x2 t04 02 curve sketching - transformations

(C) Transformations

(C) TransformationsGiven that the graph y = f(x) can be sketches, then it is possible to build other sketches through appropriate transformations.


( ) axfy ±=•(a is grouped with y: shift up or down by a)

( )xfayOR =



( )xfayOR =

y

x

y=f(x)



( )xfayOR =

y

x

y=f(x)a

a

a

a

( ) axfy +=



( )xfayOR =

( )axfy ±=• (a is grouped with x: shift left or right by a)y

x

y=f(x)a

a

a

a

( ) axfy +=



( )xfayOR =

( )axfy ±=• (a is grouped with x: shift left or right by a)y

x

y=f(x)a

a

a

a

( ) axfy +=

b

b

b

b

b

( )bxfy +=

( )xfy −=• (reflect f(x) in the x axis)


y

x

y=f(x)


y

x

y=f(x)

( )xfy −=


y

x

y=f(x)

( )xfy −=

( )xfy −=• (reflect f(x) in the y axis)


y

x

y=f(x)

( )xfy −=


( )xfy −=



( )xfy =• (reflect the part of f(x) where f(x)<0 in the x axis)


y

x

y=f(x)



( )xfy =




( )xfy =• (reflect the part of f(x) where x>0 in the y axis)


y=f(x)



( )xfy =


y

x





( )xkfy =• (stretch f(x) vertically, k<1 shallower,k>1 steeper)


y=f(x)




y

x



y=f(x)



( ) 1, >= kxkfy


y

x



y=f(x)



( ) 1, >= kxkfy


y

x


( ) 1, <= kxkfy


y=f(x)



( ) 1, >= kxkfy


y

x


( ) 1, <= kxkfy

Note: •domain remains same•x intercepts remain same






( )kxfy =• (stretch f(x) horizontally, k<1 shallower,k>1 steeper)


y=f(x)




y

x




y=f(x)



( ) 1, >= kkxfy


y

x




y=f(x)



( ) 1, >= kkxfy


y

x


( ) 1, <= kkxfy



y=f(x)



( ) 1, >= kkxfy


y

x


( ) 1, <= kkxfyNote: •range remains same•y intercepts remain same


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