AP®MathsCollege BoardPrecalculusStudy GuidesPolynomial & Rational FunctionsLinear, Quadratic & Polynomial FunctionsEnd Behavior of Polynomial Functions

End Behavior of Polynomial Functions (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 9 April 2026

End behavior of polynomial functions

What is end behavior?

The end behavior of a function describes
- what happens to the output values
- as the input values become very large (in the positive or negative direction)
For a polynomial function $p$ this describes what happens to $p (x)$
- as $x$ increases without bound ( $x \to \infty$ )
- or decreases without bound ( $x \to - \infty$ )
For a nonconstant polynomial function, the output values will always
- either increase without bound (go to $\infty$ )
- or decrease without bound (go to $- \infty$ )
- in each direction
Polynomial functions never "level off" to a horizontal asymptote
- they always increase or decrease without limit

How is end behavior expressed in limit notation?

End behavior is described using limit notation:
- $\lim_{x \to \infty} p (x) = \infty$ means "as $x$ increases without bound, $p (x)$ increases without bound"
- $\lim_{x \to \infty} p (x) = - \infty$ means "as $x$ increases without bound, $p (x)$ decreases without bound"
- $\lim_{x \to - \infty} p (x) = \infty$ means "as $x$ decreases without bound, $p (x)$ increases without bound"
- $\lim_{x \to - \infty} p (x) = - \infty$ means "as $x$ decreases without bound, $p (x)$ decreases without bound"
A complete description of a polynomial's end behavior includes both limits
- i.e. what happens as $x \to \infty$ and as $x \to - \infty$

Examiner Tips and Tricks

In limit notation, the ' $\lim$ ' stands for 'limit'. So, e.g., $\lim_{x \to \infty} p (x)$ may be read as 'the limit of $p (x)$ as $x$ goes to infinity'.

How do I determine the end behavior of a polynomial?

First, identify the leading term
- i.e. the term with the highest power of $x$
- Remember that the leading term might not be the first term written in the expression (see the Polynomial Functions study guide)
Then use the degree of the polynomial (odd or even) and the sign of the leading coefficient (positive or negative) to determine the end behavior

degree of polynomial	sign of leading coefficient	$\lim_{x \to - \infty} p (x)$	$\lim_{x \to \infty} p (x)$	description of graph
odd	positive	$- \infty$	$\infty$	Left end goes down, right end goes up
odd	negative	$\infty$	$- \infty$	Left end goes up, right end goes down
even	positive	$\infty$	$\infty$	Both ends go up
even	negative	$- \infty$	$- \infty$	Both ends go down

Four graphs of polynomial functions showing odd and even degrees with positive and negative leading coefficients, illustrating end behavior differences. — **Graphs showing end behavior of polynomial functions**

Examiner Tips and Tricks

Always check that you have identified the degree and sign of the leading coefficient correctly.

The leading term may not be the first term written in the expression

Why does the leading term determine end behavior?

As the input values increase or decrease without bound
- the values of the leading term dominate the values of all lower-degree terms
- This is because higher powers of $x$ grow much faster than lower powers
E.g. for $p (x) = 2 x^{4} - 50 x^{3} + 100 x$
- when $x$ is very large, the $2 x^{4}$ term is far larger in magnitude than the $- 50 x^{3}$ and $100 x$ terms
Therefore, the degree of the polynomial and the sign of the leading coefficient are all you need to determine the end behavior

How can I remember the four cases?

For odd degree polynomials, the two ends behave in opposite ways:
- Positive leading coefficient → falls left, rises right (think of $x^{3}$ )
- Negative leading coefficient → rises left, falls right (think of $- x^{3}$ )
For even degree polynomials, both ends behave the same way
- Positive leading coefficient → both ends go up (think of $x^{2}$ )
- Negative leading coefficient → both ends go down (think of $- x^{2}$ )

For the negative coefficient cases
- note that multiplying by a negative number flips the end behavior compared to the positive coefficient case

Examiner Tips and Tricks

When an exam question asks you to describe the end behavior of a polynomial, always express your answer using limit notation (e.g. $\lim_{x \to \infty} p (x) = - \infty$ ). To earn all the points in a free-response question, you need to include all four components of a correct limit statement:

the " $\lim$ " symbol
the direction ( $x \to \infty$ or $x \to - \infty$ )
the function, e.g. $p (x)$
and the result ( $\infty$ or $- \infty$ )

Worked Example

The function $f$ is given by $f (x) = - 3 x^{7} + 4 x^{2} - 2$ . Which of the following describes the end behavior of $f$ ?

(A) $\lim_{x \to - \infty} f (x) = - \infty$ and $\lim_{x \to \infty} f (x) = - \infty$

(B) $\lim_{x \to - \infty} f (x) = \infty$ and $\lim_{x \to \infty} f (x) = \infty$

(D) $\lim_{x \to - \infty} f (x) = \infty$ and $\lim_{x \to \infty} f (x) = - \infty$

Answer:

Consider the leading term (i.e. the term with the highest power of $x$ )

This is $- 3 x^{7}$
The leading term will determine the end behavior of the function

First consider $x^{7}$ , which is an odd power of $x$

When $x$ is positive, $x^{7}$ is positive
- and as $x$ increases in the positive direction, $x^{7}$ increases in the positive direction
When $x$ is negative, $x^{7}$ is negative
- and as $x$ decreases in the negative direction (i.e. becomes more and more negative), $x^{7}$ decreases in the negative direction

Then consider the effect of the coefficient, -3

This will make $- 3 x^{7}$ negative whenever $x^{7}$ is positive
and will make $- 3 x^{7}$ positive whenever $x^{7}$ is negative

Combining these means that

as $x$ increases in the positive direction, $- 3 x^{7}$ decreases in the negative direction (i.e. becomes more and more negative)
and as $x$ decreases in the negative direction, $- 3 x^{7}$ increases in the positive direction

In limit notation, this end behavior is written as

(D) $\lim_{x \to - \infty} f (x) = \infty$ and $\lim_{x \to \infty} f (x) = - \infty$

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End Behavior of Polynomial Functions (College Board AP® Precalculus): Study Guide

End behavior of polynomial functions

What is end behavior?

How is end behavior expressed in limit notation?

Examiner Tips and Tricks

How do I determine the end behavior of a polynomial?

Examiner Tips and Tricks

Why does the leading term determine end behavior?

How can I remember the four cases?

Examiner Tips and Tricks

Worked Example

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis