Section 6: Horizontal Geometric Transformations

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

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Section 2.6 Horizontal Transformations 191 Version: Fall 2007 2.6 Horizontal Transformations In the previous section, we introduced the concept of transformations. We made a change to the basic equation y = f ( x ) , such as y = af ( x ) , y = f ( x ) , y = f ( x ) c , or y = f ( x ) + c , then studied how these changes affected the shape of the graph of y = f ( x ) . In that section, we concentrated strictly on transformations that applied in th vertical direction. In this section, we will study transformations that will affect the shape of the graph in the horizontal direction. 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 10 y 10 f x f ( x ) ( x, f ( x )) 4 2 0 2 4 (a) The graph of f . (b) The table. Figure 1. Reading key values from the graph of f . To compute f ( 2) , for example, we would first locate 2 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 ( 2) . That is, f ( 2) = 4 . 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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192 Chapter 2 Functions Version: Fall 2007 x 10 y 10 f 2 4 x f ( x ) ( x, f ( x )) 4 2 4 ( 2 , 4) 0 2 4 x f ( x ) ( x, f ( x )) 4 0 ( 4 , 0) 2 4 ( 2 , 4) 0 0 (0 , 0) 2 2 (2 , 2) 4 0 (4 , 0) (a) The graph of f . (b) Recording f ( 2) = 4 . (c) Completed table. Figure 2. Recording coordinates of points on the graph of f in the tables. Horizontal 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 10 y 10 f x f ( x ) ( x, f ( x )) 4 0 ( 4 , 0) 2 4 ( 2 , 4) 0 0 (0 , 0) 2 2 (2 , 2) 4 0 (4 , 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 horizontal direction. l⚏ Example 2. If y = f ( x ) has the graph shown in Figure 3 (a), sketch the graph of y = f (2 x ) . In the previous section, we investigated the graph of y = 2 f ( x ) . The number 2 was outside the function notation and as a result we stretched the graph of y = f ( x ) vertically by a factor of 2. However, note that the 2 is now inside the function notation
Section 2.6 Horizontal Transformations 193 Version: Fall 2007 y = f (2 x ) . Intuition would demand that this might have something to do with scaling in the x -direction (horizontal direction), but how?

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