For the function f ( x ) = − 5x^2 − 2x , evaluate and fully simplify each of the following

Question

For the function f ( x ) = − 5x^2 − 2x , evaluate and fully simplify each of the following

f(x+h)=

f(x+h)−f(x)/h=

Answer 1

Firstly, you want to identify f(x+h). Back in algebra 2, we learned that f(x+h) means we add h to every x in our function.

f(x)= -5x² - 2x

f(x+h) = -5(x+h)² -2(x+h)

Now our difference quotient is (f(x+h)-f(x))/(h). Previously we found f(x+h), and we know f(x), so lets plug it in.

[f(x+h)-f(x)] / (h) = [-5(x+h)² -2(x+h) - (-5x²- 2x)] / h

Lets simplify:

1. -5(x+h)² = -5(x+h)(x+h) = -5(x² +2xh+h²) = -5x²- 10xh- 5h²
2. -2(x+h) = -2x - 2h

So that,

[f(x+h)-f(x)] / (h) = [-5x²- 10xh- 5h² - 2x - 2h - (-5x²- 2x)] / (h) FOIL and Distribute -2

= [-5x²- 10xh- 5h² - 2x - 2h + 5x²+2x] / (h) Distribute the negitive

= [- 10xh - 5h² - 2h ] / (h) Add like terms to cancel out terms that do not contain an h

= [h(- 10x-5h-2)] / (h) Factor out an h

= - 10x-5h-2 Simplify

The difference quiotent for f(x)=-5x²-2 is [f(x+h)-f(x)]/(h) = -10x-5h-2

If anything is incorrect, please let me know and I will promptly correct it. :)

Answer 2

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

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

(f(x+h)-f(x))/h = -10x-5h-2

Answer 3

f ( x ) = − 5x^2 − 2x

f ( x+h ) = − 5(x+h)^2 − 2(x+h)

(f ( x+h )-f(x))/h=(− 5(x+h)^2 − 2(x+h)-(− 5x^2 − 2x))/h

= (-5x^2-10xh-5h^2-2x-2h+5x^2+2x)/h

=(-10xh-5h^2-2h)/h=-10x-5h-2