Section 2.5 Transformation of FunctionsSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Section 2.5Transformation of FunctionsGraphs of Common Functions                xyReciprocal Function( ) ( )( ) ( )( ) ( )Domain: - ,0 0,Range: - ,0 0,Decreasing on - ,0 0,Odd functionand�ȥ�ȥ� �1( )f xx=Vertical Shifts( ) ( )( ) ( )Vertical ShiftsLet be a function and be a positive real number. The graph of is the graph of shifted units vertically upward. The graph of is the graph of shifted f cy f x c y f x cy f x c y f x c� = + =� = - = units vertically downward.Vertical ShiftsExampleUse the graph of f(x)=|x| to obtain g(x)=|x|-2                xyHorizontal Shifts( ) ( )( ) ( )Horizontal ShiftsLet be a function and a positive real number. The graph of is the graph of shifted to the left units. The graph of is the graph of shifted to thef cy f x c y f xcy f x c y f x� = + =� = - = right units.cHorizontal ShiftsExampleUse the graph of f(x)=x2 to obtain g(x)=(x+1)2                xyCombining Horizontal and Vertical ShiftsExampleUse the graph of f(x)=x2 to obtain g(x)=(x+1)2+2                xyReflections of Graphs( ) ( )Refection about the -AxisThe graph of is the graph of reflected about the -axis.xy f x y f xx=- =Reflections about the x-axis( ) ( )Reflection about the y-AxisThe graph of is the graph of reflected about - axis.y f x y f xy= - =ExampleUse the graph of f(x)=x3 to obtain the graph of g(x)= (-x)3.                xyExample                xyUse the graph of f(x)= x to graph g(x)=- xVertical Stretching and ShrinkingVertically ShrinkingVertically Stretching                xy                xyGraph of f(x)=x3Graph of g(x)=3x3This is vertical stretching – each y coordinate is multiplied by 3 to stretch the graph.ExampleUse the graph of f(x)=|x| to graph g(x)= 2|x|                xyHorizontal Stretching and ShrinkingHorizontal ShrinkingHorizontal StretchingExample                xyUse the graph of f(x)= to obtain the1graph of g(x)=3xx(a) (b)(c)(d)                                  xyUse the graph of f(x)= x to graph g(x)= -x.The graph of g(x) will appear in which quadrant?Quadrant IQuadrant IIQuadrant IIIQuadrant IV( )f x x=(a) (b)(c)(d)                                  xyWrite the equation of the given graph g(x). The original function was f(x) =x2g(x)2222( ) ( 4) 3( ) ( 4) 3( ) ( 4) 3( ) ( 4) 3g x xg x xg x xg x x= + -= - -= + += - +(a) (b)(c)(d)Write the equation of the given graph g(x). The original function was f(x) =|x|g(x)( ) 4( ) 4( ) 4( ) 4g x xg x xg x xg x x=- -=- -=- +=- -                                  