The difference quotient of a function $f(x)$ is given by:

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

Given the function:

$f(x) = -7x + 7$

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

To find $f(x+h)$, substitute $x+h$ into the function $f(x)$:

$f(x+h) = -7(x+h) + 7$

Simplify:

$f(x+h) = -7x - 7h + 7$

Step 2: Compute $f(x+h) - f(x)$

Now, subtract $f(x)$ from $f(x+h)$:

$f(x+h) - f(x) = \left(-7x - 7h + 7\right) - \left(-7x + 7\right)$

Simplify:

$f(x+h) - f(x) = -7h$

Step 3: Compute the difference quotient

Now, divide by $h$:

$\frac{f(x+h) - f(x)}{h} = \frac{-7h}{h}$

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

$\frac{f(x+h) - f(x)}{h} = -7$

Thus, the difference quotient is $\boxed{-7}$.

Would you like more details or explanations?

Related Questions:

1. What is the interpretation of the difference quotient?
2. How does the difference quotient relate to the derivative of the function?
3. What is the derivative of $f(x) = -7x + 7$?
4. What happens to the difference quotient as $h$ approaches 0?
5. Can the difference quotient method be used for nonlinear functions?

Tip:

For linear functions like $f(x) = -7x + 7$, the difference quotient will always be the constant slope of the line, which is the coefficient of $x$.