The difference quotient is a method to find the average rate of change of a function, which leads to the derivative when the limit is taken. For the function $f(x) = -2x + 9$, the difference quotient formula is:

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

Step-by-step:

1. Find $f(x + h)$: $f(x+h) = -2(x+h) + 9 = -2x - 2h + 9$

2. Difference $f(x + h) - f(x)$: $f(x+h) - f(x) = (-2x - 2h + 9) - (-2x + 9)$ Simplify the expression:

   = -2h$$

3. Divide by $h$: $\frac{f(x+h) - f(x)}{h} = \frac{-2h}{h} = -2$

Final result:

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

Thus, the difference quotient for the function $f(x) = -2x + 9$ is $-2$, which is also the derivative of the function.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the geometric interpretation of the derivative $-2$ in this function?
2. How does the difference quotient change for a quadratic function like $f(x) = x^2 + 3x + 1$?
3. Why does the difference quotient give us the slope of the tangent line?
4. How would the difference quotient look for a function with a higher degree polynomial?
5. Can you use the difference quotient to estimate the derivative of a non-linear function numerically?

Tip: The difference quotient can be seen as a method to compute the slope between two points on a curve, which becomes more precise as $h$ approaches 0.