Parabola of the third degree

Parabola of the third degree

The third-degree polynomial:

Simple case

This equation has a graph in the form of a curve called a parabola of the third degree. We shall consider the simplest case of this curve:

For a positive, the signs of x and y are the same; for a negative, they are different. In the first case, the curve lies in the first and third quadrants of the coordinate axes, and in the second case, it lies in the second and fourth quadrants. Figure 17 illustrates the form of the curve for various values of a.

If x and y change simultaneously to (-x) and (-y), both sides of equation (8) change sign, and the equation is not essentially altered, i.e. if the point (x, y) lies on the curve (8), the point (-x, -y) also lies on this curve. The points (x, y) and (-x, -y) are evidently symmetrical relative to the origin O, i.e. the line joining them is bisected at O. It follows from this that every chord of the curve (8) that passes through the origin of coordinates O is bisected there. In other words: the origin of coordinates O is the center of the curve (8).

Another case

A further particular case of a third-degree parabola will be mentioned:

The right-hand side of this equation is the sum of two terms, and consequently, to construct the curve, it is sufficient to draw the straight line:

and take the sums of corresponding ordinates (y value) of graphs (8) and (10) directly from the figure. Figure 18 illustrates the various forms that the curve (9) assumes (with a = 1, and various c).

If the curve

is drawn, we obtain a convenient, though not too accurate, method for solving graphically an equation of the third degree:

the roots of this equation being, in fact, the abscissa of the points of intersection of the curve y = x^3 with the line

As can be seen from the figure (Fig.19), there may be one, two, or three points of intersection, but there must be at least one, i.e. an equation of the third degree must have at least one real root.

the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek