End Behavior of Functions
End behavior describes what f(x) does as x → +∞ and x → −∞. Master the Leading Term Test for polynomials, asymptotes for rational functions, and the behavior of exponential and logarithmic functions.
Quick Reference — End Behavior Rules
Polynomials — Leading Term Test
- Even degree, positive coeff → ↑ ↑ (both tails up)
- Even degree, negative coeff → ↓ ↓ (both tails down)
- Odd degree, positive coeff → ↓ ↑ (down-left, up-right)
- Odd degree, negative coeff → ↑ ↓ (up-left, down-right)
Rational Functions — Degree Test
- num deg < den deg → y = 0 (HA)
- num deg = den deg → y = a/b (HA)
- num deg = den deg + 1 → slant asymptote
- num deg ≥ den deg + 2 → no HA, polynomial growth
What End Behavior Means
End behavior answers two questions about a function's graph:
Left End
As x → −∞
What does f(x) approach as x decreases without bound — as you move left on the graph forever?
Right End
As x → +∞
What does f(x) approach as x increases without bound — as you move right on the graph forever?
Arrow notation
End behavior is written using limit-style arrow notation. The four possible outcomes for f(x) are:
Polynomial End Behavior — The Leading Term Test
For any polynomial, end behavior is determined entirely by the leading term (the term with the highest power). All other terms become negligible as x → ±∞. The four cases depend on whether the degree is even or odd, and whether the leading coefficient is positive or negative.
| Degree | Leading Coeff. | As x → −∞ | As x → +∞ | Arrows | Example |
|---|---|---|---|---|---|
| Even | Positive (+) | f(x) → +∞ | f(x) → +∞ | ↑ both ends | f(x) = x², 2x⁴ |
| Even | Negative (−) | f(x) → −∞ | f(x) → −∞ | ↓ both ends | f(x) = −x², −3x⁴ |
| Odd | Positive (+) | f(x) → −∞ | f(x) → +∞ | ↓ left, ↑ right | f(x) = x³, 5x⁵ |
| Odd | Negative (−) | f(x) → +∞ | f(x) → −∞ | ↑ left, ↓ right | f(x) = −x³, −2x⁵ |
Memory anchor
Think of the four parent functions: y = x² (even, positive → ↑ ↑), y = −x² (even, negative → ↓ ↓), y = x³ (odd, positive → ↓ ↑), y = −x³ (odd, negative → ↑ ↓). Every other polynomial is a scaled or shifted version of one of these four cases.
Worked Polynomial Examples
For each function below, identify the degree, the sign of the leading coefficient, and state the end behavior in arrow notation.
f(x) = x⁴ − 3x² + 1
Degree: 4 (even)
Leading coeff: +1 (positive)
As x → −∞, f(x) → +∞ and as x → +∞, f(x) → +∞
Both tails rise. The −3x² and +1 terms are irrelevant for end behavior.
f(x) = −2x⁶ + 5x³ − 7
Degree: 6 (even)
Leading coeff: −2 (negative)
As x → −∞, f(x) → −∞ and as x → +∞, f(x) → −∞
Even degree, negative leading coefficient — both tails fall.
f(x) = x³ + 4x − 2
Degree: 3 (odd)
Leading coeff: +1 (positive)
As x → −∞, f(x) → −∞ and as x → +∞, f(x) → +∞
Odd degree, positive leading coefficient — left tail down, right tail up.
f(x) = −x⁵ + 3x²
Degree: 5 (odd)
Leading coeff: −1 (negative)
As x → −∞, f(x) → +∞ and as x → +∞, f(x) → −∞
Odd degree, negative leading coefficient — left tail up, right tail down.
f(x) = 4x² − 9
Degree: 2 (even)
Leading coeff: +4 (positive)
As x → −∞, f(x) → +∞ and as x → +∞, f(x) → +∞
Classic parabola opening upward. End behavior same as y = x².
f(x) = −3x⁷ + 100x⁶ − x + 2
Degree: 7 (odd)
Leading coeff: −3 (negative)
As x → −∞, f(x) → +∞ and as x → +∞, f(x) → −∞
Despite the large +100x⁶ term, only −3x⁷ controls end behavior.
Rational Function End Behavior
A rational function is a ratio of two polynomials: f(x) = p(x)/q(x). End behavior depends on the relationship between the degrees of the numerator and denominator. Compare them first — this tells you which of four cases applies before you do any calculation.
| Degree Comparison | Result | Description | Example |
|---|---|---|---|
| deg(numerator) < deg(denominator) | Horizontal asymptote: y = 0 | The denominator grows faster; the fraction shrinks to zero. | f(x) = 3x / (x² + 1) → y = 0 |
| deg(numerator) = deg(denominator) | Horizontal asymptote: y = a/b | Where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator. | f(x) = (3x² − 2) / (2x² + 5) → y = 3/2 |
| deg(numerator) = deg(denominator) + 1 | Oblique (slant) asymptote | Use polynomial long division. The quotient (ignoring remainder) is the slant asymptote equation. | f(x) = (x² + 1) / (x − 1) → slant asymptote y = x + 1 |
| deg(numerator) ≥ deg(denominator) + 2 | No horizontal asymptote | The function behaves like a polynomial; it grows without bound. | f(x) = x³ / (x + 1) → behaves like x² as x → ±∞ |
Worked Rational Function Examples
f(x) = (2x + 3) / (x² − 1)
Numerator degree: 1·Denominator degree: 2
Degree of numerator (1) < degree of denominator (2).
Horizontal asymptote: y = 0
As x → ±∞, f(x) → 0
f(x) = (5x² − 2x) / (2x² + 7)
Numerator degree: 2·Denominator degree: 2
Equal degrees. Leading coefficients: 5 (numerator) and 2 (denominator).
Horizontal asymptote: y = 5/2
As x → ±∞, f(x) → 5/2
f(x) = (x² + 3x − 2) / (x − 1)
Numerator degree: 2·Denominator degree: 1
Numerator degree (2) = denominator degree (1) + 1 → oblique asymptote. Long division: x² + 3x − 2 ÷ (x − 1) = x + 4, remainder 2. The remainder is ignored for the asymptote.
Oblique asymptote: y = x + 4
As x → ±∞, f(x) → x + 4 (the curve approaches the line)
Exponential and Logarithmic End Behavior
Exponential — Growth (a > 1)
f(x) = aˣ, a > 1 (e.g., 2ˣ, eˣ)
The graph rises steeply to the right and hugs the x-axis from above on the left. The x-axis (y = 0) is a horizontal asymptote.
Exponential — Decay (0 < a < 1)
f(x) = aˣ, 0 < a < 1 (e.g., (½)ˣ)
This is the mirror image of the growth case. The graph falls toward y = 0 on the right and rises steeply on the left.
Natural Log
f(x) = ln(x), domain x > 0
There is no left end (x must be positive). The y-axis (x = 0) is a vertical asymptote. The function grows without bound but very slowly.
Log Base a (0 < a < 1)
f(x) = log_a(x), 0 < a < 1
The mirror of ln(x). When the base is between 0 and 1, the log function is decreasing. Example: log₍½₎(x).
Absolute Value and Special Cases
Absolute Value
f(x) = |x|
As x → +∞: f(x) → +∞
As x → −∞: f(x) → +∞
Both ends rise. |x| = x for x ≥ 0 and |x| = −x for x < 0, so as x → −∞, −x → +∞. Same ↑ ↑ end behavior as an even-degree polynomial.
Square Root
f(x) = √x, domain x ≥ 0
As x → +∞: f(x) → +∞ (slowly)
No left end (x must be non-negative)
Only has right-end behavior. Grows without bound but much more slowly than any polynomial — it grows like x^(1/2).
Sine / Cosine
f(x) = sin(x), cos(x)
As x → ±∞: f(x) oscillates between −1 and 1
No end behavior limit exists — the limit at ±∞ is undefined (DNE). The function oscillates forever and never settles toward a single value.
Constant Function
f(x) = c
As x → ±∞: f(x) → c
End behavior equals the constant. The graph is a horizontal line — horizontal asymptote and the function value are the same thing here.
End Behavior Summary — All Function Types
Use this reference table to quickly identify the end behavior of any common function type using arrow notation.
| Function Type | As x → −∞ (left) | As x → +∞ (right) | Symbol |
|---|---|---|---|
| Polynomial (even deg, + coeff) | f(x) → +∞ | f(x) → +∞ | ↑ ↑ |
| Polynomial (even deg, − coeff) | f(x) → −∞ | f(x) → −∞ | ↓ ↓ |
| Polynomial (odd deg, + coeff) | f(x) → −∞ | f(x) → +∞ | ↓ ↑ |
| Polynomial (odd deg, − coeff) | f(x) → +∞ | f(x) → −∞ | ↑ ↓ |
| Rational (num deg < den deg) | f(x) → 0 | f(x) → 0 | HA y = 0 |
| Rational (num deg = den deg) | f(x) → a/b | f(x) → a/b | HA y = a/b |
| Rational (num deg = den + 1) | f(x) → slant | f(x) → slant | Oblique |
| Exponential aˣ (a > 1) | f(x) → 0 | f(x) → +∞ | HA y = 0 left |
| Exponential aˣ (0 < a < 1) | f(x) → +∞ | f(x) → 0 | HA y = 0 right |
| Logarithm ln(x) | f(x) → −∞ (x → 0⁺) | f(x) → +∞ | No HA |
| Absolute value |x| | f(x) → +∞ | f(x) → +∞ | ↑ ↑ |
Exam Tips
Ignore all terms but the leading term
For polynomials, only the highest-degree term matters for end behavior. f(x) = −7x⁵ + 300x⁴ − 1000x behaves exactly like −7x⁵ as x → ±∞. The other terms are irrelevant.
Even degree = same direction both ends
If the degree is even, both tails point the same way — both up (positive leading coeff) or both down (negative leading coeff). If the degree is odd, the tails point opposite directions.
For rational functions, always check degrees first
Before doing any algebra on a rational function, count the degrees. This tells you immediately whether you'll get y = 0, a ratio of leading coefficients, a slant asymptote, or polynomial growth. Only do long division when num degree = den degree + 1.
Frequently Asked Questions
What is end behavior of a function?
End behavior describes what happens to the output f(x) as x grows without bound in either direction — as x approaches positive infinity (+∞) and as x approaches negative infinity (−∞). For polynomials, end behavior is determined entirely by the degree and the sign of the leading coefficient. For rational functions, compare the degrees of the numerator and denominator to find horizontal or oblique asymptotes. Arrow notation summarizes end behavior: for example, as x → +∞, f(x) → +∞ means the right tail of the graph rises forever.
How do you determine end behavior of a polynomial?
Use the Leading Term Test: look only at the term with the highest power (the leading term). Even degree with positive leading coefficient: both tails go up (↑ ↑). Even degree with negative leading coefficient: both tails go down (↓ ↓). Odd degree with positive leading coefficient: left tail down, right tail up (↓ ↑). Odd degree with negative leading coefficient: left tail up, right tail down (↑ ↓). All other terms are irrelevant for end behavior. Example: f(x) = −3x⁵ + 7x² − 2 has odd degree and negative leading coefficient, so as x → −∞ f(x) → +∞ and as x → +∞ f(x) → −∞.
What is the end behavior of a rational function?
Compare the degree of the numerator (n) to the degree of the denominator (d). If n < d: horizontal asymptote at y = 0 (the function approaches zero in both directions). If n = d: horizontal asymptote at y = (leading coefficient of numerator) / (leading coefficient of denominator). If n = d + 1: oblique (slant) asymptote — use polynomial long division to find it. If n > d + 1: no horizontal asymptote; the function grows without bound like a polynomial. Example: f(x) = (3x² + 1) / (2x² − 5) has equal degrees, so horizontal asymptote y = 3/2.
Related Topics
Practice End Behavior Problems
Work through polynomial, rational, exponential, and logarithmic end behavior with step-by-step solutions. Free to start.
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