118 CHAPTER 2 Functions and Their Graphs SUMMARY OF GRAPHING TECHNIQUES Each graphing technique has a different effect on the graph of a function. Compressions and stretches change the proportions of a graph, and reflections change the orientation, but not its proportions. Vertical and horizontal shifts change the location of the graph, without changing its proportions or orientation. To Graph: Draw the Graph of f and: Functional Change to ( ) f x Vertical shifts ( ) = + > y f x k k , 0 Raise the graph of f by k units. Add k to ( ) f x . ( ) = − > y f x k k , 0 Lower the graph of f by k units. Subtract k from ( ) f x . Horizontal shifts ( ) = + > y f x h h , 0 Shift the graph of f to the left h units. Replace x by +x h. ( ) = − > y f x h h , 0 Shift the graph of f to the right h units. Replace x by −x h. In Words For ( ) =− y f x , multiply each y -coordinate on the graph of ( ) = y f x by −1. For ( ) = − y f x , multiply each x -coordinate by −1. Exploration Reflection about the y -axis: (a) Graph Y x , 1 = followed by = − Y x. 2 (b) Graph Y x 1, 1 = + followed by Y x 1. 2 =− + (c) Graph Y x x, 1 4 = + followed by Y x x x x. 2 4 4 ( ) ( ) = − + − = − Result See Tables 11(a), (b), and (c) and Figures 64(a), (b), and (c). For each point x y , ( ) on the graph of Y 1 , the point x y , ( ) − is on the graph of Y2 . Put another way, Y2 is the reflection of Y1 about the y -axis. (b) (c) (a) Table 11 (a) Figure 64 Reflection about the y -axis Y2 5 √2x Y1 5 √x 28 25 5 8 (c) Y1 5 x4 1x Y 1 5 x4 2x 23 22 2 3 (b) Y2 5 2x 11 Y1 5 x 11 28 25 5 8 Based on the previous Exploration, we have the following result: Reflection about the y -Axis When the graph of the function f is known, the graph of the new function y f x ( ) = − is the reflection about the y-axis of the graph of the function f.
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