Polynomial End Behavior
This page follows the JoeCMath video Master the 4 End Behaviors of Polynomials in 6 Minutes. The video keeps the attention on what the graph does far to the left and far to the right before worrying about every turn in the middle.
Main Idea
For end behavior, the leading term is the part of the polynomial that wins far away from the origin.
If a polynomial is written as
\[f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\]then the leading term is
\[a_nx^n.\]The degree $n$ tells you whether the ends go in the same direction or opposite directions. The leading coefficient $a_n$ tells you whether the right end goes up or down.
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 |
The right end tells you the sign of the leading coefficient, and the left end tells you whether the degree is even or odd.
Example 1: Even Degree, Positive Leading Coefficient
The first worked example in the video is
\[f(x)=6x^4+5x^3-2x^2+x-20.\]The leading term is
\[6x^4.\]Since the degree is even and the leading coefficient is positive, both ends of the graph go up:
\[f(x)\to \infty \quad \text{as} \quad x\to -\infty\]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
The next polynomial is
\[g(x)=-\frac{1}{2}x^5+5x^2-x+1.\]The leading term 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
The video also uses
\[h(x)=2x^3-x^6+2x^7+2.\]Before deciding the end behavior, put the terms in descending degree order:
\[h(x)=2x^7-x^6+2x^3+2.\]Now the leading term is clear:
\[2x^7.\]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.
Working Backwards From A Graph
The last section of the video starts with the graph shape and works backward. The same two questions still drive the decision:
- Do the ends go the same direction or opposite directions?
- Does the right end go up or down?
If both ends go the same direction, the degree is even. If the ends go opposite directions, the degree is odd. If the right end goes up, the leading coefficient is positive. If the right end goes down, the leading coefficient is negative.
That is why a graph with the left end down and the right end up must come from an odd-degree polynomial with a positive leading coefficient.
Watch this part: starting with the graph and working backwards at 4:47.
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: