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
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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