Math II: Comparing Characteristics of Functions

Question

The graph of the quadratic function f has x-intercepts −3 and 5 and y-intercept 5.

The quadratic function g is represented by the equation g(x)=−3x2+4x−5 .

What is the maximum of f minus the maximum of g?

Answer 1

f(x) = C(x+3)(x-5)

f(0) = 5 since the y-intercept is 5.

5 = C(3)(-5)

C = -1/3

So f(x) = (-1/3)(x+3)(x-5)

The extrema occurs at the average of the solutions, which is (-3+5)/2 = 1

f(1) = 16/3, so the max point is (1, 16/3)

FOr g(x) ,A=-3, b=4 and c=-5

The max occurs at -b/(2a) = -4/(2(-3)) = -4/-6 = 2/3

g(2/3) = -3(2/3)^2 + 4(2/3) - 5 = (-3)(4/9) + 8/3 - 5 = -4/3 + 8/3 - 5 = 4/3 - 5 = 4/3 - 15/3 = -11/3

So the max point is (2/3, -11/3)

The distance between these max points are sqrt ( (1 - 2/3)^2 + ( 16/3 - - 11/3)^2 )

= sqrt( (1/3)^2 + (27/3)^2 )

= sqrt( 1/9 + 729/9)

= sqrt( 730/9)

= sqrt(730)/3

or approximately 9