Polynomial Function Grapher

Enter Polynomial p(x)

Polynomial expression

Use ^ for powers or Unicode superscripts (x², x³, x⁴). Coefficients can be decimals.

Examples

Analysis

Enter a polynomial above and press Graph Polynomial to see the graph, zeros, turning points, and end behavior.

End Behavior

Real Zeros

Turning Points (estimated)

End Behavior Rules

Even degree, positive leading coefficient: Both ends point up (→ +∞ as x → ±∞).

Even degree, negative leading coefficient: Both ends point down (→ −∞ as x → ±∞).

Odd degree, positive leading coefficient: Left end down, right end up (→ −∞ as x → −∞, → +∞ as x → +∞).

Odd degree, negative leading coefficient: Left end up, right end down (→ +∞ as x → −∞, → −∞ as x → +∞).

Only the degree and the sign of the leading coefficient determine end behavior — all other terms become insignificant for large |x|.

Turning Points

degree n → at most n − 1 turning points

A turning point is a local maximum or local minimum — a place where the graph changes from increasing to decreasing or vice versa.

Key facts about turning points:

Between any two consecutive real zeros there is at least one turning point.
A degree n polynomial has at most n − 1 turning points.
Local max: graph rises to the left, falls to the right of the point.
Local min: graph falls to the left, rises to the right of the point.
Turning points occur where the derivative p′(x) = 0 (calculus); estimated here by sampling.