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.