Question

Help solve this question and show how to do it

Answer 1

We have the function:

$f(x)=-x^2+6x+8.$

We must find a simplify the difference quotient:

$\frac{f(x+h)-f(x)}{h},h\ne0.$

1) First, we compute f(x + h):

$\begin{gathered} f(x+h) \\ =-\mleft(x+h\mright)^2+6\mleft(x+h\mright)+8 \\ =-(x^2+2xh+h^2)+6(x+h)+8 \end{gathered}$

2) Now, we compute the difference f(x + h) - f(x):

$\begin{gathered} f(x+h)-f(x) \\ =\lbrack-(x^2+2xh+h^2)+6(x+h)+8\rbrack-\lbrack-x^2+6x+8\rbrack \\ =-2xh-h^2+6h \\ =(-2x+6-h)\cdot h^{} \end{gathered}$

3) Finally, the quotient between difference and h is:

$\frac{f(x+h)-f(x)}{h}=\frac{(-2x+6-h)\cdot h^{}}{h}=-2x+6-h$

Answer

$\frac{f(x+h)-f(x)}{h}=-2x+6-h$