Power functions

Power functions

Power function is general function

The functions linear, parabola of second degree, parabola ofthird degree and inverse function which we have studied, are particular cases of a function of the form:

where a and n are any given constants. Function (13) is called in general a power function. We shall confine ourselves to drawing the curves for x positive and a = 1. Figures 22 and 23 show the graphs corresponding to various values of n.

note the position of origin firstly to understand the graphs correctly.

The equation y = x^n gives y = 1 at x = 1 for all values of n, i.e. all the curves pass through the point (1, 1). The curves rise for n positive and x > 1, the rate of rise increasing with the size of n (Fig. 22).

The function y = x^n is equivalent to a fraction for negative n (Fig. 23). Instead of y = x^-2, for instance, we can write y = 1/x^2. In these cases, the ordinate y diminishes with increase of x. The curves corresponding to equation (13) are sometimes referred to as polytropic. They are often encountered in thermodynamics.

It may be noted here, that we take the value of the radical as positive,for fractional n; for example, we take as positive x^1/2 = 

The two constants a and n appearing in equation (13) are defined, provided two points of the curve M1(x1, y2) and M2(x2, y2) are given, in which case we have:

we eliminate a by dividing one equation by the other:

Taking logarithms, we obtain n as

having found n, we obtain a from either of equations (14).  Figure 24 illustrates a graphical method of obtaining any required number of points of the curve (13), given two of its points M1(x1, y1) and M2(x2, y2).

We draw two arbitrary lines through the point O at angles  to axes X and Y respectively; we then take perpendiculars to the axes from the given points M1, M2 intersecting the arbitrary lines in points S1, S2, T1, T2 and intersecting the axes in points Q1, Q2, R1, R2. We now draw R2T3 through R2 parallel to R1T2, and S2Q3 through S2 parallel to S1Q2. Having finally drawn lines parallel to the corresponding axes through T3 and Q3, the intersection of these gives us the point M3(x3, y3) of the curve. Taking similar triangles, we have:

the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek