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Characteristics of polynomial graphs

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Characteristics of Polynomial Graphs

A polynomial's graph has predictable features: zeros where it crosses the x-axis, turning points where it changes direction, and end behavior controlled by the leading term. Learn to identify all three, with a worked cubic example showing three zeros and two turning points.

What to look for in a polynomial graph

A polynomial's graph has predictable features you can read directly from its equation: its zeros (where it crosses the x-axis), its turning points (local highs and lows), and its end behavior (what the graph does far to the left and far to the right).

Characteristics of a polynomial graph: y = x³ − 3x The cubic y = x^3 - 3x crosses the x-axis at x = -root3, 0, and root3. It has a local maximum at (-1, 2) and a local minimum at (1, -2). As x goes to negative infinity y falls; as x goes to positive infinity y rises. local max (−1, 2) local min (1, −2) falls left rises right −√3 √3
y = x³ − 3x: three real zeros, a local max, a local min, and end behavior that falls left and rises right.

The zeros are the x-values where the graph crosses or touches the x-axis — the roots of the polynomial. For y = x³ − 3x = x(x² − 3), the zeros are x = 0, x = √3, and x = −√3, exactly where the curve crosses.

Turning points are local maximums and minimums, where the graph changes from rising to falling or falling to rising. A polynomial of degree n has at most n − 1 turning points. This cubic has two: a local max at (−1, 2) and a local min at (1, −2), matching the maximum of 3−1 = 2 turning points a cubic can have.

End behavior describes what happens as x moves toward positive or negative infinity, and it's controlled entirely by the leading term. For this cubic (odd degree, positive leading coefficient), the graph falls to the left and rises to the right — the same pattern as any odd-degree polynomial with a positive leading coefficient. Even-degree polynomials instead have both ends pointing the same direction.

The number of zeros (counted with multiplicity) and turning points both hint at the polynomial's degree: more turns and more crossings generally mean a higher-degree polynomial, using the polynomial function of a given degree as a reference.

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