Polynomial End Behavior
Simple overview of the four end behaviors of polynomial functions.
Polynomial End Behavior Core Concept
First off, the end behavior of a polynomial is about what our function does as $x$ approaches $\pm \infty$.
There are four potential end behaviors and we can easily classify how a polynomial will behave by identifying the term with the highest power/degree in our polynomial (also referred to as leading term or “Boss Hog”).
For general form of a polynomial function is
\[f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\]then the term with the highest power/degree is
\[a_nx^n\]The power/degree of this term $n$ tells us whether our end behavior behaves the same or opposite to one another at either $\infty$.
The sign of coefficient $a_n$ locks in the direction for both sides of our function.
Classifying The Four End Behaviors
| Leading term | Left end | Right end | Graph language |
|---|---|---|---|
| $a_nx^n$, where $n$ is even and $a_n>0$ | $f(x)\to \infty$ as $x\to -\infty$ | $f(x)\to \infty$ as $x\to \infty$ | Both ends up |
| $a_nx^n$, where $n$ is even and $a_n<0$ | $f(x)\to -\infty$ as $x\to -\infty$ | $f(x)\to -\infty$ as $x\to \infty$ | Both ends down |
| $a_nx^n$, where $n$ is odd and $a_n>0$ | $f(x)\to -\infty$ as $x\to -\infty$ | $f(x)\to \infty$ as $x\to \infty$ | Left down, right up |
| $a_nx^n$, where $n$ is odd and $a_n<0$ | $f(x)\to \infty$ as $x\to -\infty$ | $f(x)\to -\infty$ as $x\to \infty$ | Left up, right down |
Example 1: Even Degree, Positive Leading Coefficient
Consider
\[f(x)=6x^4+5x^3-2x^2+x-20.\]The term with the highest power/degree is
\[6x^4\]Since the degree is even ($4$) and the leading coefficient is positive ($+6$), both ends of the graph go up (we have same end behavior as $g(x)=x^2$)
\[f(x)\to \infty \quad \text{as} \quad x\to -\infty\]and
\[f(x)\to \infty \quad \text{as} \quad x\to \infty\]Watch this part: end behavior of $6x^4+5x^3-2x^2+x-20$ at 0:20.
Example 2: Odd Degree, Negative Leading Coefficient
Consider
\[g(x)=-\frac{1}{2}x^5+5x^2-x+1.\]The term with the highest power/degree is
\[-\frac{1}{2}x^5\]Since the degree is odd, the ends go in opposite directions. Since the leading coefficient is negative, the right end goes down. That means the left end goes up
\[g(x)\to \infty \quad \text{as} \quad x\to -\infty\]and
\[g(x)\to -\infty \quad \text{as} \quad x\to \infty\]Watch this part: end behavior of $-0.5x^5+5x^2-x+1$ at 3:14.
Example 3: Put The Polynomial In Order First
Consider the following
\[h(x)=2x^3-x^6+2x^7+2.\]Before deciding the end behavior, put the terms in descending degree order
\[h(x)=2x^7-x^6+2x^3+2.\]Now it is easy to find the term term with the highest power/degree
\[2x^7\]The degree is odd and the leading coefficient is positive, so the graph goes down on the left and up on the right:
\[h(x)\to -\infty \quad \text{as} \quad x\to -\infty\]and
\[h(x)\to \infty \quad \text{as} \quad x\to \infty.\]Watch this part: end behavior of $2x^3-x^6+2x^7+2$ at 4:03.
Timestamp Guide
More Polynomial Videos
This lesson is part of the graphing-polynomials group on the JoeCMath channel. Use the full playlist when you want to move from end behavior into sign charts and graphing from those sign charts: