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# Transformations

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### Transformations

1. 1. 1Oct 29­3:50 PMTransformations of graphsTo explore geometrical transformations and theireffects on the graphs of functionsTo relate transformations with the variations in theequation of a function.General objectives:
2. 2. 2Oct 29­3:50 PMTransformations of graphsTo explore vertical and horizontal translations andtheir effects on the graphs of functionsTo relate translations with the variations in theequation of a function.TRANSLATIONSLesson 1
3. 3. 3Oct 29­3:50 PMBased on the graph of f(x) , draw the graph of f(x)+3
4. 4. 4Oct 29­3:50 PMWhat will the graph of   f(x) ­ 2   be ?Vertical translation.ggb
5. 5. 5Oct 29­3:50 PMBased on the graph of f(x) , draw the graph of f(x) ­ 1y=f(x)
6. 6. 6Oct 29­3:50 PMThis is the graph of y = f(x)  .Sketch the graph off(x) ­ 4 y=f(x)
7. 7. 7Oct 29­3:50 PM Conclusions: If y = f (x)y = f (x ) + c   translates vertically the graph of  y = f (x),  c  units.• If it moves upwards.• If it moves downwards.c > 0c < 0translation vector:
8. 8. 8Oct 29­3:50 PMBased on the graph of             , draw the graph of y = f (x ­ 2)    f(x­2)=  y=f(x)If  , find  f (x ­ 2)
9. 9. 9Oct 29­3:50 PMBased on the graph of  , draw the graph of y=f(x­2)    f(x­2)=  y=f(x)Horizontal translation.ggb
10. 10. 10Oct 29­3:50 PMy=f(x)Use your GDC to draw the graph ofand
11. 11. 11Oct 29­3:50 PMy=f(x)andCompareWhat can you predict about the graph of g?
12. 12. 12Oct 29­3:50 PMasymptotes?
13. 13. 13Oct 29­3:50 PMConclusions:y= f( x ­ b)y= f( x + b)y= f( x )Ifb >0translates horizontally the graph of  y= f( x ), b units to the right.translates horizontally the graph of  y= f( x ), b units to the left.translation vector:translation vector:
14. 14. 14Oct 29­3:50 PM Exercise 1: Sketch the graph of:  f(x) ­ 2 ;  f (x ­ 3)f(x)
15. 15. 15Oct 29­3:50 PM Exercise 2: Sketch the graph of the  function Hence, sketch the graph of
16. 16. 16Oct 29­3:50 PMExercise 3: • Sketch the graph of • Sketch the graph of  • Find g(x) in its simplest form.
17. 17. 17Oct 29­3:50 PMExercise 4:• Sketch the graph of  and indicate clearly any asymptote.• Sketch the graph of  and indicate clearly any asymptote.
18. 18. 18Oct 29­3:50 PMExercise 5:• Sketch the graph of  and indicate clearly any asymptote.• Sketch the graph of  and indicate clearly any asymptote.
19. 19. 19Oct 29­3:50 PMLesson 2Given  sketch the graphs of the following functions on the  same set of axes.
20. 20. 20Oct 29­3:50 PMTransformations of graphsTo explore stretches and reflections and theireffects on the graphs of functionsTo relate stretches and reflections with thevariations in the equation of a function.STRETCHLesson 2REFLECTIONS
21. 21. 21Oct 29­3:50 PMWhat is the effect on the graph of thatwill produce ify= f(x)y= a f(x)  ?"a"Set up your GDC in "degrees".Prepare domain :With your calculator , plot the graphs of
22. 22. 22Oct 29­3:50 PMy = sin xVertical stretch.ggb
23. 23. 23Oct 29­3:50 PMVerify your conclusion for the graphs of
24. 24. 24Oct 29­3:50 PMConclusions:y= a f( x )y= f( x )Ifa >1stretches vertically the graph of  y= f( x ),  scale factor:stretches vertically the graph of  y= f( x ), scale factor:ay=      f( x )1a1a
25. 25. 25Oct 29­3:50 PMUse your calculator to draw the graph of , forOn the same grid drawWhat effect produces the "2" of f(2x) on the graph of f(x)?
26. 26. 26Oct 29­3:50 PMy= sin (2x) horizontal stretch  ,   scale factor y= sin  x  y= sin (2x)Now , draw andHorizontal stretch.ggb
27. 27. 27Oct 29­3:50 PMy= sin (½x) horizontal stretch  , scale factor  2y= sin x y= sin (½x)
28. 28. 28Oct 29­3:50 PMConclusions: y= f( x )Ifa >1y= f( ax ) stretches horizontally the graph of  y= f( x ), scale factor: 1ay= f(      x )1astretches horizontally the graph of  y= f( x ),   scale factor: a
29. 29. 29Oct 29­3:50 PMExercise 1:• The function f is defined by • Sketch the function with the help of your GDC.• Describe the geometric transformation  that will apply to the graph of f.
30. 30. 30Oct 29­3:50 PMUse your calculator to draw the graphs of andWhat geometrical transformation does -f(x) represent?
31. 31. 31Oct 29­3:50 PMBased on the graph of y= f(x) , draw the graph of y = ­ f(x).
32. 32. 32Oct 29­3:50 PMWrite down the equations of both lines .
33. 33. 33Oct 29­3:50 PMReflect the following function about the y-axis.
34. 34. 34Oct 29­3:50 PMComplete:g(2) = f (.....) g(1) = f (.....) g(­1) = f (.....)g(­2) = f (.....) g(x) = f (.......)y = f(x)y =g(x)
35. 35. 35Oct 29­3:50 PMConclusions:y = ­f( x )y= f(­ x )y= f( x )Ifreflects the graph of   y= f( x ),  about the x-axis.reflects the graph of  y= f( x ),  about the y-axis.
36. 36. 36Oct 29­3:50 PMf(x) f(x­2)f(x)+32f(x)Match graphs and formulae:
37. 37. 37Oct 29­3:50 PM f(x) f(x)+ab f(x)+c b f(x)f(x+a)
38. 38. 38Oct 29­3:50 PMy = f(x)y = a f(x)y = f(ax)
39. 39. 39Oct 29­3:50 PMTranslationsvertical translationhorizontal translationStretchsvertical stretch (a)horizontal stretch (      )Reflectionsreflection x-axisreflection y-axiscALibraryOfFunctionsWithTransformations.nbp
40. 40. 40Oct 29­3:50 PMTo revise this topic at home:http://enlvm.usu.edu/ma/nav/activity.jsp?sid=__shared&cid=emready@trfns&lid=136http://archives.math.utk.edu/visual.calculus/0/shifting.7/index.htmlWhen you feel ready, self-assessment:
41. 41. AttachmentsVertical translation exponential.ggbHorizontal translation quadratic.ggbALibraryOfFunctionsWithTransformations.nbpHorizontal stretch.ggbVertical translation.ggbHorizontal translation.ggbVertical stretch.ggb