Asymptotes are interesting parts of graphs that help us see how functions act, especially when we look at very large or very small numbers (like infinity). You might wonder if asymptotes can change when we change a function's equation. The answer is yes! Let’s dive into how that works.

1. Types of Asymptotes:

   • Vertical Asymptotes: These happen when a function goes towards infinity as it gets close to a specific $x$-value. A good example is the function $f(x) = \frac{1}{x}$. It has a vertical asymptote at $x=0$.

   • Horizontal Asymptotes: These deal with what happens when $x$ gets very large. For instance, if we look at $g(x) = \frac{2x + 1}{x + 2}$, the horizontal asymptote is $y=2$. This means that as $x$ gets really big, the function gets closer to the value of 2.

2. Changing the Function:

   • When we change the function, the asymptotes can change too. For example, if we switch $f(x) = \frac{1}{x}$ to $f(x) = \frac{2}{x}$, the horizontal asymptote stays at $y=0$, but how the graph approaches it changes.

   • If we add a number, like in $f(x) = \frac{1}{x} + 1$, the horizontal asymptote now moves to $y=1$.

To sum up, every time we tweak the function's equation, we can get different asymptotic behaviors. This helps us create and study various kinds of graphs!