We are tasked with finding the difference quotient for the function $f(x) = x^2 - 5x + 1$. The difference quotient is given by:

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

Step-by-step Solution:

1. Substitute $f(x+h)$: Replace $x$ with $x+h$ in $f(x)$: $f(x+h) = (x+h)^2 - 5(x+h) + 1.$ Expand each term: $f(x+h) = x^2 + 2xh + h^2 - 5x - 5h + 1.$

2. Subtract $f(x)$: Write $f(x+h) - f(x)$: $f(x+h) - f(x) = \left(x^2 + 2xh + h^2 - 5x - 5h + 1\right) - \left(x^2 - 5x + 1\right).$ Simplify by canceling terms: $f(x+h) - f(x) = 2xh + h^2 - 5h.$

3. Divide by $h$: Substitute $f(x+h) - f(x)$ into the difference quotient: $\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 - 5h}{h}.$ Simplify each term by dividing through by $h$: $\frac{f(x+h) - f(x)}{h} = 2x + h - 5.$

Final Answer:

$\frac{f(x+h) - f(x)}{h} = 2x + h - 5.$

Would you like me to expand further on any part of this solution or provide more context?

Here are five related questions to expand your understanding:

1. How is the difference quotient related to the derivative of a function?
2. What does the term $h \to 0$ mean in calculus?
3. Can the difference quotient be used for functions with discontinuities? Why or why not?
4. What would $f(x+h) - f(x)$ look like for a cubic function $f(x) = x^3$?
5. How do you interpret $2x + h - 5$ geometrically for the given quadratic?

Tip:

When simplifying expressions, always organize like terms step by step to avoid mistakes!