Understanding of end behavior of functions

End behavior definition:
With respect to polynomial functions, the end behavior of the function refers to the direction and position of the two ends of the function. For example the graph of the function y = x^2, would be as follows:

Note that both the ends of the graph are going upwards. However if we see the graph of the function y = x^3, it would look as follows:

In the above graph, both the ends of the graph point in different directions.

End behavior of polynomial functions:
Polynomial functions have primarily 4 types of end-behaviors. They are as follows:
(a) Both ends going up.

(b) Both ends going down.

(c) Left end going up and right end going down.

(d) Right end going up and left end going down.

It is possible to know the end behavior of a polynomial function without actually graphing it. We use the end behavior model for that. The following are the end behavior rules that help us model the end behavior of polynomial function.

(1) If the degree of the polynomial function is even, then both the ends would point the same direction. Therefore for an even degree polynomial function, both ends would either go up or both the ends would go down.

(2) If the degree of the polynomial function is odd, then both the ends would point in opposite direction. Therefore for an odd degree polynomial function there are two possibilities, either left ends goes up and right end goes down or left end goes down and right end goes up.

(3) If the coefficient of the leading term is positive, then the right end of the polynomial function would always point up. Therefore if the polynomial is of even degree with positive leading coefficient, then both ends would point up, and if the polynomial is odd with positive leading coefficient, then as right end points upward, the left end would point down wards.

(4) If the coefficient of the leading term is negative, then the right end of the polynomial function would always go down. Therefore if it is an even degree polynomial with negative leading coefficient, then both the ends would go down whereas, if it is an odd degree polynomial with negative leading coefficient, then as right end goes down, left end would go up.

End behavior asymptotes:
Polynomial functions do not have asymptotes. However, if it is a rational function, then it would have asymptotes.