https://file.aiquickdraw.com/mathbot/file/1725999552382m6iwxjfd.jpg

The graph in the image shows a function with both increasing and decreasing intervals, and the question asks for the intervals where the function is increasing and decreasing.

### Steps to solve:

1. **Determine critical points**: The graph appears to have turning points at \(x = -2\), \(x = 1\), and \(x = 3\). These are the points where the function changes from increasing to decreasing or vice versa.
   
2. **Identify increasing intervals**:
    - From \(x = -\infty\) to \(x = -2\), the function is decreasing.
    - From \(x = -2\) to \(x = 1\), the function is increasing.
    - From \(x = 1\) to \(x = 3\), the function is still increasing.
   
3. **Identify decreasing intervals**:
    - From \(x = 3\) onward to \(x = \infty\), the function is decreasing.

### Correct intervals:
- **Increasing** on the interval \((-2, 1)\) and \((1, 3)\).
- **Decreasing** on the interval \((-\infty, -2)\) and \((3, \infty)\).

You entered:
- For increasing: \((-\infty, -2)\) and \((1, 3)\), which should be corrected to \((-2, 1)\) and \((1, 3)\).
- For decreasing: \((-\infty, 1)\), which should be corrected to \((-\infty, -2)\) and \((3, \infty)\).

Would you like more details, or do you have any questions about this?

Here are 5 follow-up questions:
1. How do you find critical points in a graph?
2. What is the significance of turning points in a function?
3. How do you determine whether a function is increasing or decreasing in an interval?
4. What role does the first derivative play in finding intervals of increase or decrease?
5. Can a function have multiple intervals where it increases or decreases?

**Tip**: When analyzing the behavior of a function, pay attention to where the slope of the tangent line (or the derivative) changes signs.

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

Learn how to analyze increasing and decreasing intervals of a function from its graph. This problem uses turning points at x = -2, x = 1, and x = 3 to identify where the function is increasing and decreasing. Suitable for high school students.

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