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To solve this, we need to calculate the slope of the secant line between the two points on the curve defined by the function \( f(x) = -3x^2 + 1 \) at \( x = 3 \) and \( x = 3 + h \).

### Step-by-step solution:

1. **Find \( f(3) \):**
   \[
   f(3) = -3(3)^2 + 1 = -3(9) + 1 = -27 + 1 = -26
   \]

2. **Find \( f(3 + h) \):**
   \[
   f(3 + h) = -3(3 + h)^2 + 1
   \]
   Expand \( (3 + h)^2 \):
   \[
   (3 + h)^2 = 9 + 6h + h^2
   \]
   Substituting this back:
   \[
   f(3 + h) = -3(9 + 6h + h^2) + 1 = -27 - 18h - 3h^2 + 1 = -26 - 18h - 3h^2
   \]

3. **Find the slope of the secant line:**
   The slope of the secant line between \( x = 3 \) and \( x = 3 + h \) is given by the difference in the function values divided by the difference in \( x \)-values:
   \[
   \text{slope} = \frac{f(3 + h) - f(3)}{(3 + h) - 3}
   \]
   Substitute \( f(3 + h) = -26 - 18h - 3h^2 \) and \( f(3) = -26 \):
   \[
   \text{slope} = \frac{(-26 - 18h - 3h^2) - (-26)}{h} = \frac{-18h - 3h^2}{h}
   \]
   Simplify:
   \[
   \text{slope} = \frac{-18h - 3h^2}{h} = -18 - 3h
   \]

### Final Answer:
The slope of the secant line is:
\[
\boxed{-18 - 3h}
\]

Would you like a detailed explanation or have any other questions about this process?

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Here are some related questions you might find helpful:
1. How does the formula for the secant line relate to the derivative of the function?
2. What happens to the slope as \( h \) approaches zero?
3. How would you compute the instantaneous rate of change (the derivative) of \( f(x) = -3x^2 + 1 \) at \( x = 3 \)?
4. What is the geometric interpretation of a secant line in the context of this problem?
5. Can you verify the slope by using a different approach, such as a numerical method?

**Tip**: When solving for the slope of a secant line, always make sure to simplify both the numerator and the denominator before dividing. This reduces errors and helps with clearer insights into the behavior of the function.

A secant line intersects the graph of f(x) = -3x^2 + 1 at two points with x-coordinates 3 and 3 + h, where h ≠ 0. What is the slope of the secant line in terms of h?

Learn how to find the slope of the secant line between two points on the curve of f(x) = -3x^2 + 1 using algebraic and calculus concepts. The solution includes a fully expanded and simplified form of the slope in terms of h.

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