Question

find the difference quotient \\( \frac { f ( x + h ) - f ( x ) } { h } \\), where \\( h \
eq 0 \\), for the function below.

\\( f ( x ) = - 4 x ^ { 2 } - 6 x + 7 \\)

simplify your answer as much as possible.

\\( \frac { f ( x + h ) - f ( x ) } { h } = \square \\)

Explanation:

Step1: Find \(f(x + h)\)

Given \(f(x)=-4x^{2}-6x + 7\), then \(f(x + h)=-4(x + h)^{2}-6(x + h)+7\).
Using \((a + b)^{2}=a^{2}+2ab + b^{2}\), we have \(f(x + h)=-4(x^{2}+2xh+h^{2})-6x-6h + 7=-4x^{2}-8xh-4h^{2}-6x-6h + 7\).

Step2: Calculate \(f(x + h)-f(x)\)

\(f(x + h)-f(x)=(-4x^{2}-8xh-4h^{2}-6x-6h + 7)-(-4x^{2}-6x + 7)\)
\(=-4x^{2}-8xh-4h^{2}-6x-6h + 7 + 4x^{2}+6x - 7\)
\(=-8xh-4h^{2}-6h\).

Step3: Find \(\frac{f(x + h)-f(x)}{h}\)

\(\frac{f(x + h)-f(x)}{h}=\frac{-8xh-4h^{2}-6h}{h}\), since \(h
eq0\), we can factor out \(h\) from the numerator: \(\frac{h(-8x - 4h-6)}{h}=-8x-4h - 6\).

Answer:

\(-8x-4h - 6\)