360 CHAPTER 3 Polynomial and Rational Functions Compared to the graph of ƒ1x2 =axn, the following also hold true. • The graph of ƒ1x2 = −axn is reflected across the x-axis. • The graph of ƒ1x2 =axn +k is translated (shifted) up k units if k 70 and down k units if k 60. • The graph of ƒ1x2 =a1x −h2n is translated to the right h units if h 70 and to the left h units if h 60. • The graph of ƒ1x2 =a1x −h2n +k shows a combination of these translations. EXAMPLE 1 Examining Vertical and Horizontal Translations Graph each polynomial function. Determine the largest open intervals of the domain over which each function is increasing or decreasing. (a) ƒ1x2 = x5 - 2 (b) ƒ1x2 = 1x + 126 (c) ƒ1x2 = -21x - 123 + 3 SOLUTION (a) The graph of ƒ1x2 = x5 - 2 will be the same as that of ƒ1x2 = x5, but translated down 2 units. See Figure 22. This function is increasing on its entire domain 1-∞, ∞2. (b) In ƒ1x2 = 1x + 126, function ƒ has a graph like that of ƒ1x2 = x6, but because x + 1 = x - 1-12, it is translated to the left 1 unit. See Figure 23. This function is decreasing on 1-∞, -12 and increasing on 1-1, ∞2. x y f(x) = x5 – 2 0 2 4 6 –2 –2 2 Figure 22 x y f(x) = (x + 1)6 –2–1 0 1 2 2 4 6 Figure 23 x y f(x) = –2(x – 1)3 + 3 0 1 2 3 Figure 24 (c) The negative sign in -2 causes the graph of ƒ1x2 = -21x - 123 + 3 to be reflected across the x-axis when compared with the graph of ƒ1x2 = x3. Because -2 71, the graph is stretched vertically when compared to the graph of ƒ1x2 = x3. As shown in Figure 24, the graph is also translated to the right 1 unit and up 3 units. This function is decreasing on its entire domain 1-∞, ∞2. S Now Try Exercises 13, 15, and 19. Unless otherwise restricted, the domain of a polynomial function is the set of all real numbers. Polynomial functions are smooth, continuous curves on the interval 1-∞, ∞2. The range of a polynomial function of odd degree is also the set of all real numbers.
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