When trying to understand how a function behaves as it goes towards infinity, I like to break it down into simpler steps. Here’s how I usually do it:

1. Check the Degree: For polynomial functions, the highest power of $x$ is really important.

   • If the degree is even, the ends of the graph point in the same direction.
   • If it's odd, the ends point in opposite directions.

2. Leading Coefficient: The leading coefficient is the number in front of the highest power of $x$.

   • If it's positive, the function will go up towards infinity as $x$ increases.
   • If it's negative, the function goes down.

3. Rational Functions: These can be a bit more complicated, but you can still figure things out by comparing the degrees of the top part (numerator) and the bottom part (denominator):

   • If the degree on the top is less, the function will get closer to zero.
   • If they are the same, the function will approach the ratio of the leading coefficients.

4. Asymptotic Behavior: Remember to think about horizontal asymptotes! These show you the value the function gets closer to as $x$ goes off towards infinity.

By putting all these pieces together, you can understand where the graph is heading!