https://file.aiquickdraw.com/mathbot/file/1728521933605tndb3gur.jpg

The function is positive: Response area, negative: Response areaand Response area ,increasing: Response areaand Response area, decreasing: Response area and Response area .

The function given is:

\[
y = -x^4 + 8x^2 + 9
\]

From the graph, let's analyze the different intervals for positivity, negativity, increasing, and decreasing behavior.

### 1. Positive and Negative Intervals

- **Positive**: The function is above the x-axis (positive \( y \)-values) in the intervals where the curve is above the x-axis. From the graph, this happens approximately between:
  - \( -2 < x < 2 \).

- **Negative**: The function is below the x-axis (negative \( y \)-values) when the curve dips below the x-axis. This happens when \( |x| > 2 \), i.e., the intervals:
  - \( x < -2 \) and \( x > 2 \).

### 2. Increasing and Decreasing Intervals

- **Increasing**: A function is increasing when its slope is positive (the graph rises). Based on the graph, the function is increasing in the intervals:
  - \( x < -2 \) and \( 0 < x < 2 \).

- **Decreasing**: The function is decreasing when its slope is negative (the graph falls). From the graph, the function is decreasing in the intervals:
  - \( -2 < x < 0 \) and \( x > 2 \).

### Final summary:
- Positive: \( -2 < x < 2 \)
- Negative: \( x < -2 \) and \( x > 2 \)
- Increasing: \( x < -2 \) and \( 0 < x < 2 \)
- Decreasing: \( -2 < x < 0 \) and \( x > 2 \)

Would you like further details or have any questions on this?

Here are some related questions:
1. How can we confirm where the function crosses the x-axis?
2. How do we determine the local maxima and minima of the function?
3. What are the second derivative properties of this function?
4. Can this polynomial have any inflection points?
5. What is the behavior of the function as \( x \to \infty \) or \( x \to -\infty \)?

**Tip**: To analyze a polynomial function, the first derivative can help determine where it's increasing or decreasing, and the second derivative gives insights into concavity.

The function is positive: Response area, negative: Response area, increasing: Response area, decreasing: Response area.

Learn how to analyze the polynomial function y = -x^4 + 8x^2 + 9 for positive, negative, increasing, and decreasing intervals. The graph of this function provides insights into where the curve crosses the x-axis and where it changes its slope. Suitable for Grades 9-12.

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