) Is the graph of y equals negative f left parenthesis x right parenthesis ​increasing, decreasing, or neither on the interval left bracket 8 comma 9 right bracket​?
decreasing
increasing
neither

Given that \( f(x) \) is increasing on the interval \([8, 9]\), we need to determine the behavior of the graph of \( y = -f(x) \) on the same interval.

### Analysis:
When a function \( f(x) \) is increasing, it means that as \( x \) increases, \( f(x) \) also increases. Mathematically, this is expressed as:
\[
\text{If } x_1 < x_2, \text{ then } f(x_1) < f(x_2).
\]

For the function \( y = -f(x) \), we take the negative of \( f(x) \). This operation will reverse the order of the values of \( f(x) \):
\[
\text{If } x_1 < x_2 \text{ and } f(x_1) < f(x_2), \text{ then } -f(x_1) > -f(x_2).
\]

This means that if \( f(x) \) is increasing on \([8, 9]\), then \( -f(x) \) will be decreasing on the same interval.

### Conclusion:
The graph of \( y = -f(x) \) is decreasing on the interval \([8, 9]\).

If you have any questions or need further details, please let me know!

Here are some more questions you might find interesting:
1. How does the graph of \( y = -f(x) \) compare to the graph of \( y = f(x) \)?
2. What happens to the increasing interval if we consider \( y = f(-x) \)?
3. Can you explain the transformation of the graph \( y = -f(x - c) \)?
4. What is the interval of \( y = 2f(x) \) if \( f(x) \) is increasing on \([1, 2]\)?
5. How does a horizontal shift affect the increasing interval of a function?

**Tip:** When dealing with transformations, multiplying the function by \(-1\) reflects the graph across the x-axis, reversing the direction of any increasing or decreasing behavior.

Explore how the graph of y = -f(x) behaves when f(x) is increasing on the interval [8, 9]. Detailed analysis includes the transformation of functions and the impact of multiplying by -1. Suitable for high school students and beyond.

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