https://file.aiquickdraw.com/mathbot/file/1726683038616ap9dwjej.jpg

To determine over what intervals the function is positive or negative using the provided graph, we need to focus on the areas where the function crosses the x-axis.

### Key Observations:
- The function crosses the x-axis at \( x = 0 \) and \( x = 4 \).
- The graph opens upwards, meaning it represents a parabola.

### Intervals:
1. **Positive Intervals** (where the function is above the x-axis):
   - The function is positive when the y-values are above the x-axis. This occurs between \( x = 4 \) and \( x = 6 \). Therefore, the function is positive on the interval \( (4, 6) \).

2. **Negative Intervals** (where the function is below the x-axis):
   - The function is negative when the y-values are below the x-axis. This happens between \( x = 0 \) and \( x = 4 \). Therefore, the function is negative on the interval \( (0, 4) \).

### Final Answer:
- Positive interval: \( (4, 6) \)
- Negative interval: \( (0, 4) \)

Would you like more details on how these intervals were determined or have any further questions? Here are some related questions you might want to explore:

1. How do you determine the x-intercepts of a function from a graph?
2. What does the shape of the parabola tell you about the nature of its function?
3. How do you find the vertex of a parabola?
4. How do you determine whether a function is increasing or decreasing from its graph?
5. What role does the axis of symmetry play in understanding the behavior of a quadratic function?

**Tip:** When analyzing graphs, always look for the x-intercepts to identify sign changes in the function!

Use the graph of the function. Determine over what interval(s) the function is positive or negative.

Learn how to analyze a quadratic function graph and identify intervals where the function is positive or negative. Suitable for Grades 9-12 students, this guide focuses on graph analysis and understanding key features of parabolas.

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