To find the difference quotient of the function $f(x) = -9x + 3$, we need to compute the expression:

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

where $h \neq 0$.

First, let's find $f(x+h)$:

$f(x+h) = -9(x+h) + 3$ $f(x+h) = -9x - 9h + 3$

Now, we need to substitute $f(x+h)$ and $f(x)$ into the difference quotient:

$\frac{f(x+h) - f(x)}{h} = \frac{(-9x - 9h + 3) - (-9x + 3)}{h}$

Simplify the numerator:

$= \frac{-9x - 9h + 3 + 9x - 3}{h}$ $= \frac{-9h}{h}$

Since $h \neq 0$, we can cancel $h$ in the numerator and denominator:

$= -9$

Thus, the difference quotient is:

$\boxed{-9}$

Would you like further details or have any questions? Here are some related questions for further exploration:

1. What is the definition of a difference quotient?
2. How does the difference quotient relate to the derivative of a function?
3. Compute the derivative of $f(x) = -9x + 3$ and verify it using the limit definition.
4. How would you find the difference quotient for a quadratic function, such as $f(x) = x^2$?
5. What is the geometric interpretation of the difference quotient?
6. Can the difference quotient be used for non-linear functions? Provide an example.
7. How does the difference quotient change if $f(x)$ is a constant function?
8. What are some common applications of the difference quotient in calculus?

Tip: The difference quotient is a fundamental concept in calculus, forming the basis for the definition of the derivative. Understanding it deeply enhances your grasp of more advanced topics in calculus.