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We are tasked with identifying the correct graph of the polynomial function \( y = f(x) \) based on the provided conditions:

### Given Information:
1. \( f(x) \) is positive on intervals \( (-3, -2) \) and \( (0, \infty) \).
2. \( f(x) \) is negative on intervals \( (-\infty, -3) \) and \( (-2, 0) \).
3. \( f(x) \) is increasing on intervals \( (-\infty, -2.5) \) and \( (-1, \infty) \).
4. \( f(x) \) is decreasing on interval \( (-2.5, -1) \).

### Key Analysis:
- **Sign of \( f(x) \)**:
   - \( f(x) \) is **positive** in the intervals \( (-3, -2) \) and \( (0, \infty) \), which means the graph should be **above the x-axis** in these intervals.
   - \( f(x) \) is **negative** in the intervals \( (-\infty, -3) \) and \( (-2, 0) \), so the graph should be **below the x-axis** in these ranges.

- **Increasing/Decreasing Behavior**:
   - \( f(x) \) is **increasing** on \( (-\infty, -2.5) \) and \( (-1, \infty) \). This suggests that the slope of the graph is upward in these intervals.
   - \( f(x) \) is **decreasing** on \( (-2.5, -1) \), meaning the slope is downward in this region.

### Matching with the Graphs:
Based on these observations, we can eliminate graphs that do not follow the sign and behavior changes correctly:
- **Graph a**: Shows a positive value between \( (-3, -2) \) and \( (0, \infty) \), negative value in \( (-\infty, -3) \) and \( (-2, 0) \), and the correct increasing/decreasing pattern. 
- **Graph b, c, and d** do not match all the required increasing/decreasing patterns and signs.

### Correct Answer:
**Graph a** matches all the given conditions.

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Do you want more details about the analysis or have any further questions?

Here are some related questions to deepen understanding:
1. How do we determine when a function is increasing or decreasing on a graph?
2. What role do critical points play in determining the behavior of a polynomial function?
3. How can the sign of \( f(x) \) help in sketching polynomial functions?
4. How do turning points affect the shape of polynomial graphs?
5. What other features of polynomials (like concavity) could help in graph sketching?

**Tip**: When analyzing graphs, always track both the signs and slopes of the function to understand its behavior over different intervals.

Use the information below to sketch a graph of the polynomial function y = f(x).

- f(x) is positive on intervals (-3, -2) and (0, ∞).
- f(x) is negative on intervals (-∞, -3) and (-2, 0).
- f(x) is increasing on intervals (-∞, -2.5) and (-1, ∞).
- f(x) is decreasing on interval (-2.5, -1).

Identify the correct graph from the options a, b, c, and d.

Learn how to identify the correct graph of a polynomial function using given intervals for positive, negative, increasing, and decreasing behaviors. This exercise involves understanding polynomial functions and their behavior across different intervals, suitable for students in Grades 10-12.

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