We are asked to find the difference quotient of the function $f(x) = x^2 - 6x + 4$. The difference quotient is given by:

$\frac{f(x+h) - f(x)}{h}, \quad h \neq 0$

Step 1: Find $f(x+h)$

To do this, replace every occurrence of $x$ in $f(x)$ with $x+h$.

$f(x+h) = (x+h)^2 - 6(x+h) + 4$

Now expand and simplify each term:

$(x+h)^2 = x^2 + 2xh + h^2$ $-6(x+h) = -6x - 6h$

Thus,

$f(x+h) = (x^2 + 2xh + h^2) - (6x + 6h) + 4$

Simplifying:

$f(x+h) = x^2 + 2xh + h^2 - 6x - 6h + 4$

Step 2: Subtract $f(x)$ from $f(x+h)$

Now subtract $f(x) = x^2 - 6x + 4$ from $f(x+h)$:

$f(x+h) - f(x) = \left(x^2 + 2xh + h^2 - 6x - 6h + 4\right) - \left(x^2 - 6x + 4\right)$

Distribute the negative sign and subtract:

$f(x+h) - f(x) = x^2 + 2xh + h^2 - 6x - 6h + 4 - x^2 + 6x - 4$

Now cancel out the like terms:

• $x^2$ cancels with $-x^2$
• $-6x$ cancels with $+6x$
• $4$ cancels with $-4$

This leaves:

$f(x+h) - f(x) = 2xh + h^2 - 6h$

Step 3: Divide by $h$

Now, divide the result by $h$:

$\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 - 6h}{h}$

Simplify each term:

$\frac{f(x+h) - f(x)}{h} = 2x + h - 6$

Final Answer

Thus, the simplified difference quotient is:

$\frac{f(x+h) - f(x)}{h} = 2x + h - 6, \quad h \neq 0$

Would you like more details on this process, or do you have any questions?

Here are five related questions to expand on this topic:

1. How does the difference quotient relate to the derivative of a function?
2. Can you compute the difference quotient for $f(x) = 3x^2 - 2x + 1$?
3. What happens to the difference quotient as $h$ approaches zero?
4. How would the result change if $f(x)$ were a cubic function?
5. How can the difference quotient be interpreted geometrically?

Tip: The difference quotient provides an average rate of change of the function over an interval and is a crucial step in finding the derivative.