Identify the vertex, axis of symmetry, and min/max value of each

Question

11) f (x) = 3x^2 − 54x + 241

12) f (x) = x^2 − 18x + 86

Answer 1

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

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

11)

a)

f (x) = 3x^2 − 54x + 241

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

Comapare above two equations

a = 3, b = -54 and c = 241 

x-value of the vertex

x = ( - b / 2a) 

x = ( - (-54) / 2 (3) ) 

x = ( 54 / 6) 

x = 9 

y - value of the vertex :

y = ( c - b^2/4a) 

y = ( 241 - 54^2/ 4* 3) 

y = (241 - 2916/ 12) 

y = (241 - 243) 

y = - 2 

Hence, Vertex is (x, y)  =  (9, -2)
 

b)

Axis of symmetry is the vertex's x value, so Axis of symmetry is x = 9

c)

Since a is bigger than zero, the parabola of the function opens upwards and has a minimum.

To get the minimum value Differentiate f(x) with respect to x and equate it to zero

f'(x)  =  0

d/dx [ 3x^2 - 54x + 241 ]  =  0

6x - 54  =  0

6x  =  54

x  =  54/6

x = 9 

Substitute x = 9 in f(x)

f (9)  =  3(9)^2 − 54(9) + 241

        =  243 - 486 + 241

        =  - 2
 
The minimum value of the function is - 2

Answer 2

Contd........

12)

f (x) = x^2 − 18x + 86

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

Comapare above two equations

a = 1, b = -18 and c = 86 

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  =  86 - [ (-18)^2 / 4 X 1] 

y  =  86 - [ 324 / 4] 

y  =  86 - 81

y  =  5

Hence, Vertex is (x, y)  =  (9, 5)
 

b)

Axis of symmetry is the vertex's x value, so Axis of symmetry is x = 9

c)

Since a is bigger than zero, the parabola of the function opens upwards and has a minimum.

To get the minimum value Differentiate f(x) with respect to x and equate it to zero

f'(x)  =  0

d/dx [ x^2 − 18x + 86 ]  =  0

2x - 18  =  0

2x  =  18

x  =  18/2

x  =  9

Substitute x = 9 in f(x)

f (9)  =   9^2 − 18(9) + 86

        =   81 − 162 + 86

        =  5

The minimum value of the function is 5

Answer :

11)  Vertex is (x, y)  =  (9, -2)

       Axis of symmetry is x = 9 

       The minimum value of the function is -2

12)  Vertex is (x, y)  =  (9, 5)

       Axis of symmetry is x = 9 

       The minimum value of the function is 5.