SECTION 2.5 Graphing Techniques: Transformations 117 3 Graph Functions Using Reflections about the x -Axis or y -Axis Figure 60 y f x( ) = ( ( ( ( ( ( x y y 5 f(x) 1 2 21 2p 3p p p , 1 2 5p , 1 2 3p , 21 2 p 2 3p 2 5p Figure 61 y f x 2 ( ) = ; Vertical stretch by a factor of 2 ( ( ( ( ( ( x y 3p 1 2 3 21 22 2 p p 2 p , 2 2 5p , 2 2 3p, 22 2 p 2 3p 2 5p Figure 62 y f x3( ) = ; Horizontal compression by a factor of 1 3 ( ( ( ( ( ( x y 3p 2p 1 2 21 p 3 p 2p 3 6 p , 1 6 5p , 1 2 p , 21 Now Work PROBLEMS 65(e) AND (g) Based on the previous Exploration, we have the following result: Reflection about the x -Axis When a function f is multiplied by 1, − the graph of the new function y f x( ) = − is the reflection about the x-axis of the graph of the function f. Now Work PROBLEM 51 Exploration Reflection about the x -axis: (a) Graph and create a table of Y x 1 2 = and Y x . 2 2 =− (b) Graph and create a table of = Y x 1 and =− Y x . 2 (c) Graph and create a table of Y x 4 1 2 = − and Y x x 4 4. 2 2 2 ( ) =− − =− + Result See Tables 10(a), (b), and (c) and Figures 63(a), (b), and (c). For each point x y , ( ) on the graph of Y1 , the point x y , ( ) − is on the graph of Y2 . Put another way, Y2 is the reflection of Y1 about the x -axis. Table 10 (a) (c) (c) 28 25 5 8 Y2 5 2x2 1 4 Y1 5 x2 2 4 (b) 5 25 28 8 Y2 5 2|x| Y1 5 |x| (a) Figure 63 Reflection about the x -axis 28 25 5 8 Y2 5 2x2 Y1 5 x2 (b)
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