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.