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.