112 CHAPTER 2 Functions and Their Graphs At this stage, if you were asked to graph any of the functions defined by yxyxyxy xy xy x y x , , , , , 1 , or , 2 3 3 = = = = = = = your response should be, “Yes, I recognize these functions and know the general shapes of their graphs.” (If this is not your answer, review the previous section, Figures 41 through 47.) Sometimes we are asked to graph a function that is “almost” like one that we already know how to graph. In this section, we develop techniques for graphing such functions. Collectively, these techniques are referred to as transformations . We introduce the method of transformations because it is a more efficient method of graphing than point plotting. 1 Graph Functions Using Vertical and Horizontal Shifts Figure 54 Vertical shift 26 22 6 6 Y2 5 x2 1 2 Y3 5 x2 2 2 Y1 5 x2 In Words For y f x k k , 0 ( ) = + > , add k to each y -coordinate on the graph of ( ) = y f x to shift the graph up k units. For y f x k k , 0 ( ) = − > , subtract k from each y -coordinate to shift the graph down k units. 2.5 Graphing Techniques: Transformations OBJECTIVES 1 Graph Functions Using Vertical and Horizontal Shifts (p. 112) 2 Graph Functions Using Compressions and Stretches (p. 115) 3 Graph Functions Using Reflections about the x -Axis or y -Axis (p. 117) Exploration On the same screen, graph each of the following functions: Y x Y x Y x , 2, 2 1 2 2 2 3 2 = = + = − What do you observe? Now create a table of values for Y Y , 1 2 , and Y3 . What do you observe? Result Figure 54 illustrates the graphs using a TI-84 Plus CE. You should have observed a general pattern. With Y x 1 2 = on the screen, the graph of Y x 2 2 2 = + is identical to that of Y x , 1 2 = except that it is shifted vertically up 2 units. The graph of Y x 2 3 2 = − is identical to that of Y x , 1 2 = except that it is shifted vertically down 2 units. From Table 8(a), we see that the y -coordinates on Y x 2 2 2 = + are 2 units larger than the y -coordinates on Y x 1 2 = for any given x -coordinate. From Table 8(b), we see that the y -coordinates on Y x 2 3 2 = − are 2 units smaller than the y -coordinates on Y x 1 2 = for any given x -coordinate. Notice a vertical shift only affects the range of a function, not the domain. For example, the range of Y1 is [ )∞0, while the range of Y2 is [ )∞2, . The domain of both functions is all real numbers. (b) (a) Table 8 We are led to the following conclusions: Vertical Shifts • If a positive real number k is added to the output of a function y f x , ( ) = the graph of the new function y f x k ( ) = + is the graph of f shifted vertically up k units. • If a positive real number k is subtracted from the output of a function y f x , ( ) = the graph of the new function y f x k ( ) = − is the graph of f shifted vertically down k units.
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