Question

For the quadratic function f(x)=5x^2-2x+3, find f(x+2), f(x+2)-f(x), and f(x+h)

Answer 1

f(x + 2) = 5x² + 18x + 19

f(x + 2) - f(x) = 20x + 16

f(x + h) = 5x² + 10hx + 5h² - 2x - 2h + 3

Explanation:

1. f(x + 2)

`f(x)=5x^2-2x+3\\f(x+2)=5(x+2)^2-2(x+2)+3\\f(x+2)=5(x+2)(x+2)-2(x+2)+3\\f(x+2)=5(x^2+4x+4)-2(x+2)+3\\f(x+2)=5x^2+20x+20-2(x+2)+3\\f(x+2)=5x^2+20x+20-2x-4+3\\f(x+2)=5x^2+18x+20-4+3\\f(x+2)=5x^2+18x+19`

2. f(x + 2) - f(x)

`f(x+2)-f(x)\\(5x^2+18x+19)-(5x^2-2x+3)\\5x^2-5x^2+18x-(-2x)+19-3\\0x^2+18x+2x+16\\20x+16`

3. f(x + h)

`f(x)=5x^2-2x+3\\f(x+h)=5(x+h)^2-2(x+h)+3\\f(x+h)=5(x+h)(x+h)-2(x+h)+3\\f(x+h)=5(x^2+2hx+h^2)-2(x+h)+3\\f(x+h)=5x^2+10hx+5h^2-2(x+h)+3\\f(x+h)=5x^2+10hx+5h^2-2x-2h+3`