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2.6 Function Transformations

2.6 Function Transformations. 1. Transformations. To graph: Identify parent function and adjust key points. 1. Translations (Shift). Vertical Shift (or translation) shifts UP k units shifts DOWN k units . Horizontal shift (or translation) shifts LEFT h units

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2.6 Function Transformations

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  1. 2.6 Function Transformations

  2. 1. Transformations To graph: Identify parent function and adjust key points.

  3. 1. Translations (Shift) • Vertical Shift (or translation) • shifts UP k units • shifts DOWN k units • Horizontal shift (or translation) • shifts LEFT h units • shifts RIGHT h units

  4. a. Vertical Shift Parent function : Shift Down 2 units

  5. b. Horizontal Shift Parent function : Shift left 3 units

  6. 2. Reflections Reflectsgraph about the x-axis Reflectsgraph about the y-axis

  7. 2a. Reflection about the x-axis Parent function : Reflect over x-axis.

  8. 2b. Reflects graph about the y-axis Parent function : Reflect over y-axis.

  9. 3. Vertical Dilation (Scale) • If a > 1, stretches graph vertically • If 0 < a < 1, compresses graph vertically

  10. 3a. Stretch (dilate) the graph vertically Parent function : Stretch vertically by : 2

  11. 3b. Horizontal Dilation (Scale) • Horizontal Scale • If b > 1, compresses graph horizontally • If 0 < b < 1, stretches graph horizontally When the scale is “inside” the parent function, it is preferable to pull it OUTSIDE the parent function and apply vertical dilation

  12. 3b. Horizontal Stretch/Compress

  13. 4. Practice with single Transformations Practice: p. 127 A - L Make a table, describing the parent and transformations applied

  14. Practice p. 127 #30 Graph the transformations as described Write what you think the equation will be from the description graph your equation on the calculator to check your result. Did it work out like expected?

  15. 4. Sequence of Transformations When a function has multiple transformatinos applied, does the order of the transformations matter? Which operation is first: Reflection or Shift ? How about this one? Does the order matter.

  16. 5. a) Rewrite function in standard form Step 1:Factor out coefficients When a function is written in the standard form, Perform operations from left to right! Examples

  17. 6. Describe sequence of Transformations

  18. 6. Describe sequence of Transformations

  19. 6. Describe sequence of Transformations

  20. 6. More Practice… For each function, describe (in order) the sequence of transformations and sketch the final graph. 1) 4) 2) 3)

  21. 7. Domain How is the domain of a function affected by the transformations?

  22. 8. A second method for sequence of transformations • Method 2: Less Preferred method • When a function is not in the standard form, perform transformations in this order: • Horizontal shift • Stretch/shrink • Reflect • Vertical stretch Shrink

  23. 11. Write an equation from the graph • Identify parent function (look at shape) • Compare key points of parent function with your graph to determine if y values are scaled. • Observe translations and reflections and adjust equation accordingly.

  24. 11. Write an equation from the graph

  25. Perform the transformations in this order 1. Vertical scale by a If a is negative, reflects across x-axis 4. Vertical shift +k: shift up k -k : shift down k 2. 3. Horizontal scale by If b is negative, reflects across y-axis Horizontal shift -h : shift to right +h : shift to left

  26. Transformations 1) 2) 3)

  27. Warm-up. • a) List the sequence of transformations and sketch • b) List the transformations that are made to each key point of the parent function. Even or Odd ?

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