https://file.aiquickdraw.com/mathbot/file/1726896864078l0rajcdv.jpg

Let's go through each question based on the graph:

1. **For what values of \( x \) is the function positive?**
   - The function is positive when its graph is above the x-axis. From the graph, this occurs approximately between \( x \in (-5, -2) \) and \( x \in (2, 5) \).

2. **For what values of \( x \) is the function negative?**
   - The function is negative when the graph is below the x-axis. This happens when \( x \in (-\infty, -5) \) and \( x \in (-2, 2) \), and again for \( x \in (5, \infty) \).

3. **For what values of \( x \) does the function reach a relative minimum?**
   - The function reaches a relative minimum at the lowest points of the valleys. From the graph, this occurs around \( x = 0 \) (approximately at \( y = -5 \)).

4. **For what value of \( x \) does the function reach a relative maximum?**
   - A relative maximum occurs at the peak of the curve. This happens around \( x = -3.5 \) and \( x = 3.5 \), where the graph reaches its highest points between the valleys.

5. **For what values of \( x \) is the function increasing?**
   - The function is increasing when the graph moves upwards as \( x \) increases. From the graph, this occurs for \( x \in (-\infty, -3.5) \), \( x \in (0, 3.5) \), and for \( x > 5 \).

6. **For what values of \( x \) is the function decreasing?**
   - The function is decreasing when the graph moves downwards as \( x \) increases. This happens for \( x \in (-3.5, 0) \) and \( x \in (3.5, 5) \).

7. **Describe what happens to the value of \( f(x) \) as \( x \) gets very large.**
   - As \( x \to \infty \), the graph rises sharply, indicating that \( f(x) \to \infty \).

8. **Describe what happens to the value of \( f(x) \) as \( x \) gets very negative.**
   - As \( x \to -\infty \), the graph also rises sharply, implying that \( f(x) \to \infty \).

If you have any questions or want further details, feel free to ask!

Here are 5 follow-up questions:
1. What does it mean for a function to have a relative minimum and maximum?
2. How can we algebraically determine intervals of increase or decrease in a function?
3. Why does the function behave symmetrically as \( x \to \infty \) and \( x \to -\infty \)?
4. How do relative extrema help us understand the overall shape of a function?
5. What types of functions exhibit similar behavior at \( x \to \infty \) and \( x \to -\infty \)?

**Tip:** Always check the signs of the function's derivative to confirm where it is increasing or decreasing!

For what values of x is the function positive? For what values of x is the function negative? For what values of x does the function reach a relative minimum? For what value of x does the function reach a relative maximum? For what values of x is the function increasing? For what values of x is the function decreasing? Describe what happens to the value of f(x) as x gets very large. Describe what happens to the value of f(x) as x gets very negative.

Understand how to analyze a function graph by identifying intervals of positivity, negativity, and relative extrema. This guide covers key calculus concepts such as increasing and decreasing intervals, as well as end behavior for a function's graph. Suitable for high school students in Grades 10-12.

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