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Explain your answer and reasoning. I believe the 'h' component is different ( X horizontal translation )

The problem asks whether the transformations that occur for \( y = f(5(x+1)) \) and \( y = f(5(x+2)) \) are the same. The key transformations here involve the horizontal translations and how they affect the function. 

### Let's break it down:

1. **General form of a function transformation**:
   \[
   y = f(a(x - h)) + k
   \]
   where:
   - \( a \) affects the horizontal stretch/compression.
   - \( h \) represents the horizontal shift (translation).
   - \( k \) represents the vertical shift (translation).

2. **For \( y = f(5(x+1)) \)**:
   - This can be rewritten as \( y = f(5(x - (-1))) \), indicating a **horizontal shift** to the left by 1 unit (since \( h = -1 \)).
   - The factor of 5 inside the function also affects the **horizontal compression** by a factor of \( \frac{1}{5} \).

3. **For \( y = f(5(x+2)) \)**:
   - This can be rewritten as \( y = f(5(x - (-2))) \), indicating a **horizontal shift** to the left by 2 units (since \( h = -2 \)).
   - Again, the factor of 5 results in a **horizontal compression** by a factor of \( \frac{1}{5} \).

### Are the transformations the same?
- **Horizontal Compression**: Both functions have the same compression by a factor of 5.
- **Horizontal Translation**: The two functions have different horizontal shifts. The first function, \( y = f(5(x+1)) \), is shifted left by 1 unit, while the second function, \( y = f(5(x+2)) \), is shifted left by 2 units.

### Conclusion:
The transformations are **not the same** because the horizontal translations differ. While both functions undergo the same horizontal compression (due to the factor of 5), the horizontal shifts are different: one moves left by 1 unit, and the other by 2 units.

Let me know if you'd like further details or clarifications!

Here are some related questions:
1. What is the effect of the horizontal compression on the graph of a function?
2. How do vertical translations affect the function \( y = f(x) \)?
3. What happens to the graph of \( f(x) \) when you replace \( x \) with \( x-h \)?
4. How would the transformations change if the factor inside the function was \( \frac{1}{5} \) instead of 5?
5. How do multiple transformations, like translation and stretching, combine when applied to a function?

**Tip**: Always rewrite transformations clearly in terms of the standard form \( a(x-h) \) to identify shifts and compressions/stretching correctly.

Are the transformations that occur to y=f(x) the same for y=f(5(x+1)) and y=f(5(x+2))?

Understand the differences between the transformations of y = f(5(x+1)) and y = f(5(x+2)), focusing on horizontal translation and compression. Explore how function transformations affect graph behavior, including horizontal shifts and compression. Suitable for Grades 10-12.

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