When we look at rational functions in math, understanding asymptotes is very important. They help us draw the graph and see what the function does.

There are three main types of asymptotes you should know about:

1. Vertical Asymptotes
2. Horizontal Asymptotes
3. Oblique (or slant) Asymptotes

Let’s go through each type to make it easier to understand.

Vertical Asymptotes

Vertical asymptotes show up when the function goes really high (to infinity) or really low (to negative infinity) as the input (or x-value) gets close to a certain value.

This usually happens when the bottom part (denominator) of the rational function equals zero, but the top part (numerator) doesn’t also equal zero at the same time.

To find vertical asymptotes, just set the denominator equal to zero and solve for that x-value.

Example:
For the function $f(x) = \frac{1}{x - 3}$, if we set the denominator $x - 3 = 0$, we find $x = 3$. Thus, there is a vertical asymptote at $x = 3$.

Horizontal Asymptotes

Horizontal asymptotes help us see what happens with the function when x gets really big (positive or negative). They give us a sense of the end behavior of the graph.

To find horizontal asymptotes, we compare the degree of the top and bottom parts of the function:

1. If the degree of the top is less than the bottom, the horizontal asymptote is $y = 0$.
2. If they are equal, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading numbers (coefficients) of the numerator and denominator.
3. If the degree of the top is greater than the bottom, there’s no horizontal asymptote (but there could be an oblique asymptote, which we will talk about next).

Example:
For $f(x) = \frac{2x^2 + 3}{x^2 + 5}$, the degrees are equal. So, the horizontal asymptote is $y = \frac{2}{1} = 2$.

Oblique Asymptotes

Oblique asymptotes happen when the degree of the top part is one greater than the degree of the bottom part. To find these, we use polynomial long division.

The result of this division (ignoring any leftovers) will show us the equation of the oblique asymptote.

Example:
For $f(x) = \frac{x^3 + 2x^2 + 3}{x^2 + 1}$, if we do long division, we might end up with something like $y = x + 2$, which shows us an oblique asymptote.

By keeping track of these different types of asymptotes, you will better understand how rational functions work. This will help you draw their graphs with a lot more confidence!