How to Find End Behavior of a Function?

Behavior of any function can be understood by drawing its graph. Its behavior at the end of graph i.e. the values at which function approaches + ve infinity or – ve infinity (making asymptotes), can be checked for values in Domain. Suppose we have a polynomial function F (X), its end behavior can be explained by considering the values of F (x) i.e. the range of function as 'x' approaches + ve infinity or - ve infinity.

So, how to find end behavior of a function? We decide end behavior on the basis of degree of function and also the most prominent coefficient in the polynomial function.

Most prominent or noticeable coefficient in the polynomial is defined as that noteworthy figure when compared to additional coefficients in the function for having either a very huge or very insignificant value. The sign of coefficient is enough to guess the end behavior of the function. Let us consider an example to understand this concept:

Q. Judge the end behavior of the polynomial function given as follows: u⁴ – 4 u³ + 3 u + 25?
Solution: In the given polynomial function we can find the degree to be 4, which is an even figure and so leading coefficient will be of the term with power 4 only i.e. 1. We see that leading coefficient is also + ve. So, end behavior of polynomial function can be defined as:
F (u) → + ve infinity, as u → - ve infinity and
F (u) → + ve infinity, as u → + ve infinity
Graph of u⁴ – 4u³ + 3 u + 25 also shows the end behavior of function as: