uppose that a function yequals​f(x) is increasing on the interval left bracket 8 comma 9 right bracket. Answer parts​ (a) through​ (d).
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Part 1
​(a) Over what interval is the graph of y equals f left parenthesis x plus 2 right parenthesis ​increasing?
The graph of y equals f left parenthesis x plus 2 right parenthesis is increasing over the interval 
  
left bracket 6 comma 7 right bracket. 
​(Type your answer in interval​ notation.)
Part 2
​(b) Over what interval is the graph of y equals f left parenthesis x minus 6 right parenthesis ​increasing?
The graph of y equals f left parenthesis x minus 6 right parenthesis is increasing over the interval 
  
enter your response here.
​(Type your answer in interval​ notation.)

Given that \( f(x) \) is increasing on the interval \([8, 9]\), we need to determine the intervals over which the graphs of \( f(x+2) \) and \( f(x-6) \) are increasing.

### Part 1: \( y = f(x + 2) \)

To find the interval where \( f(x + 2) \) is increasing, we set \( u = x + 2 \). Thus, if \( u \) ranges from 8 to 9, then:
\[
8 \leq u \leq 9
\]
Replacing \( u \) with \( x + 2 \):
\[
8 \leq x + 2 \leq 9
\]
Solving for \( x \):
\[
6 \leq x \leq 7
\]

Therefore, the graph of \( y = f(x + 2) \) is increasing over the interval \([6, 7]\).

### Part 2: \( y = f(x - 6) \)

To find the interval where \( f(x - 6) \) is increasing, we set \( v = x - 6 \). Thus, if \( v \) ranges from 8 to 9, then:
\[
8 \leq v \leq 9
\]
Replacing \( v \) with \( x - 6 \):
\[
8 \leq x - 6 \leq 9
\]
Solving for \( x \):
\[
14 \leq x \leq 15
\]

Therefore, the graph of \( y = f(x - 6) \) is increasing over the interval \([14, 15]\).

### Summary:
- The graph of \( y = f(x + 2) \) is increasing over the interval \([6, 7]\).
- The graph of \( y = f(x - 6) \) is increasing over the interval \([14, 15]\).

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

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

**Tip:** Remember, when dealing with function transformations, horizontal shifts involve changes inside the function argument \( x \), while vertical shifts affect the function directly \( f(x) \).

Explore how function transformations affect increasing intervals with a detailed explanation. Learn about shifting functions like y = f(x+2) and y = f(x-6) and determine their increasing intervals. Suitable for high school students.

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