Question

9. Find different quotient of ()=4−22+5. Recall the difference quotient is =(+ℎ)−()ℎ . Simplify the expression so that the h on the bottom is gone. Be sure to show your work.

Answer 1

The given function is expressd as

f(x) = 4 - 2x^2 + 5x

The first step is to find f(x + h) by substituting x = x + h into the function. Thus, we have

f(x + h) = 4 - 2(x + h)^2 + 5x

f(x + h) = 4 - 2(x^2 + hx + hx + h^2) + 5x

f(x + h) = 4 - 2x^2 - 2hx - 2hx - 2h^2 + 5x

f(x + h) = - 2x^2 - 4h^2 + 5x + 4

Difference quotient = [f(x + h) - f(x)]/h

Thus, we have

Difference quotient = [- 2x^2 - 4h^2 + 5x + 4 - (4 - 2x^2 + 5x)]/h

Difference quotient = [(- 2x^2 - 4h^2 + 5x + 4 - 4 + 2x^2 - 5x)]/h

Collecting like terms, we have

Difference quotient = [(- 2x^2 + 2x^2 - 4h^2 + 5x - 5x + 4 - 4)]/h

Difference quotient = - 4h^2/h

h cancels out

Difference quotient = - 4h