How-to · Updated October 2, 2026
How does the leading coefficient test work?
Direct answer
x³ is odd with a positive leading coefficient, so the left end is -∞ and the right end is +∞. x² is even and positive, so both ends are +∞.
The four patterns
Even degree, both ends match. Odd degree, the ends oppose each other. A negative leading coefficient reverses the positive case.
| Leading term | Degree | As x → −∞ | As x → +∞ |
|---|---|---|---|
| x² | 2 | +∞ | +∞ |
| −x² | 2 | -∞ | -∞ |
| x³ | 3 | -∞ | +∞ |
| −x³ | 3 | +∞ | -∞ |
As x → -∞, f(x) → -∞ and as x → +∞, f(x) → +∞. As x → -∞, f(x) → -∞ and as x → +∞, f(x) → -∞.
Limits
- This test describes the ends of a polynomial. It does not locate the turning points in the middle of the graph.
- Write the leading term first, or read its sign yourself. A string that hides the leading minus sign will be read as positive.
- Rational functions use a degree comparison, not this four-row table alone.
Use the calculator
When the function is a fraction: end behavior of a rational function. Enter a polynomial in the end behavior calculator.