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?
The graph of y equals f left parenthesis x plus 9 right parenthesis is increasing over the interval

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If the function \( y = f(x) \) is increasing on the interval \( [8, 9] \), the transformed function \( y = f(x + 9) \) will shift the graph 9 units to the left. 

This transformation affects the interval of increase as follows:

- The interval \( [8, 9] \) will shift 9 units to the left, resulting in \( [8 - 9, 9 - 9] = [-1, 0] \).

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

Would you like more details on this transformation or have any questions?

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Here are five related questions:
1. How does the transformation \( y = f(x - c) \) affect the graph of \( f(x) \)?
2. What happens to the interval of increase if we have \( y = f(x - 5) \)?
3. How do we determine the interval of increase for \( f(x) \) from its derivative?
4. How would the interval change for the transformation \( y = f(2x) \)?
5. Can a function be increasing on multiple disjoint intervals?

**Tip:** Horizontal shifts \( y = f(x + c) \) move the graph to the left if \( c > 0 \), and to the right if \( c < 0 \).

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?

Understand how the graph of a function y = f(x + 9) shifts and how to find the new interval of increase when the original function is increasing on [8, 9]. This step-by-step explanation covers function transformations and is suitable for Grades 10-12.

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