https://file.aiquickdraw.com/mathbot/file/1725414578592x0t9fpm7.jpg

To solve the problem of determining the intervals where the function is increasing or decreasing, let's analyze the graph provided.

### Identifying the Intervals:

1. **Increasing Intervals:**
   - A function is increasing where its graph rises as you move from left to right.
   - From the graph, the function is increasing on the intervals:
     - From approximately \( x \approx -3.5 \) to \( x \approx -2 \)
     - From \( x \approx 1 \) to \( x \approx 3.5 \)

2. **Decreasing Intervals:**
   - A function is decreasing where its graph falls as you move from left to right.
   - From the graph, the function is decreasing on the intervals:
     - From \( x \approx -5 \) to \( x \approx -3.5 \)
     - From \( x \approx -2 \) to \( x \approx 1 \)
     - From \( x \approx 3.5 \) onward

### Final Answer:

- **Increasing on the interval(s):**
  \[
  (-3.5, -2), \quad (1, 3.5)
  \]

- **Decreasing on the interval(s):**
  \[
  (-5, -3.5), \quad (-2, 1), \quad (3.5, \infty)
  \]

Would you like more details or have any questions?

Here are some related questions:
1. What is the significance of the critical points on the graph?
2. How can the first derivative test be applied to determine increasing or decreasing intervals?
3. How does concavity relate to the intervals of increase and decrease?
4. What role do inflection points play in changing the nature of the function's graph?
5. How can you determine the exact coordinates of the local maxima and minima?

**Tip:** To find where a function is increasing or decreasing algebraically, take its derivative and analyze the sign changes on different intervals.

The function graphed above is: Increasing on the interval(s) _______. Decreasing on the interval(s) _______.

Learn how to identify the intervals where a function is increasing or decreasing by analyzing its graph. This problem involves key calculus concepts such as the first derivative test and is suitable for students in Grades 11-12.

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