# Polynomial End Behavior

Polynomial end behavior is the direction the graph of a polynomial function goes as the input value goes "to infinity" on the left and right sides of the graph. There are four possibilities, as shown below.

## Basic rules

With end behavior, the only term that matters with the polynomial is the one that has an exponent of largest degree. For example, if you have the polynomial $5x^4 + 12x^2 - 3x ,$ only the $5x^4$ matters in terms of end behavior.

This term will be of the form $ax^n .$

A) When $a$ is positive and $n$ is an even number, the left and right sides of the graph both go to +infinity.

B) When $a$ is negative and $n$ is an even number, the left and right sides of the graph both go to -infinity.

C) When $a$ is positive and $n$ is an odd number, the left side goes to -infinity and the right side goes to +infinity.

D) When $a$ is negative and $n$ is an odd number, the left side goes to +infinity and the right side goes to -infinity.

## What is the end behavior of $f(x) = -55x^4 - 3x^3 + 2x - 1 ?$

The largest exponent is 4, so the relevant term is $-55x^4 .$ $-55$ is negative and $4$ is an even number so the end behavior matches that of B above.

**Cite as:**Polynomial End Behavior.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/polynomial-end-behavior/