Question

find the difference quotient of f; that is, find \\(\frac{f(x + h) - f(x)}{h}\\), \\(h \
eq 0\\), for the following function.

\\(f(x) = x^2 - 8x + 1\\)

\\(\frac{f(x + h) - f(x)}{h} =\\) (simplify your answer.)

Explanation:

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<reasoning>

Evaluate the function at x + h

Using the Function Evaluation knowledge point
\[

$$\begin{aligned} f(x+h) &= (x+h)^2 - 8(x+h) + 1 \\ &= x^2 + 2xh + h^2 - 8x - 8h + 1 \end{aligned}$$

\]

Set up the difference quotient numerator

Using the Difference Quotient knowledge point
\[

$$\begin{aligned} f(x+h) - f(x) &= (x^2 + 2xh + h^2 - 8x - 8h + 1) - (x^2 - 8x + 1) \\ &= 2xh + h^2 - 8h \end{aligned}$$

\]

Divide by h and simplify

Using the Difference Quotient knowledge point
\[

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

\]
</reasoning>

<answer>
Find the difference quotient of f; that is, find \(\frac{f(x+h)-f(x)}{h}\), \(h
eq 0\), for the following function.
\(f(x) = x^2 - 8x + 1\)

\(\frac{f(x+h)-f(x)}{h} =\) <blank>\(2x + h - 8\)</blank> (Simplify your answer.)
</answer>

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Answer:

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<reasoning>

Evaluate the function at x + h

Using the Function Evaluation knowledge point
\[

$$\begin{aligned} f(x+h) &= (x+h)^2 - 8(x+h) + 1 \\ &= x^2 + 2xh + h^2 - 8x - 8h + 1 \end{aligned}$$

\]

Set up the difference quotient numerator

Using the Difference Quotient knowledge point
\[

$$\begin{aligned} f(x+h) - f(x) &= (x^2 + 2xh + h^2 - 8x - 8h + 1) - (x^2 - 8x + 1) \\ &= 2xh + h^2 - 8h \end{aligned}$$

\]

Divide by h and simplify

Using the Difference Quotient knowledge point
\[

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

\]
</reasoning>

<answer>
Find the difference quotient of f; that is, find \(\frac{f(x+h)-f(x)}{h}\), \(h
eq 0\), for the following function.
\(f(x) = x^2 - 8x + 1\)

\(\frac{f(x+h)-f(x)}{h} =\) <blank>\(2x + h - 8\)</blank> (Simplify your answer.)
</answer>

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