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We are tasked with determining the rate of change (or difference quotient) of the given function \( f(x) = (x - 3)^2 + 5 \) using the formula:

\[
\frac{\Delta y}{\Delta x} = \frac{f(x+h) - f(x)}{h}
\]

Let's compute this step by step.

### 1. Function Evaluation:
Given \( f(x) = (x - 3)^2 + 5 \), first calculate \( f(x + h) \).

\[
f(x + h) = ((x + h) - 3)^2 + 5 = (x + h - 3)^2 + 5 = (x - 3 + h)^2 + 5
\]

Now expand \( (x - 3 + h)^2 \):

\[
(x - 3 + h)^2 = (x - 3)^2 + 2(x - 3)h + h^2 = (x^2 - 6x + 9) + 2(x - 3)h + h^2
\]

Thus:

\[
f(x + h) = (x^2 - 6x + 9) + 2(x - 3)h + h^2 + 5 = x^2 - 6x + 14 + 2(x - 3)h + h^2
\]

### 2. Difference Quotient:
Now, we compute \( \frac{f(x+h) - f(x)}{h} \).

We already know:

\[
f(x) = (x - 3)^2 + 5 = x^2 - 6x + 14
\]

Thus:

\[
f(x+h) - f(x) = \left[ x^2 - 6x + 14 + 2(x - 3)h + h^2 \right] - (x^2 - 6x + 14)
\]

Simplifying this:

\[
f(x+h) - f(x) = 2(x - 3)h + h^2
\]

Finally, dividing by \( h \):

\[
\frac{f(x+h) - f(x)}{h} = \frac{2(x - 3)h + h^2}{h} = 2(x - 3) + h
\]

Thus, the difference quotient is:

\[
\frac{\Delta y}{\Delta x} = 2(x - 3) + h = 2x - 6 + h
\]

### Answer:
The correct option is **c**:

\[
\frac{\Delta y}{\Delta x} = 2x - 6 + h
\]

Would you like more details on any part of this? Here are five related questions:

1. How do you interpret the difference quotient in terms of the slope of a function?
2. What does \( h \) represent in the context of the difference quotient?
3. What happens to the difference quotient as \( h \) approaches 0?
4. Can you apply the same process to compute the difference quotient for a cubic function?
5. What is the geometric significance of the difference quotient for a quadratic function like \( f(x) \)?

**Tip:** The difference quotient gives an approximation of the slope of the tangent line at a point on the curve for small values of \( h \).

Seleccione la opción que corresponde a la razón de cambio de la función f(x) = (x - 3)^2 + 5 utilizando la fórmula de cociente de diferencias.

This problem involves calculating the rate of change for the quadratic function f(x) = (x - 3)^2 + 5 using the difference quotient formula. It's an essential exercise in understanding how to work with quadratic functions and applying algebraic methods. Suitable for students in Grades 10-12.

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