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Transformations
 

Transformations

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    Transformations Transformations Document Transcript

    • 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:
    • 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
    • 3Oct 29­3:50 PMBased on the graph of f(x) , draw the graph of f(x)+3 
    • 4Oct 29­3:50 PMWhat will the graph of   f(x) ­ 2   be ?Vertical translation.ggb
    • 5Oct 29­3:50 PMBased on the graph of f(x) , draw the graph of f(x) ­ 1y=f(x)
    • 6Oct 29­3:50 PMThis is the graph of y = f(x)  .Sketch the graph off(x) ­ 4 y=f(x)
    • 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:
    • 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)   
    • 9Oct 29­3:50 PMBased on the graph of  , draw the graph of y=f(x­2)    f(x­2)=  y=f(x)Horizontal translation.ggb
    • 10Oct 29­3:50 PMy=f(x)Use your GDC to draw the graph ofand
    • 11Oct 29­3:50 PMy=f(x)andCompareWhat can you predict about the graph of g?
    • 12Oct 29­3:50 PMasymptotes?
    • 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:
    • 14Oct 29­3:50 PM Exercise 1: Sketch the graph of:  f(x) ­ 2 ;  f (x ­ 3)f(x)
    • 15Oct 29­3:50 PM Exercise 2: Sketch the graph of the  function Hence, sketch the graph of  
    • 16Oct 29­3:50 PMExercise 3: • Sketch the graph of • Sketch the graph of  • Find g(x) in its simplest form.
    • 17Oct 29­3:50 PMExercise 4:• Sketch the graph of  and indicate clearly any asymptote.• Sketch the graph of  and indicate clearly any asymptote.
    • 18Oct 29­3:50 PMExercise 5:• Sketch the graph of  and indicate clearly any asymptote.• Sketch the graph of  and indicate clearly any asymptote.
    • 19Oct 29­3:50 PMLesson 2Given  sketch the graphs of the following functions on the  same set of axes.
    • 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
    • 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
    • 22Oct 29­3:50 PMy = sin xVertical stretch.ggb
    • 23Oct 29­3:50 PMVerify your conclusion for the graphs of
    • 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
    • 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)?
    • 26Oct 29­3:50 PMy= sin (2x) horizontal stretch  ,   scale factor y= sin  x  y= sin (2x)Now , draw andHorizontal stretch.ggb
    • 27Oct 29­3:50 PMy= sin (½x) horizontal stretch  , scale factor  2y= sin x y= sin (½x)
    • 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
    • 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.
    • 30Oct 29­3:50 PMUse your calculator to draw the graphs of andWhat geometrical transformation does -f(x) represent?
    • 31Oct 29­3:50 PMBased on the graph of y= f(x) , draw the graph of y = ­ f(x).
    • 32Oct 29­3:50 PMWrite down the equations of both lines .
    • 33Oct 29­3:50 PMReflect the following function about the y-axis.
    • 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)
    • 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.
    • 36Oct 29­3:50 PMf(x) f(x­2)f(x)+32f(x)Match graphs and formulae:
    • 37Oct 29­3:50 PM f(x) f(x)+ab f(x)+c b f(x)f(x+a)
    • 38Oct 29­3:50 PMy = f(x)y = a f(x)y = f(ax)
    • 39Oct 29­3:50 PMTranslationsvertical translationhorizontal translationStretchsvertical stretch (a)horizontal stretch (      )Reflectionsreflection x-axisreflection y-axiscALibraryOfFunctionsWithTransformations.nbp
    • 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:
    • AttachmentsVertical translation exponential.ggbHorizontal translation quadratic.ggbALibraryOfFunctionsWithTransformations.nbpHorizontal stretch.ggbVertical translation.ggbHorizontal translation.ggbVertical stretch.ggb