Function \( f(x) = ax^{2} + bx + c \), where \( a, b, \) and \( c \) are constants.

Define functions \( g \) and \( h \) as follows:
\[ g(x) = f(x+1) - f(x) \]
\[ h(x) = g(x+1) - g(x) \]

Find the algebraic form of \( h(x) \).

Can anyone explain how to make it step by step?

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

Answered by leonghw | 2024-06-10

We derived g ( x ) from the given quadratic function and then found h ( x ) by taking the difference of g ( x + 1 ) and g ( x ) . Ultimately, we concluded that h ( x ) = 2 a .
;

Answered by leonghw | 2024-12-24