SECTION 2.5 Graphing Techniques: Transformations 119 The previous examples demonstrated how to graph a given function using transformations. In the next example, we build a function based on transformations. It is helpful to consult the “Functional Change to f x( )” column in the summary table above. To Graph: Draw the Graph of f and: Functional Change to ( ) f x Compressing or stretching ( ) = > y af x a , 0 Multiply each y-coordinate of ( ) = y f x by a. Multiply ( ) f x by a. Stretch the graph of f vertically if > a 1. Compress the graph of f vertically if < < a 0 1. ( ) = > y f ax a , 0 Multiply each x-coordinate of ( ) = y f x by a 1 . Replace x by ax. Stretch the graph of f horizontally if < < a 0 1. Compress the graph of f horizontally if > a 1. Reflection about the x-axis ( ) = − y f x Reflect the graph of f about the x-axis. Multiply ( ) f x by −1. Reflection about the y-axis ( ) = − y f x Reflect the graph of f about the y-axis. Replace x by −x. Determining the Function Obtained from a Series of Transformations Find the function that is finally graphed after the following three transformations are applied one after the other to the graph of ( ) = f x x . 1. Shift left 2 units. Followed by . . . 2. Shift up 3 units. Followed by . . . 3. Reflect about the x-axis. Solution EXAMPLE 4 1. Shift left 2 units: Replace x by +x 2. ( ) ( ) = = + = + y g x f x x 2 2 2. Shift up 3 units: Add 3. ( ) ( ) = = + = + + y G x g x x 3 2 3 3. Reflect about the x-axis: Replace G x( ) by G x . ( ) − ( ) ( ) [ ] = =− =−++=−+− y H x G x x x 2 3 2 3 Now Work PROBLEM 33 Using Graphing Techniques Graph the function f x x 3 2 1 ( ) = − + . Find the domain and range of f . Solution EXAMPLE 5 It is helpful to write f as f x x 3 1 2 1 ( ) = ⋅ − + . Now use the following steps to obtain the graph of f . Step 1 y x 1 = Reciprocal function Step 2 y x 1 2 = − Replace x by x 2; − horizontal shift to the right 2 units. Step 3 y x x 3 1 2 3 2 = ⋅ − = − Multiply by 3; vertical stretch by a factor of 3. Step 4 y x 3 2 1 = − + Add 1; vertical shift up 1 unit. (continued) Hint Although the order in which transformations are performed can be altered, consider using the following order for consistency: 1. Horizontal shift 2. Reflections 3. Compressions and stretches 4. Vertical shift j
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