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Part 1
Suppose that a function yequals​f(x) is increasing on the interval left bracket 8 comma 9 right bracket .
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Part 1
​(a)  Over what interval is the graph of y equals f left parenthesis x plus 9 right parenthesis ​increasing?

If the function \( y = f(x) \) is increasing on the interval \([8, 9]\), we need to determine the interval where the function \( y = f(x + 9) \) is increasing.

To solve this, consider the transformation \( x + 9 \). This shifts the graph of the original function \( f(x) \) to the left by 9 units.

### Step-by-step process:
1. The original function \( f(x) \) is increasing on the interval \([8, 9]\).
2. The transformation \( f(x + 9) \) moves the graph to the left by 9 units.
3. To find the new interval where \( f(x + 9) \) is increasing, subtract 9 from both endpoints of the interval \([8, 9]\).

Thus:
\[
\text{New interval} = [8 - 9, 9 - 9] = [-1, 0]
\]

### Conclusion:
The function \( y = f(x + 9) \) is increasing on the interval \([-1, 0]\).

Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:
1. How does shifting a function horizontally affect its graph?
2. What happens if you shift a function vertically instead of horizontally?
3. What interval would \( f(x - 5) \) be increasing if \( f(x) \) is increasing on \([8, 9]\)?
4. How would the increasing behavior change if you had a negative shift, like \( f(x - 9) \)?
5. How do you determine whether a function is increasing or decreasing in general?

**Tip:** When shifting functions horizontally, remember that adding to \( x \) shifts the graph to the left, while subtracting from \( x \) shifts it to the right.

Suppose that a function y = f(x) is increasing on the interval [8, 9]. Over what interval is the graph of y = f(x + 9) increasing?

Learn how to determine the interval where the function y = f(x + 9) is increasing, given that y = f(x) is increasing on [8, 9]. Understand key concepts in function transformations, interval analysis, and increasing functions. Suitable for high school students in Grades 10-12.

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