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Find the difference quotient of \(\displaystyle f(x)=- 7 x^{2} - 4 x + 8\) .

The difference quotient is \(\displaystyle \frac{f(x+h)-f(x)}{h}\). Evaluating \(\displaystyle f(x+h)=- 4 h - 4 x - 7 \left(h + x\right)^{2} + 8\) and expanding gives \(\displaystyle f(x+h)=- 7 h^{2} - 14 h x - 4 h - 7 x^{2} - 4 x + 8\) Evaluating the difference quotient gives \(\displaystyle \frac{(- 7 h^{2} - 14 h x - 4 h - 7 x^{2} - 4 x + 8)-(- 7 x^{2} - 4 x + 8)}{h}\) Simplifying gives \(\displaystyle \frac{- 7 h^{2} - 14 h x - 4 h}{h}=- 7 h - 14 x - 4\)

Download \(\LaTeX\) 

\begin{question}Find the difference quotient of $f(x)=- 7 x^{2} - 4 x + 8$ . 
    \soln{9cm}{The difference quotient is $\frac{f(x+h)-f(x)}{h}$. Evaluating $f(x+h)=- 4 h - 4 x - 7 \left(h + x\right)^{2} + 8$ and expanding gives $f(x+h)=- 7 h^{2} - 14 h x - 4 h - 7 x^{2} - 4 x + 8$  Evaluating the difference quotient gives $\frac{(- 7 h^{2} - 14 h x - 4 h - 7 x^{2} - 4 x + 8)-(- 7 x^{2} - 4 x + 8)}{h}$ Simplifying gives $\frac{- 7 h^{2} - 14 h x - 4 h}{h}=- 7 h - 14 x - 4$ }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Find the difference quotient of  <img class="equation_image" title=" \displaystyle f(x)=- 7 x^{2} - 4 x + 8 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D-%207%20x%5E%7B2%7D%20-%204%20x%20%2B%208%20" alt="LaTeX:  \displaystyle f(x)=- 7 x^{2} - 4 x + 8 " data-equation-content=" \displaystyle f(x)=- 7 x^{2} - 4 x + 8 " />  . </p> </p>
HTML for Canvas
<p> <p>The difference quotient is  <img class="equation_image" title=" \displaystyle \frac{f(x+h)-f(x)}{h} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7Bf%28x%2Bh%29-f%28x%29%7D%7Bh%7D%20" alt="LaTeX:  \displaystyle \frac{f(x+h)-f(x)}{h} " data-equation-content=" \displaystyle \frac{f(x+h)-f(x)}{h} " /> . Evaluating  <img class="equation_image" title=" \displaystyle f(x+h)=- 4 h - 4 x - 7 \left(h + x\right)^{2} + 8 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%2Bh%29%3D-%204%20h%20-%204%20x%20-%207%20%5Cleft%28h%20%2B%20x%5Cright%29%5E%7B2%7D%20%2B%208%20" alt="LaTeX:  \displaystyle f(x+h)=- 4 h - 4 x - 7 \left(h + x\right)^{2} + 8 " data-equation-content=" \displaystyle f(x+h)=- 4 h - 4 x - 7 \left(h + x\right)^{2} + 8 " />  and expanding gives  <img class="equation_image" title=" \displaystyle f(x+h)=- 7 h^{2} - 14 h x - 4 h - 7 x^{2} - 4 x + 8 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%2Bh%29%3D-%207%20h%5E%7B2%7D%20-%2014%20h%20x%20-%204%20h%20-%207%20x%5E%7B2%7D%20-%204%20x%20%2B%208%20" alt="LaTeX:  \displaystyle f(x+h)=- 7 h^{2} - 14 h x - 4 h - 7 x^{2} - 4 x + 8 " data-equation-content=" \displaystyle f(x+h)=- 7 h^{2} - 14 h x - 4 h - 7 x^{2} - 4 x + 8 " />   Evaluating the difference quotient gives  <img class="equation_image" title=" \displaystyle \frac{(- 7 h^{2} - 14 h x - 4 h - 7 x^{2} - 4 x + 8)-(- 7 x^{2} - 4 x + 8)}{h} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%28-%207%20h%5E%7B2%7D%20-%2014%20h%20x%20-%204%20h%20-%207%20x%5E%7B2%7D%20-%204%20x%20%2B%208%29-%28-%207%20x%5E%7B2%7D%20-%204%20x%20%2B%208%29%7D%7Bh%7D%20" alt="LaTeX:  \displaystyle \frac{(- 7 h^{2} - 14 h x - 4 h - 7 x^{2} - 4 x + 8)-(- 7 x^{2} - 4 x + 8)}{h} " data-equation-content=" \displaystyle \frac{(- 7 h^{2} - 14 h x - 4 h - 7 x^{2} - 4 x + 8)-(- 7 x^{2} - 4 x + 8)}{h} " />  Simplifying gives  <img class="equation_image" title=" \displaystyle \frac{- 7 h^{2} - 14 h x - 4 h}{h}=- 7 h - 14 x - 4 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B-%207%20h%5E%7B2%7D%20-%2014%20h%20x%20-%204%20h%7D%7Bh%7D%3D-%207%20h%20-%2014%20x%20-%204%20" alt="LaTeX:  \displaystyle \frac{- 7 h^{2} - 14 h x - 4 h}{h}=- 7 h - 14 x - 4 " data-equation-content=" \displaystyle \frac{- 7 h^{2} - 14 h x - 4 h}{h}=- 7 h - 14 x - 4 " />  </p> </p>