End behavior and infinity notation
End Behavior and Infinity Notation
End Behavior and Infinity Notation is a topic in Functions in the California Common Core State Standards. It is aligned to Standard F-IF.7, which requires students to graph functions and describe their end behavior.
End behavior describes what a function does as \(x\to\pm\infty\), written with infinity notation; for a polynomial it is set by the leading term, and for a rational function by the degree comparison.
Theory
For a polynomial, only the leading term matters at the extremes. Its degree (even/odd) and the sign of its coefficient fix both ends:
- Even degree: both ends go the same way (up if the coefficient is positive, down if negative).
- Odd degree: the ends go opposite ways.
For a rational function, compare the degrees of numerator and denominator to find the horizontal asymptote that governs end behavior.
Horizontal asymptote of a rational function \(\dfrac{p(x)}{q(x)}\) by degree:
where \(a,b\) are the leading coefficients of \(p,q\).
How to determine end behavior
- Polynomial: find the leading term; use its degree (even/odd) and sign to state both ends.
- Rational: compare \(\deg p\) and \(\deg q\) using the rule above.
- Write it with infinity notation for each direction \(x\to+\infty\) and \(x\to-\infty\).
End behavior is set by the leading term \(-2x^3\) (odd degree, negative coefficient).
| \(x\to+\infty\) | \(\Rightarrow\) | \(f(x)\to-\infty\) |
| \(x\to-\infty\) | \(\Rightarrow\) | \(f(x)\to+\infty\) |
The graph falls on the right and rises on the left.
The leading term \(3x^4\) has even degree and positive coefficient, so both ends go the same way — upward.
| \(x\to\pm\infty\) | \(\Rightarrow\) | \(f(x)\to+\infty\) |
When numerator and denominator have the same degree, the horizontal asymptote is the ratio of leading coefficients.
| \(y\) | \(=\) | \(\dfrac{2}{1}=2\) |
So \(f(x)\to 2\) as \(x\to\pm\infty\).
The numerator degree (1) is less than the denominator degree (2), so the function tends to 0 at the ends.
| \(x\to\pm\infty\) | \(\Rightarrow\) | \(f(x)\to 0\) |
The horizontal asymptote is \(y=0\).
Common pitfalls
Frequently asked questions
What is end behavior?
How a function behaves as \(x\to+\infty\) and \(x\to-\infty\) — whether the outputs rise, fall, or level off at the far left and right.
How do you find the end behavior of a polynomial?
Look only at the leading term. Its degree (even or odd) and the sign of its coefficient determine what both ends do.
How do you find the horizontal asymptote of a rational function?
Compare degrees: smaller top gives \(y=0\); equal degrees give the ratio of leading coefficients; larger top gives no horizontal asymptote.
What does f(x) tends to infinity mean?
The output grows without bound as \(x\) moves toward some value or toward infinity. Infinity is a direction of unbounded growth, not a reachable number.