https://file.aiquickdraw.com/mathbot/file/1733539352718vx4ptyoy.jpg

The graph depicts a parabolic curve that opens upwards, and the vertex is at \( (0, -1) \). Here's how we can analyze it:

### Part (a): Intervals where the function is **increasing**
The function is increasing on intervals where the graph slopes upward as \( x \) increases. Observing the graph, the function starts increasing from the vertex (\( x = 0 \)) onwards to the right. Hence, the interval is:
\[
(0, \infty)
\]

### Part (b): Intervals where the function is **decreasing**
The function is decreasing on intervals where the graph slopes downward as \( x \) increases. This occurs from the left end of the graph up to the vertex (\( x = 0 \)). Thus, the interval is:
\[
(-\infty, 0)
\]

### Final Answer:
1. **Increasing interval**: \( (0, \infty) \)
2. **Decreasing interval**: \( (-\infty, 0) \)

Would you like a detailed breakdown of how to write this in interval notation or more explanation on the graph's behavior? 

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### Related Questions:
1. What is the vertex of this parabola and how is it calculated?
2. How can we determine if a function is increasing or decreasing algebraically?
3. What is the domain and range of this function based on the graph?
4. How can symmetry in a graph help identify intervals of increase and decrease?
5. How does this graph represent a quadratic function?

### Tip:
For any quadratic function \( f(x) = ax^2 + bx + c \), the function decreases until the vertex and increases thereafter if \( a > 0 \) (upward parabola).

What are the intervals where the function represented by the graph is increasing and decreasing?

Learn how to identify the intervals where a quadratic function is increasing and decreasing based on its graph. The problem involves analyzing a parabola with vertex (0, -1), providing insights into the behavior of quadratic functions. Suitable for students in Grades 9-11.

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