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Section2.5 Transformations Transformations Translations - PowerPoint PPT Presentation

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Section2.5 Transformations Transformations Translations Horizontal Translations: Vertical Translations: The graph of f ( x c ) is f ( x ) shifted c units to the right. f ( x ) f ( x c ) Translations Horizontal Translations: Vertical

  1. Section2.5 Transformations

  2. Transformations

  3. Translations Horizontal Translations: Vertical Translations: The graph of f ( x − c ) is f ( x ) shifted c units to the right. f ( x ) f ( x − c )

  4. Translations Horizontal Translations: Vertical Translations: The graph of f ( x − c ) is f ( x ) shifted c units to the right. f ( x ) f ( x − c ) The graph of f ( x + c ) is f ( x ) shifted c units to the left. f ( x + c ) f ( x )

  5. Translations Horizontal Translations: Vertical Translations: The graph of f ( x − c ) is f ( x ) The graph of f ( x ) + c is f ( x ) shifted c units to the right. shifted c units up. f ( x ) + c f ( x ) f ( x − c ) f ( x ) The graph of f ( x + c ) is f ( x ) shifted c units to the left. f ( x + c ) f ( x )

  6. Translations Horizontal Translations: Vertical Translations: The graph of f ( x − c ) is f ( x ) The graph of f ( x ) + c is f ( x ) shifted c units to the right. shifted c units up. f ( x ) + c f ( x ) f ( x − c ) f ( x ) The graph of f ( x + c ) is f ( x ) The graph of f ( x ) − c is f ( x ) shifted c units to the left. shifted c units down. f ( x ) f ( x + c ) f ( x ) f ( x ) − c

  7. Examples 1. Graph y = x 2 and y = ( x +3) 2 .

  8. Examples 1. Graph y = x 2 and y = ( x +3) 2 . 8 6 4 2 − 4 − 2 2 4 8 6 4 2 − 6 − 4 − 2 2 4

  9. Examples 2. Graph y = √ x and y = √ x − 2 1. Graph y = x 2 and y = ( x +3) 2 . 8 6 4 2 − 4 − 2 2 4 8 6 4 2 − 6 − 4 − 2 2 4

  10. Examples 2. Graph y = √ x and y = √ x − 2 1. Graph y = x 2 and y = ( x +3) 2 . 3 8 2 . 5 6 2 4 1 . 5 1 2 0 . 5 − 4 − 2 2 4 − 4 − 2 2 4 6 8 8 1 0 . 5 6 − 4 − 2 2 4 6 8 4 − 0 . 5 2 − 1 − 1 . 5 − 6 − 4 − 2 2 4 − 2

  11. Stretching/Shrinking Horizontal Stretch/Shrink: Vertical Stretch/Shrink: If c is positive, f ( cx ) is f ( x ) stretched or shrunk by a factor of 1 c from/towards the y -axis. f ( x ) f ( cx ) f ( cx ) f ( x )

  12. Stretching/Shrinking Horizontal Stretch/Shrink: Vertical Stretch/Shrink: If c is positive, f ( cx ) is f ( x ) If c is positive, cf ( x ) is f ( x ) stretched or shrunk by a factor stretched or shrunk by a factor of 1 c from/towards the y -axis. of c from/towards the x -axis. f ( x ) f ( x ) f ( cx ) cf ( x ) cf ( x ) f ( cx ) f ( x ) f ( x )

  13. Examples 1. Graph y = x 3 and y = ( 1 2 x ) 3 .

  14. Examples 1. Graph y = x 3 and y = ( 1 2 x ) 3 . 2 − 4 − 2 2 4 − 2 2 − 4 − 2 2 4 − 2

  15. Examples 1. Graph y = x 3 and y = ( 1 2. Graph y = x 3 and y = 2 x 3 . 2 x ) 3 . 2 − 4 − 2 2 4 − 2 2 − 4 − 2 2 4 − 2

  16. Examples 1. Graph y = x 3 and y = ( 1 2. Graph y = x 3 and y = 2 x 3 . 2 x ) 3 . 2 2 − 4 − 2 2 4 − 4 − 2 2 4 − 2 − 2 6 4 2 2 − 4 − 2 2 4 − 4 − 2 2 4 − 2 − 4 − 2 − 6

  17. Reflection Horizontal Reflection: Vertical Reflection: f ( − x ) is f ( x ) reflected across the y -axis.

  18. Reflection Horizontal Reflection: Vertical Reflection: f ( − x ) is f ( x ) reflected across − f ( x ) is f ( x ) reflected across the y -axis. the x -axis.

  19. Examples Graph y = √ x and y = √− x and y = −√ x . 3 2 . 5 2 1 . 5 1 0 . 5 − 2 2 4 6 8 10

  20. Examples Graph y = √ x and y = √− x and y = −√ x . 3 3 2 . 5 2 . 5 2 2 1 . 5 1 . 5 1 1 0 . 5 0 . 5 − 2 2 4 6 8 10 − 10 − 8 − 6 − 4 − 2 2

  21. Examples Graph y = √ x and y = √− x and y = −√ x . 3 3 2 . 5 2 . 5 2 2 1 . 5 1 . 5 1 1 0 . 5 0 . 5 − 2 2 4 6 8 10 − 10 − 8 − 6 − 4 − 2 2 − 2 2 4 6 8 10 − 0 . 5 − 1 − 1 . 5 − 2 − 2 . 5 − 3

  22. “Starting” Functions y = 1 y = x 2 y = x 3 x y = √ x √ x y = y = | x | 3

  23. Graphing Equations with Multiple Transformations √ 4 − x + 1 Let’s say we want to graph y = 1 2 1. Identify the “starting” function. y = √ x

  24. Graphing Equations with Multiple Transformations √ 4 − x + 1 Let’s say we want to graph y = 1 2 1. Identify the “starting” function. y = √ x 2. Compare the “starting” function with the final function. Anything that’s changed where the x is in the starting function will correspond to a horizontal transformation. Anything else will correspond to a vertical transformation. Vertical y = 1 � � y = 4 − x + 1 x vs 2 Horizontal

  25. Graphing Equations with Multiple Transformations (continued) 3. Graph the horizontal transformations, by going against the order of operations. 3 3 3 2 . 5 2 . 5 2 . 5 2 2 2 1 . 5 1 . 5 1 . 5 1 1 1 0 . 5 0 . 5 0 . 5 − 10 y = √ x + 4 = − 5 5 10 − 10 − 5 5 10 − 10 − 5 5 10 y = √ 4 − x y = √ x √ 4 + x

  26. Graphing Equations with Multiple Transformations (continued) 4. Graph the vertical transformations, by following the order of operations. 3 3 2 . 5 2 . 5 2 2 1 . 5 1 . 5 1 1 0 . 5 0 . 5 − 10 − 5 5 10 − 10 − 5 5 10 √ 4 − x √ 4 − x + 1 y = 1 y = 1 2 2

  27. Examples 1. Graph y = ( x − 1) 2 − 3

  28. Examples 1. Graph y = ( x − 1) 2 − 3 6 4 2 − 4 − 2 2 4 − 2

  29. Examples 1. Graph y = ( x − 1) 2 − 3 6 4 2 − 4 − 2 2 4 − 2 2. Graph y = − 2 √ x + 2

  30. Examples 1. Graph y = ( x − 1) 2 − 3 6 4 2 − 4 − 2 2 4 − 2 2. Graph y = − 2 √ x + 2 − 4 − 2 2 4 6 8 10 − 2 − 4 − 6

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