https://file.aiquickdraw.com/mathbot/file/1726863879253d3qij3y2.jpg

I can now view the graphs. Let's break down each part and determine the intervals where the functions are increasing or decreasing.

### a) First graph:
- The graph shows a parabola-like shape.
- **Increasing**: The function is increasing after the local minimum (around \( x = 1 \)).
- **Decreasing**: The function is decreasing before the local minimum, to the left of \( x = 1 \).

Answer:
- **Increasing**: \( (1, \infty) \)
- **Decreasing**: \( (-\infty, 1) \)

### b) Second graph:
- The function has an inflection point and then grows rapidly.
- **Increasing**: The function increases everywhere after \( x = 0 \).
- **Decreasing**: No decreasing interval is observed.

Answer:
- **Increasing**: \( (0, \infty) \)
- **Decreasing**: NONE

### c) Third graph:
- The graph appears to be sinusoidal, with alternating increasing and decreasing intervals.
- **Increasing**: The function increases on the intervals \( (-1, 0) \) and \( (2, 3) \).
- **Decreasing**: The function decreases on the intervals \( (0, 2) \) and \( (3, 4) \).

Answer:
- **Increasing**: \( (-1, 0) \cup (2, 3) \)
- **Decreasing**: \( (0, 2) \cup (3, 4) \)

Would you like further details or explanations on any of the intervals? Here are some related questions to expand:

1. How can you determine increasing or decreasing behavior from the derivative?
2. What role do critical points play in analyzing a function’s intervals?
3. How does concavity affect the increasing/decreasing nature of a function?
4. What is the significance of inflection points in determining function behavior?
5. Can a function have more than one increasing and decreasing interval?

**Tip**: Use the first derivative of a function to confirm increasing or decreasing intervals by checking its sign (positive for increasing, negative for decreasing).

For the following functions, determine on what intervals the function is increasing and on what intervals the function is decreasing. Assume the entire function is shown in the provided graph. Give answers as open intervals, and if no interval exists, type the word NONE in the associated blank.

Learn how to determine the intervals where functions are increasing or decreasing by analyzing graphs. This problem involves interpreting graphs and using calculus concepts like the first derivative and critical points. Suitable for high school students (Grades 11-12).

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