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1. Transformations

1. Transformations. To graph: Identify parent function and adjust key points. Warm-up . For each function below, a) state the domain b) even/odd/neither c) symmetry. Warm-up. Suppose. 1) If , what is x? 2) Find all intercepts of the graph of f. Warm-up.

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1. Transformations

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

  2. Warm-up. For each function below, a) state the domain b) even/odd/neither c) symmetry

  3. Warm-up. Suppose 1) If , what is x? 2) Find all intercepts of the graph of f

  4. Warm-up. Suppose and are points on a line. Write the equation of the line containing these 2 points.

  5. Warm-up. 1. Evaluate the following: State the domain for this function Sketch the graph

  6. 2.6 Function Transformations

  7. 2.6 Function Transformations

  8. a. Vertical Shift Parent function : Shift Down 2 units • Vertical Shift (or translation) • shifts UP k units • shifts DOWN k units

  9. b. Horizontal Shift Parent function : Shift left 3 units • Horizontal shift (or translation) • shifts LEFT h units • shifts RIGHT h units

  10. 2a. Reflection about the x-axis Parent function : Reflect over x-axis. Reflectsgraph about the x-axis

  11. 2b. Reflects graph about the y-axis Parent function : Reflect over y-axis. Reflectsgraph about the y-axis

  12. 3a. Stretch (dilate) the graph vertically Parent function : Stretch vertically by : 2 • If a > 1, stretches graph vertically • If 0 < a < 1, compresses graph vertically

  13. 3b. Horizontal Stretch/Compress • Horizontal Scale • If b > 1, compresseshorizontally (x-values by 1/b) • If 0 < b < 1, stretcheshorizontally (x-values by 1/b)

  14. 3b. Horizontal Dilation (Scale) When scale is “inside” the parent function, it is preferable to pull it OUTSIDE the parent function and apply vertical dilation

  15. Practice

  16. 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 ?

  17. 5. a) Rewrite function in standard form Step 1:Always, factor out coefficients and write in standard form, before doing transformations! Rewrite in standard form:

  18. Horizontal scale Perform the transformations in this order 1. Vertical scale 4. Vertical shift 2. 3. Horizontal shift

  19. 6. Describe sequence of Transformations

  20. 6. Describe sequence of Transformations

  21. 6. Describe sequence of Transformations

  22. 6. More Practice… For each function, describe (in order) the sequence of transformations and sketch the final graph. 1) 4) 2) 5) 3)

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

  24. 11. Write an equation from the graph • Identify parent (shape) • Compare key points to determine if y values are scaled. • Observe translations and reflections • Write in standard form

  25. 1. Library of Functions (Take note of key points) “Slope” = 1 Move: Right 1, Up 1 to next point on graph

  26. College Algebra Notes 2.6 WritetheFunctionfromtheGraph For each graph below: a)Name the parent function b) Describe the sequence of transformations (in order) c) Determine the function that describes the graph d) Verify key points by plugging into your function. 1) 2)

  27. 3) 4)

  28. 11. Write an equation from the graph

  29. Transformations 1) 2) 3)

  30. 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 ?

  31. 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

  32. 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

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