SECTION 4.1 Polynomial Functions 193 To summarize: Properties of Power Functions, ( ) = f x x ,n n Is a Positive Even Integer • f is an even function, so its graph is symmetric with respect to the y -axis. • The domain is the set of all real numbers.The range is the set of nonnegative real numbers. • The graph always contains the points 1, 1 , ( ) − 0, 0 , ( ) and 1,1 . ( ) • As the exponent n increases in magnitude, the graph is steeper when x 1 < − or x 1; > but for x near the origin, the graph tends to flatten out and lie closer to the x -axis. • End behavior: As x x f x x or , . n ( ) →−∞ →∞ = →∞ Power Functions of Odd Degree Now we consider power functions of odd degree of the form f x x ,n ( ) = n odd . Exploration Using your graphing utility and the viewing window x y 2 2, 16 16, − ≤ ≤ − ≤ ≤ graph the function Y f x x . 1 3 ( ) = = On the same screen, graph Y g x x 2 7 ( ) = = and Y h x x . 3 11 ( ) = = 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, 1 1. − ≤ ≤ − ≤ ≤ What do you notice? Result See Figures 3(a) and 3(b) using a TI-84 Plus CE. The domain and the range of f x x n , 3 n ( ) = ≥ and n odd, are the set of real numbers. Such a power function is an odd function (do you see why?), so its graph is symmetric with respect to the origin. Its graph always contains the origin 0, 0 ( ) and the points 1, 1 ( ) − − and 1,1 . ( ) It appears that the graph coincides with the x -axis near the origin, but it does not; the graph actually crosses the x -axis only at the origin. Also, it appears that as x increases 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 , →−∞ the power function f x x n , n ( ) = odd, becomes unbounded in the negative direction; that is f x . ( ) →−∞ But as x becomes unbounded in the positive direction, x , →∞ the power function becomes unbounded in the positive direction; that is f x . ( ) →∞ This describes the end behavior of the graph of f. TRACE along the graphs to verify these distinctions. Figure 3 Y x Y x Y x ; ; 1 3 2 7 3 11 = = = Y3 5 x11 Y 2 5 x7 Y1 5 x3 22 216 16 2 Y2 5 x7 Y 3 5 x11 Y1 5 x3 21 21 1 1 (b) (a) Properties of Power Functions, f x x , n = ( ) n Is a Positive Odd Integer • f is an odd function, so its graph is symmetric with respect to the origin. • The domain and the range are the set of all real numbers. • The graph always contains the points 1, 1 , ( ) − − 0, 0 , ( ) and 1,1 . ( ) • As the exponent n increases in magnitude, the graph is steeper when x 1 < − or x 1; > but for x near the origin, the graph tends to flatten out and lie closer to the x -axis. • End behavior: As x f x x , . n ( ) →−∞ = →−∞ As x f x x , . n ( ) →∞ = →∞ To summarize:
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