The difference quotient for a function \( f(x) \) is defined as:

\[
\frac{f(x+h) - f(x)}{h}
\]

for a small increment \( h \) in \( x \).

Let's find the difference quotient for \( f(x) = 3x^2 - 8x + 1 \).

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

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

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

2. **Subtract \( f(x) \) from \( f(x + h) \):**
   \[
   f(x + h) - f(x) = (3x^2 + 6xh + 3h^2 - 8x - 8h + 1) - (3x^2 - 8x + 1)
   \]
   Simplifying by canceling common terms:
   \[
   f(x + h) - f(x) = 6xh + 3h^2 - 8h
   \]

3. **Divide by \( h \):**
   \[
   \frac{f(x + h) - f(x)}{h} = \frac{6xh + 3h^2 - 8h}{h}
   \]
   Simplifying by dividing each term by \( h \):
   \[
   \frac{f(x + h) - f(x)}{h} = 6x + 3h - 8
   \]

So, the difference quotient for \( f(x) = 3x^2 - 8x + 1 \) is:

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

This expression gives the average rate of change of the function over the interval \([x, x + h]\). 

Would you like more details on any step or have any questions? Here are some related questions:

1. How does the difference quotient relate to the derivative of a function?
2. What happens to the difference quotient as \( h \) approaches zero?
3. Can you find the derivative of \( f(x) = 3x^2 - 8x + 1 \) using the limit definition?
4. How is the difference quotient used in real-world applications?
5. What is the geometric interpretation of the difference quotient?

**Tip:** The derivative of a function \( f(x) \) at a point \( x \) is the limit of the difference quotient as \( h \) approaches zero.

Learn how to compute the difference quotient for the quadratic function f(x) = 3x^2 - 8x + 1. This step-by-step guide covers key algebraic concepts, including the difference quotient formula and its relationship to the derivative. Ideal for high school students in Grades 10-12.

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