In asymptotic analysis, $n$ and $-n$ do not have the same growth rate.

1. Growth Rate Definition: The growth rate of a function typically refers to how the function behaves as $n$ approaches infinity. For $n$, it grows positively and linearly as $n$ increases.

2. Negative Growth: In contrast, $-n$ decreases linearly as $n$ increases. Although they are both linear functions, their behaviors are opposite; one increases without bound while the other decreases without bound.

3. Formal Analysis: If we analyze them in Big O notation:

   • $n$ is $O(n)$
   • $-n$ is $O(n)$ as well, but since it tends towards negative infinity, it can be misleading in terms of "growth."

4. Limit Comparison: The limit comparison of the two as $n$ approaches infinity: $\lim_{n \to \infty} \frac{-n}{n} = -1$ This indicates that while they have the same rate of increase in magnitude, they are fundamentally different in direction.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. What is the significance of growth rates in algorithm analysis?
2. How do we compare the growth rates of different functions like $n^2$ and $n$?
3. Can you explain the concept of Big O notation further?
4. What are some common functions used in complexity analysis?
5. How do we analyze growth rates in practical applications, like sorting algorithms?

Tip: Always consider the context of growth rates—positive growth can be critical in performance analysis while negative growth is usually less relevant.