Graph of y = x³ - 3x
Interactive graph of y = x³ - 3x (cubic with turning points). Explore local maxima, local minima, and the S-shaped cubic curve.
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Understanding the Function
The function y = x³ - 3x is a cubic polynomial with two turning points, creating a characteristic S-shaped curve. Unlike the simple y = x³ which increases monotonically, subtracting 3x creates a local maximum and a local minimum, giving the curve a more complex shape that dips and rises.
Setting the derivative 3x² - 3 = 0 gives critical points at x = -1 and x = 1. Evaluating the function: f(-1) = 2 (local maximum) and f(1) = -2 (local minimum). The function factors as x(x² - 3) = x(x - sqrt(3))(x + sqrt(3)), giving three real roots at x = 0 and x = +/- sqrt(3).
Key properties: an odd function (f(-x) = -f(x)), three x-intercepts at 0, +/- sqrt(3) approximately +/- 1.732, local maximum of 2 at x = -1, local minimum of -2 at x = 1, inflection point at origin. The function demonstrates how the interplay of cubic and linear terms creates turning points in polynomial graphs.