Parabola of the second degree

Parabola of the second degree

The linear function

is a particular case of an integral function of the n-th degree or a polynomial of the n-th degree:

the simplest case of which, after the linear function, is the polynomial of the second degree (n = 2):

graph the parabola

the graph of this function is called a parabola of the second degree or simply a parabola.

For the present, we shall consider only the simplest case of a parabola:

This curve can easily be plotted. Figure 13 shows the curves y =x^2 (a = 1) and y = -x^2 (a = -1).

The curve corresponding to equation (5) is situated wholly above the axis OX for a > 0, and wholly below the axis OX for a < 0. The ordinate of this curve increases in absolute value when x increases in absolute value, the increase being the faster, the greater the absolute value of a .Figure 14 shows a series of graphs of the function (5) for different values of a, these values being marked in the figure against the corresponding parabolas.

Equation (5) contains only x^2, and hence does not vary on changing x to -x, i.e. if a given point (x, y) lies on the parabola (5), the point (-x, y) also lies on it. The two points (x, y) and (-x, y) are evidently symmetrical relative to the axis OY, i.e. one of them is the mirror image of the other relative to this axis. Thus, if the right-hand portion of the plane is turned through 180° about the axis OY and combined with the left-hand portion, the part of the parabola lying to the right of the axis OY will coincide with the part of the parabola lying to the left of this axis. In other words, axis OY is an axis of symmetry of the parabola (5). The origin of coordinates is the lowest point of the curve for a > 0, and the highest point for a < 0, and is called the vertex of the parabola. The coefficient a is fully defined if one point M0 (x0, yQ) of the parabola is given, not at the vertex, since we have then:

the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek