Some further properties are discussed below. For conveniance we assume c=1.

It is easy to find the second point for which the tangent is parallel to the previous one. This happens for x = e = (a+2*r)/3. The two corresponding points A, B on the graph have parallel tangents and the middle M of the line AB coincides with the symmetry center of the cubic, thus locates its inflexion point, where the second derivative is zero, at g = (e+a)/2 = (2*a+r)/3.

Let C be the other intersection point of the qubic with the tangent at B. (r,0) and the projections of the four points A, B, M and C on the x-axis are equidistant, the common distance being (a-r)/3.

CubicReduced.html

CubicSymmetry.html

Produced with EucliDraw© |