Question

instructions - 1. find the difference quotient for the function $p(x)=3x - 2$. 2. find the difference quotient for the function $m(x)=x^{2}+4x - 1$.

Explanation:

Step1: Recall the difference - quotient formula

The difference - quotient formula for a function $y = f(x)$ is $\frac{f(x + h)-f(x)}{h}$, where $h
eq0$.

Step2: Find the difference - quotient for $p(x)=3x - 2$

First, find $p(x + h)$:
$p(x + h)=3(x + h)-2=3x+3h - 2$.
Then, calculate $\frac{p(x + h)-p(x)}{h}$:
\[

$$\begin{align*} \frac{p(x + h)-p(x)}{h}&=\frac{(3x + 3h-2)-(3x - 2)}{h}\\ &=\frac{3x+3h - 2-3x + 2}{h}\\ &=\frac{3h}{h}\\ &=3 \end{align*}$$

\]

Step3: Find the difference - quotient for $m(x)=x^{2}+4x - 1$

First, find $m(x + h)$:
$m(x + h)=(x + h)^{2}+4(x + h)-1=x^{2}+2xh+h^{2}+4x + 4h-1$.
Then, calculate $\frac{m(x + h)-m(x)}{h}$:
\[

$$\begin{align*} \frac{m(x + h)-m(x)}{h}&=\frac{(x^{2}+2xh+h^{2}+4x + 4h-1)-(x^{2}+4x - 1)}{h}\\ &=\frac{x^{2}+2xh+h^{2}+4x + 4h-1-x^{2}-4x + 1}{h}\\ &=\frac{2xh+h^{2}+4h}{h}\\ &=2x + h+4 \end{align*}$$

\]

Answer:

1. The difference - quotient for $p(x)$ is 3.
2. The difference - quotient for $m(x)$ is $2x + h+4$.