What is the answer to this problem?

Difference = 5.6 which makes C the right answer.

Step-by-step explanation:

You have to determine g(x) before doing anything

g(x) = ax^2 + bx + c

g(0) =  c

c = 9 from the table

g(x) = ax^2 + b(x) + 9

g(1) = a + b + 9 = 33           multiply this equation by 2

g(1) = 2a + 2b + 18 = 66     Subtract 18 from both sides

2a + 2b = 66 - 18

2a + 2b = 48                       (1)

=====================

g(2) = 4a + 2b + 9  = 25     Subtract 9 from both sides

4x + 2b = 25 - 9

4x + 2b = 16                        (2)

===============    

Subtract (1) from (2)

4a + 2b = 16

2a + 2b = 48                        Subtract

2a = -       32                        Divide by 2

2a/2 =   -32/2

a = - 16

=================

Solve for b

2a + 2b = 48                         Let a = - 16

2(-16) + 2b = 48                    Simplify the left

-32 + 2b = 48                        Add 32

2b = 48 + 32                         Simplify the right

2b = 80                                 Divide by 2

b = 80/2                  

b = 40

=================

Complete equation

g(x) = -16x^2 + 40x + 9

The next step is to put the two equations into max/min state by completing the square.

Completing the square for f(x)

I am going to present you with the completed square with some guiding steps. First f(x)

f(x) = - 16(x^2 - 42x/16 +    ) + 12

f(x) = - 16(x^2 - 42/16 + (42/32)^2 ) + 12 + 42^2/(32*2)

f(x) = -16(x - 42/32)^2 + 12 + 26.5625

f(x) = - 16( x - 42/32)^2 + 39.5625

Use the same procedure for g(x)

g(x) = -16x^2 + 40x + 9

g(x) = -16(x^2 + 40x/16 +  ) + 9

g(x) = -16(x^2 + 40x/16 + (40/(16*2) )^2  )

g(x) = -16(x - 40/32)^2 + 9 + 40^2 / 32*2

g(x) = -16(x - 40/32)^2 + 34

=======================

The max height for g(x) = 34

The max height for f(x) = 39.56

The difference = 5.56

Graph

Red: g(x)

blue: f(x)

Answer C

If you know calculus, the question is much much shorter.