192 CHAPTER 4 Polynomial and Rational Functions Examples of power functions are f x x f x x f x x f x x 3 5 8 5 2 3 4 ( ) ( ) ( ) ( ) = = − = = − degree 1 degree 2 degree 3 degree 4 The graph of a power function of degree 1, f x ax, ( ) = is a line with slope a that passes through the origin. The graph of a power function of degree 2, f x ax ,2 ( ) = is a parabola with vertex at the origin. The parabola is concave up if a 0 > and is concave down if a 0. < If we know how to graph a power function of the form f x x ,n ( ) = a compression or stretch and, perhaps, a reflection about the x-axis will enable us to obtain the graph of g x ax .n ( ) = Consequently, we concentrate on graphing power functions of the form f x x .n ( ) = Power Functions of Even Degree We begin with power functions of even degree of the form f x x n , 2 n ( ) = ≥ and n even. We define the behavior of the graph of a function for large values of x, either positive or negative, as its end behavior. Exploration Using your graphing utility and the viewing window x y 2 2, 4 16, − ≤ ≤ − ≤ ≤ graph the function Y f x x . 1 4 ( ) = = On the same screen, graph Y g x x . 2 8 ( ) = = Now, also on the same screen, graph Y h x x . 3 12 ( ) = = What do you notice about the graphs as the magnitude of the exponent increases? Repeat this procedure for the viewing window x y 1 1, 0 1. − ≤ ≤ ≤ ≤ What do you notice? Result See Figures 2(a) and 2(b) using a TI-84 Plus CE. NOTE Don’t forget how graphing calculators express scientific notation. In Table 2, 1E–8 means 1 10 8 × − which is equivalent to 0.00000001 (a very small number). j Figure 2 Y x Y x Y x ; ; 1 4 2 8 3 12 = = = Y3 5 x12 Y2 5 x8 Y1 5 x4 22 24 16 2 (b) Y3 5 x12 Y2 5 x8 Y1 5 x4 21 0 1 1 (a) (b) The domain of f x x n , 2 n ( ) = ≥ and n even, is the set of all real numbers, and the range is the set of nonnegative real numbers. Such a power function is an even function (do you see why?), so its graph is symmetric with respect to the y-axis. Its graph always contains the origin 0, 0 ( ) and the points 1, 1 ( ) − and 1,1 . ( ) If n 2, = the graph is the parabola y x2 = that is concave up, with vertex at the origin. For large n, it appears that the graph coincides with the x-axis near the origin, but it does not; the graph actually touches the x-axis only at the origin. See Table 2, where Y x Y x , , 1 4 2 8 = = and Y x . 3 12 = For x close to 0, the values of y are positive and close to 0. Also, for large n, it may appear that for x 1 <− or for x 1 > the graph is vertical, but it is not; it is just increasing very rapidly in these intervals. That is, as x becomes unbounded in the negative direction, x , →−∞ or as x becomes unbounded in the positive direction, x , →∞ the power function f x x n , n ( ) = even, becomes unbounded in the positive direction; that is, f x . ( ) →∞ If you TRACE along one of the graphs, these distinctions will be clear. Table 2
RkJQdWJsaXNoZXIy NjM5ODQ=