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Transformations of Functions and their Graphs

Transformations of Functions and their Graphs. Mr. Perone Precalculus, 2013. Adapted from Mary Dwyer Wolfe, Ph.D., Macon State College. Summary of Shift Transformations. Translations. Horizontal Shifts

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Transformations of Functions and their Graphs

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  1. Transformations of Functions and their Graphs Mr. Perone Precalculus, 2013 Adapted from Mary Dwyer Wolfe, Ph.D., Macon State College

  2. Summary of Shift Transformations

  3. Translations Horizontal Shifts If h is a real number and y = f(x) is a function, we say that the graph of y = f(x - h) is the graph of f(x) shifted horizontally by h units. If h follows a minus sign, then the shift is right and if h follows a + sign, then the shift is left. Vertical Shifts If k is a real number and y = f(x) is a function, we say that the graph of y = f(x) + k is the graph of f(x) shifted vertically by k units. If k > 0 then the shift is upward and if k < 0, the shift is downward.

  4. Horizontal Translation Shift to the left by h: y=a(x+h)+k Shift to the right by h: y=a(x-h)+k

  5. x |x| 2 2 1 1 0 0 1 1 2 2 Practice Horizontal Translation Exercise: Shift to the left by 1 g(x) = |x|

  6. x |x| |x + 1| 2 2 1 1 1 0 0 0 1 1 1 2 2 2 3 Practice Horizontal Translation Graph g(x) = |x + 1| What if we shifted g(x)=|x| by 2 to the right?

  7. x |x| |x - 2| 2 2 4 1 1 3 0 0 2 1 1 1 2 2 0 Practice Horizontal Translation Graph g(x) = |x - 2|

  8. Horizontal Translation y = 3x2 Shift the function: LEFT 2 & RIGHT 2 AND GRAPH

  9. Horizontal Translation y = 3x2 y = 3(x+2)2 y = 3(x-2)2

  10. Translations What translation could be applied to the left graph to obtain the right graph? y = ???

  11. Translations Graphic Representations: Following the vertex, it appears that the vertex, and hence all the points, have been shifted up 1 unit and right 3 units.

  12. x |x| 2 2 1 1 0 0 1 1 2 2 Vertical Translation Example Graphy = |x|

  13. x |x| |x|+2 2 2 4 1 1 3 0 0 2 1 1 3 2 2 4 Vertical Translation Example Graph y = |x| + 2

  14. x |x| |x| -1 2 2 1 1 1 0 0 0 -1 1 1 0 2 2 1 Vertical Translation Example Graph y = |x| - 1

  15. Practice Vertical Translations y = 3x2 Now shift the formula vertically up and down. Sketch the graph. Let’s say you chose:3 down 2 up.

  16. Practice Vertical Translations y = 3x2 y = 3x2 – 3 y = 3x2 + 2

  17. Practice Vertical Translations Parent Function y = x3

  18. What are the equations after the shift? y = x3 y = x3 – 3 y = x3 + 2

  19. Translations Getting back to our unfinished task: 1 3 The vertex has been shifted up 1 unit and right 3 units.

  20. Translations Compare the following 2 graphs by explaining what to do to the graph of the first function to obtain the graph of the second function. f(x) = x4 g(x) = (x – 3)4 - 2

  21. Summary of Shift Transformations

  22. Reflections

  23. Reflection: Reflection: (across the x-axis) The graph of the function, y = - f(x) is the reflection of the graph of the function y = f(x).

  24. Example Reflection over x-axis f(x) = x2

  25. Example Reflection over x-axis f(x) = x2 f(x) = -x2

  26. Example Reflection over x-axis f(x) = x3

  27. Example Reflection over x-axis f(x) = x3 f(x) = -x3

  28. Example Reflection over x-axis f(x) = x + 1

  29. Example Reflection over x-axis f(x) = x + 1 f(x) = -(x + 1) = -x - 1

  30. Reflection: Reflection: (across the x-axis) The graph of the function, y = - f(x) is the reflection of the graph of the function y = f(x). Reflection: (across the y-axis) The graph of the function, y = f(-x) is the reflection of the graph of the function y = f(x).

  31. Example Reflection over y-axis f(x) = x2

  32. Example Reflection over y-axis f(x) = x2 f(-x) = (-x)2 = x2

  33. Example Reflection over y-axis f(x) = x3

  34. Example Reflection over y-axis f(x) = x3 f(-x) = (-x)3 = -x3

  35. Example Reflection over y-axis f(x) = x + 1

  36. Example Reflection over y-axis f(x) = x + 1 f(-x) = -x + 1 http://www.mathgv.com/

  37. Dilations (Nonrigid)(Vertical Stretch and Shrink)

  38. Dilations (Stretches and Shrinks) Definitions: Vertical Stretching and Shrinking The graph of y = af(x) is obtained from the graph of y = f(x) by a). stretching the graph of y = f ( x) by a when a > 1, or b). shrinking the graph of y = f ( x) by a when 0 < a < 1. Vertical Stretch Vertical Shrink

  39. Example Vertical Stretching/Shrinking y = |x|

  40. Example: Vertical Stretching/Shrinking y = |x| y = 0.5|x| y = 3|x|

  41. What is this? Base Function y = |x| y = ???? y = -2|x -1| + 4

  42. Multiple Transformations Procedure: Multiple Transformations Graph a function involving more than one transformation in the following order: • Horizontal translation • Stretching or shrinking • Reflecting • Vertical translation

  43. Graphing with More than One Transformation Graph -|x – 2| + 1 First graph f(x) = |x|

  44. Graphing with More than One Transformation Graph f(x) = -|x – 2| + 1 First graph f(x) = |x| 1. Perform horizontal translation: f(x) = |x-2| The graph shifts 2 to the right.

  45. Graphing with More than One Transformation Graph f(x) = -|x – 2| + 1 First graph f(x) = |x| 1. Perform horizontal translation: f(x) = |x-2| The graph shifts 2 to the right. 2. There is no stretch 3. Reflect in x-axis: f(x) = -|x-2|

  46. Graphing with More than One Transformation Graph f(x) = -|x – 2| + 1 First graph f(x) = |x| 1. Perform horizontal translation: f(x) = |x-2| The graph shifts 2 to the right. 2. There is no stretch 3. Reflect in x-axis: f(x) = -|x-2| 4. Perform vertical translation: f(x) = -|x-2| + 1 The graph shifts up 1 unit.

  47. Graphing with More than One Transformation Graph f(x) = -|x – 2| + 1 First graph f(x) = |x| 1. Perform horizontal translation: f(x) = |x-2| The graph shifts 2 to the right. 2. There is no stretch 3. Reflect in x-axis: f(x) = -|x-2| 4. Perform vertical translation: f(x) = -|x-2| + 1 The graph shifts up 1 unit.

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