End Behavior Calculator
Describe the left and right end behavior of a polynomial.
Table of Contents
How to Use
- Enter the degree n of your polynomial (0 for constant, 1 for linear, etc.).
- Type the leading coefficient aₙ (the coefficient of xⁿ).
- Click Analyze to evaluate whether each tail rises or falls.
- Read the summary to interpret the limit direction, dominant term, and turning point information.
Key Ideas
The leading term aₙxⁿ dominates the graph of a polynomial for large |x|. Only the degree parity (even/odd) and the sign of aₙ matter when describing end behavior.
Lower-degree terms affect local wiggles but never change the ultimate trend because their growth rate is smaller than xⁿ.
Turning Points
- A polynomial of degree n has at most n − 1 turning points.
- Even-degree graphs either rise-rise or fall-fall.
- Odd-degree graphs always point in opposite directions at ±∞.
- Negative leading coefficients flip the graph vertically.
Frequently Asked Questions
- What if my polynomial is missing terms?
- Only the highest power term matters. Even if middle terms are zero, the degree and leading coefficient still determine the end behavior.
- How do I use this with factored forms?
- Expand just enough to identify the highest power of x and its coefficient. For example, (x − 3)(x + 2)^2 is degree 3 with positive leading coefficient, so it falls left and rises right.