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solve all problems algebraically unless you are specially asked to graph. you may always use graphing to check your answers.
if you are told to solve a specific way, you must do that to earn the points.
give all answers as exact answers unless you are asked to round.
notes: module 1 (answer key)
notes: module 2 (answer key)
notes: module 3 (answer key)
score on last try: 5 of 5 pts. see details for more.

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given the function ( f(x)=8 + 3x-4x^{2} ), find and simplify the difference quotient.
( 3-8x - 4h )
*algebraic work must be shown to retain points on quizzes, midterm or final
question help: video
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Explanation:

Step1: Find \(f(x + h)\)

Given \(f(x)=8 + 3x-4x^{2}\), then \(f(x + h)=8+3(x + h)-4(x + h)^{2}\).
Expand \((x + h)^{2}=x^{2}+2xh+h^{2}\), so \(f(x + h)=8 + 3x+3h-4(x^{2}+2xh + h^{2})=8 + 3x+3h-4x^{2}-8xh-4h^{2}\).

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

\(f(x + h)-f(x)=(8 + 3x+3h-4x^{2}-8xh-4h^{2})-(8 + 3x-4x^{2})\)
\(=8 + 3x+3h-4x^{2}-8xh-4h^{2}-8 - 3x + 4x^{2}=3h-8xh-4h^{2}\).

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

\(\frac{f(x + h)-f(x)}{h}=\frac{3h-8xh-4h^{2}}{h}\), factor out \(h\) from the numerator: \(\frac{h(3 - 8x-4h)}{h}\).
Cancel out \(h\) (assuming \(h
eq0\)), we get \(3-8x - 4h\).

Answer:

\(3-8x - 4h\)