Simplify the difference quotient 

StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFractionf(x+h)−f(x)h 

for the given function.

f left parenthesis x right parenthesis equals 3 x squared minus 8 x plus 1

To simplify the difference quotient \(\frac{f(x+h) - f(x)}{h}\) for the given function \(f(x) = 3x^2 - 8x + 1\), follow these steps:

### Step 1: Compute \(f(x+h)\)
Substitute \(x + h\) into the function \(f(x)\):

\[
f(x+h) = 3(x+h)^2 - 8(x+h) + 1
\]

### Step 2: Expand and Simplify \(f(x+h)\)
First, expand \((x+h)^2\):

\[
(x+h)^2 = x^2 + 2xh + h^2
\]

Now, substitute this back into the expression for \(f(x+h)\):

\[
f(x+h) = 3(x^2 + 2xh + h^2) - 8(x + h) + 1
\]

Distribute the constants:

\[
f(x+h) = 3x^2 + 6xh + 3h^2 - 8x - 8h + 1
\]

### Step 3: Compute the Difference \(f(x+h) - f(x)\)
Subtract \(f(x)\) from \(f(x+h)\):

\[
f(x+h) - f(x) = \left(3x^2 + 6xh + 3h^2 - 8x - 8h + 1\right) - \left(3x^2 - 8x + 1\right)
\]

Distribute and combine like terms:

\[
f(x+h) - f(x) = 3x^2 + 6xh + 3h^2 - 8x - 8h + 1 - 3x^2 + 8x - 1
\]

Simplify by canceling out the common terms \(3x^2\), \(-8x\), and \(1\):

\[
f(x+h) - f(x) = 6xh + 3h^2 - 8h
\]

### Step 4: Divide by \(h\)
Now, divide the simplified difference by \(h\):

\[
\frac{f(x+h) - f(x)}{h} = \frac{6xh + 3h^2 - 8h}{h}
\]

Factor out \(h\) from the numerator:

\[
\frac{f(x+h) - f(x)}{h} = \frac{h(6x + 3h - 8)}{h}
\]

Cancel out \(h\) from the numerator and denominator:

\[
\frac{f(x+h) - f(x)}{h} = 6x + 3h - 8
\]

### Final Answer
The simplified difference quotient is:

\[
\frac{f(x+h) - f(x)}{h} = 6x + 3h - 8
\]

Would you like any further explanation or have any questions?

### Related Questions
1. What happens to the difference quotient as \( h \) approaches 0?
2. How do you use the difference quotient to find the derivative of a function?
3. What is the geometric interpretation of the difference quotient?
4. Can the difference quotient be used for non-polynomial functions?
5. How does the difference quotient relate to the concept of a secant line?

### Tip
Remember, the difference quotient is a fundamental tool in calculus for finding the derivative, which represents the rate of change or the slope of the tangent line at a specific point on the curve.

Learn how to simplify the difference quotient for the function f(x) = 3x^2 - 8x + 1 with a step-by-step solution. This problem covers key calculus concepts, including the difference quotient formula and its application. Suitable for advanced high school and college students.

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