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The cube-connected cycles of order n is the Cayley graph of a group that acts on binary words of length n by rotation and flipping bits of the word. [1] The generators used to form this Cayley graph from the group are the group elements that act by rotating the word one position left, rotating it one position right, or flipping its first bit.
Firstly, if a < 0, the change of variable x → –x allows supposing a > 0. After this change of variable, the new graph is the mirror image of the previous one, with respect of the y-axis. Then, the change of variable x = x 1 – b / 3a provides a function of the form = + +.
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.
Graphs of y = b x for various bases b: base 10, base e, base 2, base 1 / 2 . Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x = 1, the value of y equals the base because any number raised to the power of 1 is the number itself.
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.
Singular cubic y 2 = x 2 ⋅ (x + 1). A parametrization is given by t ↦ (t 2 – 1, t ⋅ (t 2 – 1)). A cubic curve may have a singular point, in which case it has a parametrization in terms of a projective line. Otherwise a non-singular cubic curve is known to have nine points of inflection, over an algebraically closed field such as the ...
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.
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