 Suppose f of x goes to infinity as x goes to infinity. Then, on the graph of y equals f of x, remember, equals means replaceable, then y goes to infinity. But, as x goes to infinity, we move right on the graph of y equals f of x, and as y goes to infinity, we move upward on the graph of y equals f of x. So, if f of x goes to infinity as x goes to infinity, following the graph of y equals f of x will require us to move right and upward. Similarly, if f of x goes to negative infinity as x goes to infinity, following the graph requires moving right and downward. And the end behavior as x goes to minus infinity describes how to follow the graph to the left. For example, suppose that as x goes to infinity, f of x goes to infinity, and as x goes to minus infinity, f of x goes to infinity, we can use those to sketch the graph of y equals f of x. What's important to remember here is that what happens near the origin is irrelevant for end behavior. And so, with the given information, anything could happen near the origin, so we'll ignore what goes on there. As x goes to infinity, f of x goes to infinity, so following the graph requires moving right and upward, and as x goes to minus infinity, f of x goes to infinity, so following the graph requires moving left and upward. Or we could try to identify a graph like 27x cubed minus 5,000x squared plus 800x, and we can do that by checking end behavior. So, remember, only the highest power term matters for checking end behavior, and so that highest power term is 27x cubed. As x goes to infinity, x cubed goes to infinity, 27x cubed goes to infinity, and y goes to infinity, so following the graph requires moving right and upward, which means our graph can't be either of these two, and so we'll eliminate them from consideration. In the other direction, as x goes to minus infinity, x cubed goes to minus infinity, 27x cubed goes to minus infinity, and so y goes to minus infinity. So in this case, following the graph requires moving left and downward, and this means that it can't be this graph, and it must be this graph.