https://file.aiquickdraw.com/mathbot/file/1729084229266hp1z34l8.jpg

To analyze the graph of the polynomial function \( f(x) \) in the image and determine the intervals where the function is positive and negative, we need to observe where the graph is above or below the x-axis.

### Positive Intervals:
A function is **positive** when the graph is above the x-axis (i.e., \( f(x) > 0 \)).

From the graph:
- The function is positive between \( x = -4.5 \) and \( x = -0.5 \).
- The function becomes positive again from \( x = 1 \) to infinity.

Thus, the **positive intervals** are:
\[ (-4.5, -0.5) \cup (1, \infty) \]

### Negative Intervals:
A function is **negative** when the graph is below the x-axis (i.e., \( f(x) < 0 \)).

From the graph:
- The function is negative between \( x = -\infty \) and \( x = -4.5 \).
- The function is negative again between \( x = -0.5 \) and \( x = 1 \).

Thus, the **negative intervals** are:
\[ (-\infty, -4.5) \cup (-0.5, 1) \]

### Final Answer:
- Positive Intervals: \( (-4.5, -0.5) \), \( (1, \infty) \)
- Negative Intervals: \( (-\infty, -4.5) \), \( (-0.5, 1) \)

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Would you like more details or have any further questions? Here are five related questions that might help deepen your understanding:

1. How can you identify zeros of the function from a graph?
2. What happens to the sign of a polynomial function when it crosses the x-axis?
3. How can you determine if the function is increasing or decreasing on certain intervals?
4. How would you express the intervals if the graph touched the x-axis but didn’t cross it?
5. What kind of polynomial function might produce this kind of graph?

**Tip:** Always observe where the function intersects the x-axis to determine the exact points where the sign changes from positive to negative or vice versa.

The graph of a polynomial function f is shown. Use interval notation to describe where the function is positive and negative.

Analyze the graph of a polynomial function to identify where it is positive and negative. Using interval notation, this problem helps students practice interpreting polynomial graphs and identifying changes in sign across the x-axis. Suitable for high school students.

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