Function library
Cubic Function
f(x) = x^3 - 3*x
An S-shaped cubic with three real roots, a local hill and valley, and 180° rotational symmetry.
The cubic f(x) = x³ − 3x traces an elongated S: it rises from −∞, crests a small hill at (−1, 2), dips through the origin into a valley at (1, −2), and climbs away to +∞. Unlike a parabola it has no global maximum or minimum — the x³ term eventually overwhelms everything, dragging the left arm down forever and the right arm up forever. Between the extremes, the curve crosses the x-axis three times, at −√3, 0, and √3.
This particular cubic is a favorite in textbooks because everything about it can be computed by hand: its roots factor via x(x² − 3), its turning points come from the clean derivative 3x² − 3, and it hides a beautiful connection to triple-angle trigonometry. Cubics model volume growth, cubic equations of state, and any relationship where a quantity’s cube matters. Plot x^3 - 3*x in the calculator and zoom out to see the S straighten into its steep end behavior.
Computed properties
Calculated by the graphing calculator's own math engine at build time for x^3 - 3*x.
- Roots in [−10, 10]
- -1.7321, 0.0000, 1.7321
- y-intercept f(0)
- 0.0000
- Local extrema in [−10, 10]
-
- local maximum at (-1.0000, 2.0000)
- local minimum at (1.0000, -2.0000)
- Sample values
- f(-2) = -2 · f(-1) = 2 · f(0) = 0 · f(1) = -2 · f(2) = 2
- Derivative f′(1)
- 0.0000
- Integral ∫₀¹ f(x) dx
- -1.2500
Key facts
- Factored: x(x − √3)(x + √3); three real roots at x = −√3, 0, and √3.
- Local maximum at (−1, 2); local minimum at (1, −2); no global max or min.
- Inflection point at (0, 0); odd function with 180° rotational symmetry about the origin.
- Domain and range: all real numbers.
- End behavior: f(x) → −∞ as x → −∞ and f(x) → +∞ as x → +∞.
- Identity: with x = 2cos θ, x³ − 3x = 2cos(3θ).
What this cubic is
Factor out x and the roots appear: x³ − 3x = x(x² − 3) = x(x − √3)(x + √3), so the graph crosses the x-axis at −√3 ≈ −1.732, 0, and √3 ≈ 1.732. Three real roots is the most a cubic can display as distinct crossings — the function’s degree sets the maximum. Between consecutive roots the curve must turn around, which is exactly what the hill and valley do.
The end behavior is dictated by x³ alone: as x → +∞ the function → +∞, and as x → −∞ it → −∞. The −3x term only shapes the middle of the graph, carving out the wiggle. Every odd-degree polynomial shares this opposite-ends behavior, which guarantees at least one real root — the curve must cross the axis to get from −∞ to +∞.
Turning points and the inflection point
The derivative f′(x) = 3x² − 3 = 3(x − 1)(x + 1) vanishes at x = ±1, marking the two turning points: a local maximum at (−1, 2) and a local minimum at (1, −2). The function rises until x = −1, falls until x = 1, then rises forever — the classic up-down-up of a cubic with two critical points. These are only local extremes; the global behavior is unbounded both ways.
Halfway between them, at (0, 0), sits the inflection point, where the curve changes from concave-down to concave-up. The second derivative f″(x) = 6x confirms it: negative left of 0, positive right of 0, zero exactly at the origin. Because the cubic is an odd function, the inflection point is also the center of its 180° rotational symmetry — rotate the graph half a turn about (0, 0) and it maps onto itself.
A hidden trigonometric identity
Here is the surprise this cubic is famous for: substituting x = 2cos θ gives x³ − 3x = 2cos(3θ). You can verify it from the triple-angle formula cos(3θ) = 4cos³θ − 3cos θ: with x = 2cos θ, the left side becomes 8cos³θ − 6cos θ = 2(4cos³θ − 3cos θ) = 2cos(3θ). The cubic is secretly a tripled angle in disguise.
This identity is more than a curiosity — it is the key to solving cubic equations trigonometrically. A cubic with three real roots, like this one, can be solved by writing its roots as scaled cosines of suitable angles, a method going back to Viète. It also explains why the hill and valley have the exact heights ±2: they are 2cos(3θ) evaluated at its own peaks.
Frequently asked questions
Why does x³ − 3x cross the x-axis three times?
Its factored form x(x − √3)(x + √3) shows three distinct linear factors, each contributing one zero: x = 0, x = √3, and x = −√3. A degree-3 polynomial can have at most three real roots, and this one attains the maximum. Between each pair of roots the derivative’s turning points force the curve to reverse direction.
What are the local max and min?
Solve f′(x) = 3x² − 3 = 0 to get x = ±1. Then f(−1) = −1 + 3 = 2 is the local maximum and f(1) = 1 − 3 = −2 is the local minimum. They are “local” because the function exceeds any bound far to the right and falls below any bound far to the left.
What does this cubic have to do with trigonometry?
The identity x³ − 3x = 2cos(3θ) under the substitution x = 2cos θ links the cubic to triple-angle formulas. Historically this connection gave a trigonometric method for solving cubics with three real roots — the “casus irreducibilis” that puzzled 16th-century algebraists.