Technical Mathematics, Sixth Edition

17.4. The Power Function

17.4.1. Definition

Earlier in this chapter we saw that we could represent the statement "y varies directly as x" by Eq. 49,

y = kx

Similarly, if y varies directly as the square of x, we have

y = kx²

or if y varies directly as the square root of x,

These are all examples of the power function.

NOTE

The constants can, of course, be represented by any letter. Appendix A shows a instead of k.

The constants k and n can be any positive or negative number. This simple function gives us a great variety of relations that are useful in technology and whose forms depend on the value of the exponent, n. A few of these are shown in the following table:

When:	We Get This Function:		Whose Graph Is a:
n = 1	Linear function	y = kx (direct variation)	Straight line
n = 2	Quadratic function	y = kx2	Parabola
n = 3	Cubic function	y = kx3	Cubical parabola
n = −1			Hyperbola

For exponents of 4 and 5, we have the quartic and quintic functions, respectively.

17.4.2. Graph of the Power Function

The graph of a power function varies greatly, depending on the exponent. We first show some power functions whose exponents are positive integers.

Screen for Example 21: y = x², shown heavy, and y = x