To understand how functions behave at the ends of their graphs, we need to look at what happens when the input values (the $x$ values) get really big or really small. Here are some helpful tips:

1. Check the degree and leading coefficient: For polynomial functions, the degree is the highest power of $x$. It tells you how the ends of the graph will go.

   • If the degree is odd, one end goes up and the other goes down.
   • If the degree is even, both ends go the same way.

2. Look at rational functions: These are fractions where one polynomial is on top (the numerator) and another is on the bottom (the denominator).

   • If the numerator’s degree is higher, the graph goes up towards infinity.
   • If both degrees are the same, you can find a horizontal line that the graph gets close to, called a horizontal asymptote.

3. Watch the graph closely: When you draw the graph, pay attention to what happens as $x$ gets really big (towards $+\infty$) or really small (towards $-\infty$). The direction it goes will show you the end behavior.

By practicing these steps, you’ll get better at predicting how functions act at their ends!