Question

Identify the vertex of each.

1) y = x^2 + 16x + 64

2) y = 2x^2 − 4x − 2

3) y = −x^2 + 18x − 75

4) y = −3x^2 + 12x − 10

Answer 1

Formulas :  x-value of the vertex : x = ( - b / 2a) 

                    y - value of the vertex : y = ( c - b^2/4a) 

1)

y = x^2 + 16x + 64

This is quadratic equation of the parabola y = ax^2 + bx + c 

Comapare above two equations

a = 1, b = 16 and c = 64

x-value of the vertex

x = ( - b / 2a) 

x = ( - 16 / 2 (1) ) 

x = ( - 16 / 2) 

x = - 8 

y - value of the vertex :

y = ( c - b^2/4a) 

y = ( 64 - 16^2/ 4 X 1) 

y = ( 64 - 256/ 4) 

y =  64 - 64

y  =  0 

Hence, Vertex is (x, y)  =  (-8, 0)

 

2)

y = 2x^2 − 4x − 2

This is quadratic equation of the parabola y = ax^2 + bx + c 

Comapare above two equations

a = 2, b = -4 and c = -2

x-value of the vertex

x = ( - b / 2a) 

x = ( 4 / 2 (2) ) 

x = ( 4 / 4) 

x = 1 

y - value of the vertex :

y = ( c - b^2/4a) 

y = ( -2 - (-4)^2/ 4 X 2) 

y = ( -2 -16/8) 

y =  -2  - 2

y = -4

Hence, Vertex is (x, y)  =  (1, -4)

 

3) y = −x^2 + 18x − 75

This is quadratic equation of the parabola y = ax^2 + bx + c 

Comapare above two equations

a = -1, b = 18 and c = -75 

x-value of the vertex

x = ( - b / 2a) 

x = ( -18 / 2 (-1) ) 

x = ( -18) / (-2) 
x = 9 

y - value of the vertex :

y  =  c - b^2/4a

y  =  -75 - [ (18)^2 / 4 X (-1)] 

y  =  -75 - [ 324 / (-4)] 

y  =  -75 - (- 81)

y  =  -75 + 81

y  =  6

Hence, Vertex is (x, y)  =  (9, 6)

4) y = −3x^2 + 12x − 10

This is quadratic equation of the parabola y = ax^2 + bx + c 

Comapare above two equations

a = -3, b = 12 and c = -10 

x-value of the vertex

x = ( - b / 2a) 

x = ( -12 / 2 (-3) ) 

x = ( -12) / (-6) 
x = 2

y - value of the vertex :

y  =  c - b^2/4a

y  =  -10 - [ (12)^2 / 4 X (-3)] 

y  =  -10 - [ (12)^2 / (-12)] 

y  =  -10 - (- 12)

y  =  - 10 + 12

y  =  2

Hence, Vertex is (x, y)  =  (2, 2)

Answer :

1)  Vertex is (x, y)  =  (-8, 0)

2)  Vertex is (x, y)  =  (1, -4)

 

3)  Vertex is (x, y)  =  (9, 6)

 

4)  Vertex is (x, y)  =  (2, 2)