Question

find the difference quotient and simplify. ( f(x) = -4x^2 - 4x + 7 ) the difference quotient of ( f(x) ) is (square).

Explanation:

Step1: Recall the difference quotient formula

The difference quotient of a function \( f(x) \) is given by \( \frac{f(x + h)-f(x)}{h} \), where \( h
eq0 \).

Step2: Find \( f(x + h) \)

Given \( f(x)=-4x^{2}-4x + 7 \), substitute \( x + h \) into the function:
\[

$$\begin{align*} f(x + h)&=-4(x + h)^{2}-4(x + h)+7\\ &=-4(x^{2}+2xh+h^{2})-4x-4h + 7\\ &=-4x^{2}-8xh-4h^{2}-4x-4h + 7 \end{align*}$$

\]

Step3: Compute \( f(x + h)-f(x) \)

Subtract \( f(x) \) from \( f(x + h) \):
\[

$$\begin{align*} f(x + h)-f(x)&=(-4x^{2}-8xh-4h^{2}-4x-4h + 7)-(-4x^{2}-4x + 7)\\ &=-4x^{2}-8xh-4h^{2}-4x-4h + 7 + 4x^{2}+4x-7\\ &=-8xh-4h^{2}-4h \end{align*}$$

\]

Step4: Divide by \( h \) to get the difference quotient

Divide \( f(x + h)-f(x) \) by \( h \) (\( h
eq0 \)):
\[
\frac{f(x + h)-f(x)}{h}=\frac{-8xh-4h^{2}-4h}{h}
\]
Factor out \( h \) from the numerator:
\[
\frac{h(-8x - 4h-4)}{h}
\]
Cancel out the common factor \( h \) (since \( h
eq0 \)):
\[
-8x-4h - 4
\]

Answer:

\( -8x - 4h-4 \)