Answer:
Difference = 5.6 which makes C the right answer.
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.