Question

1. -/1 points let f(x)=-x^{2}+4x - 7. find and simplify the difference quotient, \\(\frac{f(x + h)-f(x)}{h}\\). resources ebook

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$:
\[

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

\]

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

\[

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

\]

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

\[

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

\]

Answer:

$-2x - h + 4$