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:
   \[
   = -2x - 2h + 9 + 2x - 9
   = -2h
   \]

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.

Explore how to apply the difference quotient to the linear function f(x) = -2x + 9. This step-by-step solution explains the process of finding the derivative using the difference quotient formula, suitable for high school students studying algebra and calculus concepts.

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