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The points P 1, P 2, and P 3 (in blue) are collinear and belong to the graph of x 3 + 3 / 2 x 2 − 5 / 2 x + 5 / 4 . The points T 1, T 2, and T 3 (in red) are the intersections of the (dotted) tangent lines to the graph at these points with the graph itself. They are collinear too.
For solving the cubic equation x 3 + m 2 x = n where n > 0, Omar Khayyám constructed the parabola y = x 2 /m, the circle that has as a diameter the line segment [0, n/m 2] on the positive x-axis, and a vertical line through the point where the circle and the parabola intersect above the x-axis.
The 1st equal areas cubic is the locus of a point X such that area of the cevian triangle of X equals the area of the cevian triangle of X*. Also, this cubic is the locus of X for which X* is on the line S*X, where S is the Steiner point. (S = X(99) in the Encyclopedia of Triangle Centers).
Set of affine points of elliptic curve y 2 = x 3 − x over finite field F 89. The set of points E(F q) is a finite abelian group. It is always cyclic or the product of two cyclic groups. For example, [17] the curve defined by =
cubic graph special points: Image title: Graph showing the relationship between the roots, turning or stationary points and inflection point of a cubic polynomial and its first and second derivatives by CMG Lee. The vertical scale is compressed 1:50 relative to the horizontal scale for ease of viewing.
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
According to Brooks' theorem every connected cubic graph other than the complete graph K 4 has a vertex coloring with at most three colors. Therefore, every connected cubic graph other than K 4 has an independent set of at least n/3 vertices, where n is the number of vertices in the graph: for instance, the largest color class in a 3-coloring has at least this many vertices.
Solving for y leads to the explicit form =, which imply that every real point satisfies x ≥ 0. The exponent explains the term semicubical parabola. (A parabola can be described by the equation y = ax 2.) Solving the implicit equation for x yields a second explicit form