 2.(1)
Vx= -4/2
Vx = -2

Vy= - Δ/4.a
Vy= - (4)/4.(1)
Vy= -4/4
Vy= -1
Vértices= ( -2 , -1)


g) y = x² - x - 2
x² - x - 2=0
a = 1 ; b = -1 ; c = -2
∆ = b² - 4.a.c
∆ = (-1)² - 4.(1).(-2)
∆= 1+8
∆= 9
x = -b ± √∆) / 2.a
x = - (-1)± √9/ 2.(1)
x = 1± 3/2

x' = 1 + 3/2 → 4/2
x'= 2

x" = 1 - 3/2 → -2/2
x" = -1
Raízes { 2 , -1 }

As coordenadas do vértice:
Vx= - b/2.a
Vx= - (-1)/ 2.(1)
Vx= 1/2

Vy= - Δ/4.a
Vy= - (9)/4.(1)
Vy= -9/4
Vértices= ( 1/2 , -9/4)


h) y = x² - 2x - 3
x² - 2x - 3=0
a = 1 ; b = -2 ; c = -3
∆ = b² - 4.a.c
∆ = (-2)² - 4.(1).(-3)
∆= 4+12
∆= 16
x = -b ± √∆) / 2.a
x = - (-2)± √16/ 2.(1)
x = 2 ± 4/2

x' = 2+4/2 →6/2
x'= 3

x" = 2 - 4/2 → -2/2
x" = -1
Raízes { 3 , -1 }

As coordenadas do vértice:
Vx= - b/2.a
Vx= - (-2)/ 2.(1)
Vx= 2/2
Vx= 1

Vy= - Δ/4.a
Vy= - (16)/4.(1)
Vy= -16/4
Vy= - 4
Vértices = (1 , -4)


Bons estudos.