The image contains a problem asking for the value of the difference quotient $\frac{f(a+h) - f(a)}{h}$ for a given function $f(x)$.

Problem Statement:

Given the function $f(x) = x^2 - 5x + 2$, the task is to find the value of the difference quotient: $\frac{f(a+h) - f(a)}{h}$ for $a = 1$.

Solution:

Let's go through the steps to solve the problem.

1. Substitute the function $f(x)$: $f(x) = x^2 - 5x + 2$

2. Find $f(a)$: Since $a = 1$, $f(1) = (1)^2 - 5(1) + 2 = 1 - 5 + 2 = -2$

3. Find $f(a+h)$: Substitute $a + h$ into the function: $f(1+h) = (1+h)^2 - 5(1+h) + 2$ Expand and simplify: $f(1+h) = (1 + 2h + h^2) - (5 + 5h) + 2 = h^2 - 3h - 2$

4. Calculate the difference $f(a+h) - f(a)$: $f(1+h) - f(1) = (h^2 - 3h - 2) - (-2) = h^2 - 3h$

5. Find the difference quotient: $\frac{f(a+h) - f(a)}{h} = \frac{h^2 - 3h}{h} = h - 3$

So, the difference quotient simplifies to $h - 3$.

Given the specific options provided in the image (though not clearly visible, the final solution's simplified form should match one of them), the difference quotient is likely $h - 3$.

Would you like further details or explanations on any of the steps? Here are five additional questions you might find helpful:

1. How is the difference quotient related to the derivative?
2. What happens to the difference quotient as $h$ approaches 0?
3. Can the difference quotient be used to find the slope of a tangent line?
4. What is the geometric interpretation of the function $f(x)$?
5. How does changing the value of $a$ affect the difference quotient?

Tip: The difference quotient is a fundamental concept in calculus, as it forms the basis for understanding derivatives.