Question

Function f(x) = ax^{2}+bx+c, where a, b, and c are some constants. Define functions g and h as follows:

g(x) = f(x+ 1)−f(x)
h(x) = g(x+ 1)−g(x)
Find algebraic form of h(x)
Can anyone explain how to make it step by step?

Answer 1

g(x) = f(x+1) - f(x)
=[ a(x+1)^2+b(x+1)+c ] - [ax^2+bx+c]
=[ a(x^2+2x+1) +bx + b + c ] - [ax^2 + bx + c]
=[ ax^2 + 2ax + a + bx + b + c ] - [ax^2 + bx + c]
= ax^2 + 2ax + a + bx + b + c - ax^2 - bx - c
= 2ax + a + b
Therefore g(x) = 2ax + a + b
h(x) = g(x+1) - g(x)
=2a (x+1) + a + b - [2ax+a+b]
=2ax + 1 + a +b - 2ax - a - b
Therefore h(x) = 1