1. y = f(x) y = f(x + 3) 3. y = f(x) y = f(x 1) 5. y = 3f(x) 6. y = f(3x) 7. y = f(x) 8. y = f( x) 9. y = f(x 3) + 1

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1 .7 Transformations.7. Eercises To see all of the help resources associated with this section, click OSttS Chapter b. Suppose (, ) is on the graph of = f(). In Eercises - 8, use Theorem.7 to find a point on the graph of the given transformed function.. = f() +. = f( + ). = f(). = f( ). = f() 6. = f() 7. = f() 8. = f( ) 9. = f( ) + 0. = f( + ). = 0 f(). = f(). = f( ). = f( + ) +. = f( ) 7 6. = f 7. = f() f( ) 8. = 7 For help with Eercises 9 -, click on one or more of the resources below: Horizontal and vertical shift transformations Horizontal and vertical scaling transformations Reflections across the or ais Multiple transformations The complete graph of = f() is given below. In Eercises 9-7, use it and Theorem.7 to graph the given transformed function. (, ) (, ) (0, 0) The graph for E = f() + 0. = f(). = f( + ). = f( ). = f(). = f()

2 6 Relations and Functions. = f() 6. = f( ) 7. = f( ) 8. Some of the answers to Eercises 9-7 above should be the same. Which ones match up? What properties of the graph of = f() contribute to the duplication? The complete graph of = f() is given below. In Eercises 9-7, use it and Theorem.7 to graph the given transformed function. (0, ) (, 0) (, 0) (, ) The graph for E = f() 0. = f( + ). = f(). = f(). = f(). = f( ). = f( + ) 6. = f() 7. = f( + ) The complete graph of = f() is given below. In Eercises 8-9, use it and Theorem.7 to graph the given transformed function. (0, ) (, 0) (, 0) The graph for E g() = f() + 9. h() = f() 0. j() = f ( )

3 .7 Transformations 7. a() = f( + ). b() = f( + ). c() = f(). d() = f(). k() = f ( ) 6. m() = f() 7. n() = f( ) 6 8. p() = + f( ) 9. q() = f + The complete graph of = S() is given below. (, ) (, 0) (0, 0) (, ) (, 0) The graph of = S() The purpose of Eercises 0 - is to graph = S( + ) + b graphing each transformation, one step at a time. 0. = S () = S( + ). = S () = S ( ) = S( + ). = S () = S () = S( + ). = S () = S () + = S( + ) + Let f() =. Find a formula for a function g whose graph is obtained from f from the given sequence of transformations.. () shift right units; () shift down units. () shift down units; () shift right units 6. () reflect across the -ais; () shift up unit 7. () shift up unit; () reflect across the -ais 8. () shift left unit; () reflect across the -ais; () shift up units 9. () reflect across the -ais; () shift left unit; () shift up units 60. () shift left units; () vertical stretch b a factor of ; () shift down units 6. () shift left units; () shift down units; () vertical stretch b a factor of

4 8 Relations and Functions 6. () shift right units; () horizontal shrink b a factor of ; () shift up unit 6. () horizontal shrink b a factor of ; () shift right units; () shift up unit 6. The graph of = f() = is given below on the left and the graph of = g() is given on the right. Find a formula for g based on transformations of the graph of f. Check our answer b confirming that the points shown on the graph of g satisf the equation = g() = = g() 6. For man common functions, the properties of Algebra make a horizontal scaling the same as a vertical scaling b (possibl) a different factor. For eample, we stated earlier that 9 =. With the help of our classmates, find the equivalent vertical scaling produced b the horizontal scalings = (), =, = 7 and = ( ). What about = ( ), =, = 7 and = ( )? 66. We mentioned earlier in the section that, in general, the order in which transformations are applied matters, et in our first eample with two transformations the order did not matter. (You could perform the shift to the left followed b the shift down or ou could shift down and then left to achieve the same result.) With the help of our classmates, determine the situations in which order does matter and those in which it does not. 67. What happens if ou reflect an even function across the -ais? 68. What happens if ou reflect an odd function across the -ais? 69. What happens if ou reflect an even function across the -ais? 70. What happens if ou reflect an odd function across the -ais? 7. How would ou describe smmetr about the origin in terms of reflections? 7. As we saw in Eample.7., the viewing window on the graphing calculator affects how we see the transformations done to a graph. Using two different calculators, find viewing windows so that f() = on the one calculator looks like g() = on the other.

5 .7 Transformations 9 Checkpoint Quiz.7. The complete graph of = f() is given below. (0, ) (, ) (, ) (, 0) = f() Let g() = f. Sketch the graph of = g(). From our graph, determine the domain and range of g. List the intervals over which g is increasing and the intervals over which g is decreasing. List the local maimums and local minimums, if an.. Let f() =. Find a formula for a function g whose graph is obtained from the graph of = f() after the following sequence of transformations: Shift left units. Reflection across the -ais. Shift down unit. Vertical scaling b a factor of. Reflection across the -ais. For worked out solutions to this quiz, click the links below: Quiz Solution Part Quiz Solution Part

6 0 Relations and Functions.7. Answers. (, 0). (, ). (, ). (, ). (, 9) 6. (, ) 7. (, ) 8. (, ) 9. (, ) 0. (, 6). (, ). = (, 0). (, ). (, ). (, 7) 6. (, ) 7. (, ) 8. (, ) 9. = f() + 0. = f() (, ) (, ) (0, ) (, ) (, ) (0, ). = f( + ). = f( ) (, ) (, ) (0, ) (, ) (, 0) (, 0) 6. = f(). = f() (, ) (, ) (, ) (, ) (0, 0) (0, 0)

7 .7 Transformations. = f() 6. = f( ) (0, ) (, 0) (, 0) (0, ) (, ) (, 0) 6 7. = f( ) 9. = f() (, ) (0, ) (0, 0) (, 0) 6 (, ) (, ) (, ) 0. = f( + ). = f() (, ) (0, ) (, 0) (, 0) (, ) (, 0) (, 0) (, )

8 Relations and Functions. = f(). = f() (0, ) (, 0) (, 0) (, ) (, ) (, 0) (, 0) (0, ). = f( ). = f( + ) (0, ) (, ) (, 0) (, 0) (, ) (, ) (, ) (, ) 6. = f() 7. = f( + ) (, ) (, ) (, ) (, ) (0, ) (, ) (, ) (, )

9 .7 Transformations 8. g() = f() + (0, 6) 6 9. h() = f() 0, (, ) (, ), (, ) 0. j() = f ( ) (, ). a() = f( + ) (, ) ) ( ( 7, 0, 0 ) 7 6 ( 7, 0) (, 0). b() = f( + ) (, ). c() = f() 0, 9 (, ) (, ) (, 0) (, 0). d() = f() (, 0) (, 0). k() = f ( ) (0, ) ) ( ( 9, 0 9, 0) 6 (0, 6)

10 Relations and Functions 6. m() = f() (, 0) (, 0) 0, 7. n() = f( ) 6 (, 6) (0, 6) (6, 6) 8. p() = + f( ) = f( + ) + (, 7) 7 6 (, ) (, ) 9. q() = f + = f ( + ) ( 0, ), 9 (, )

11 .7 Transformations 0. = S () = S( + ). = S () = S ( ) = S( + ) (0, ) (0, ) (, 0) (, 0) (, 0) (, 0) (, 0) (, 0) (, ) (, ). = S () = S () = S( + ). = S () = S () + = S( + ) + 0, (, 0), (, 0) (, 0) (, ) 0, (, ) (, ),. g() =. g() = 6. g() = + 7. g() = ( + ) = 8. g() = g() = ( + ) + = g() = + 6. g() = ( + ) = g() = + 6. g() = ( ) + = g() = + or g() =