https://file.aiquickdraw.com/mathbot/file/1727026469274xe2x88an.jpg

What are the intervals of increasing and decreasing?

To determine the intervals of increasing and decreasing from the provided graph, we will analyze the general behavior of the curve based on the critical points and inflection points visible in the image. 

### Key observations from the graph:
- **Critical points** (where the slope is zero or undefined) are labeled:
  - \((-1.414, -1)\)
  - \((-0.632, -2.619)\)
  - \((0, -1)\)
  - \((0.632, 0.619)\)
  - \((1, 0)\)
  - \((1.666, 0)\)

#### Intervals of Increase:
The function is increasing when the curve moves upwards (positive slope).
- Between \((-1.414, -1)\) and \((0, -1)\): The function is rising.
- Between \((0.269, 0)\) and \((1, 0)\): The function is rising.

Therefore, the function is increasing on the intervals:
- \((-1.414, 0)\)
- \((0.269, 1)\)

#### Intervals of Decrease:
The function is decreasing when the curve moves downwards (negative slope).
- From \((-0.632, -2.619)\) to \((-1.414, -1)\): The function is falling.
- From \((0, -1)\) to \((0.632, 0.619)\): The function is falling again.
- From \((1.666, 0)\) onward: The function appears to be falling once more.

Thus, the function is decreasing on the intervals:
- \((-0.632, -1.414)\)
- \((0, 0.269)\)
- \((1.666, \infty)\)

Let me know if you'd like more details on any specific part of the function!

Here are 5 questions that expand on this:
1. What are the intervals of concavity and inflection points?
2. How can you classify the critical points based on local maxima and minima?
3. Can you identify any points of symmetry from the graph?
4. How would the graph change if it represented a transformation of this function, such as a vertical shift?
5. How can derivatives help confirm the increasing and decreasing behavior of the function?

**Tip**: For analyzing the increasing and decreasing behavior of any function, use the first derivative. Where the derivative is positive, the function is increasing, and where it's negative, the function is decreasing.

Analyze the intervals of increasing and decreasing behavior of a function using critical points. Key calculus concepts such as first derivative tests and slope are explored. Ideal for Grades 10-12.

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