Question

follow the step - by - step process in parts a through c to determine the difference quotient, $\frac{f(x + h)-f(x)}{h}$. $f(x)=4x^{2}+7x - 6$ a determine $f(x + h)$ $f(x + h)=square$ (simplify your answer. do not factor). b simplify the expression $f(x + h)-f(x)$ $f(x + h)-f(x)=square$ (simplify your answer. do not factor.) c determine the difference quotient $\frac{f(x + h)-f(x)}{h}$ $\frac{f(x + h)-f(x)}{h}=square$

Explanation:

Step1: Find f(x + h)

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

$$\begin{align*} f(x + h)&=4(x + h)^2+7(x + h)-6\\ &=4(x^{2}+2xh+h^{2})+7x + 7h-6\\ &=4x^{2}+8xh+4h^{2}+7x + 7h-6 \end{align*}$$

\]

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

\[

$$\begin{align*} f(x + h)-f(x)&=(4x^{2}+8xh+4h^{2}+7x + 7h-6)-(4x^{2}+7x - 6)\\ &=4x^{2}+8xh+4h^{2}+7x + 7h-6 - 4x^{2}-7x + 6\\ &=8xh+4h^{2}+7h \end{align*}$$

\]

Step3: Determine the difference quotient \(\frac{f(x + h)-f(x)}{h}\)

\[

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

\]

Answer:

a. \(4x^{2}+8xh+4h^{2}+7x + 7h-6\)
b. \(8xh+4h^{2}+7h\)
c. \(8x + 4h+7\)