Section 5: Vertical Geometric Transformations

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 2.5 Vertical Transformations 167 Version: Fall 2007 2.5 Vertical Transformations In this section we study the art of transformations: scalings, reflections, and transla- tions. We will restrict our attention to transformations in the vertical or y -direction. Our goal is to apply certain transformations to the equation of a function, then ask what effect it has on the graph of the function. We begin our task with an example that requires that we read the graph of a function to capture several key points that lie on the graph of the function. l⚏ Example 1. Consider the graph of f presented in Figure 1 (a). Use the graph of f to complete the table in Figure 1 (b). x 5 y 5 f x f ( x ) ( x, f ( x )) 2 1 0 1 2 (a) The graph of f . (b) The table. Figure 1. Reading key values from the graph of f . To compute f ( 1) , we would locate 1 on the x -axis, draw a vertical arrow to the graph of f , then a horizontal arrow to the y -axis, as shown in Figure 2 (a). The y -value of this final destination is the value of f ( 1) . That is, f ( 1) = 2 . This allows us to complete one entry in the table, as shown in Figure 2 (b). Continue in this manner to complete all of the entries in the table. The result is shown in Figure 2 (c). Copyrighted material. See: 1
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168 Chapter 2 Functions Version: Fall 2007 x 5 y 5 f 1 2 x f ( x ) ( x, f ( x )) 2 1 2 ( 1 , 2) 0 1 2 x f ( x ) ( x, f ( x )) 2 0 ( 2 , 0) 1 2 ( 1 , 2) 0 0 (0 , 0) 1 2 (1 , 2) 2 0 (2 , 0) (a) The graph of f . (b) Recording f ( 1) = 2 . (c) Completed table. Figure 2. Recording coordinates of points on the graph of f in the tables. Vertical Scaling In the narrative that follows, we will have repeated need of the graph in Figure 2 (a) and the table in Figure 2 (c). They characterize the basic function that will be the starting point for the concepts of scaling, reflection, and translation that we develop in this section. Consequently, let’s place them side-by-side for emphasis in Figure 3 . x 5 y 5 f x f ( x ) ( x, f ( x )) 2 0 ( 2 , 0) 1 2 ( 1 , 2) 0 0 (0 , 0) 1 2 (1 , 2) 2 0 (2 , 0) (a) (b) Figure 3. The original graph of f and a table of key points on the graph of f . We are now going to scale the graph of f in the vertical direction. l⚏ Example 2. If y = f ( x ) has the graph shown in Figure 3 (a), sketch the graph of y = 2 f ( x ) . What do we do when we meet a graph whose shape we are unsure of? The answer to this question is we plot some points that satisfy the equation in order to get an
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Section 2.5 Vertical Transformations 169 Version: Fall 2007 idea of the shape of the graph. With that thought in mind, let’s evaluate the function y = 2 f ( x ) at x = 2 . The letter f refers to the original function shown in Figure 3 (a) and the table in Figure 3 (b) contains the values of that function at the given values of x . Thus, in computing y = 2 f ( 2) , the first step is to look up the value of f ( 2) in the table in Figure 3 (b). There we find that f ( 2) = 0 . Thus, we can write y = 2 f ( 2) = 2(0) = 0 .
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