GraphicalGraphicalTransformations!Transformations!

!!!!Sec. 1.5a is amazing!!!Sec. 1.5a is amazing!!!

New DefinitionsNew DefinitionsTransformationsTransformations – functions that map real numbers – functions that map real numbersto real numbersto real numbers

Rigid TransformationsRigid Transformations – leave the size and shape of – leave the size and shape ofa graph unchanged (includes translations anda graph unchanged (includes translations andreflections)reflections)

Non-rigid TransformationsNon-rigid Transformations – generally distort the – generally distort theshape of a graph (includes stretches and shrinks)shape of a graph (includes stretches and shrinks)

New DefinitionsNew DefinitionsRigid TransformationsRigid Transformations

Vertical Translation Vertical Translation – of the graph of y = f(x) is a– of the graph of y = f(x) is ashift of the graph up or down in the coordinate planeshift of the graph up or down in the coordinate plane

Horizontal Translation Horizontal Translation – a shift of the graph to the– a shift of the graph to theleft or the rightleft or the right

TranslationsTranslationsLet c be a positive real number. Then the following trans-formations result in translations of the graph of y = f(x):

Horizontal Translations

y = f(x – c) a translation to the right by c units

y = f(x + c) a translation to the left by c units

Vertical Translations

y = f(x) + c a translation up by c units

y = f(x) – c a translation down by c units

Each figure shows the graph of the original square root function,along with a translation function. Write an equation for eachtranslation.

5y x 4y x 1y x

New DefinitionsNew DefinitionsRigid TransformationsRigid Transformations

Points (x, y) and (x, –y) are Points (x, y) and (x, –y) are reflections of each other acrossreflections of each other acrossthe x-axis.the x-axis.

Points (x, y) and (–x, y) are Points (x, y) and (–x, y) are reflections of each other acrossreflections of each other acrossthe y-axis.the y-axis.

(x, y)(x, y)(–x, y)(–x, y)

(x, –y)(x, –y)

ReflectionsReflectionsThe following transformations result in the reflections ofthe graph of y = f(x):

Across the x-axis

y = – f(x)

Across the y-axis

y = f(–x)

Find an equation for the reflection of the given function acrosseach axis:

y f x

2

5 9

3

xf x

x

Across the x-axis:2

5 9

3

x

x

2

9 5

3

x

x

y f x Across the y-axis:

25 9

3

x

x

2

5 9

3

x

x

Let’s support our algebraic work graphically…

On to vertical and horizontal stretches and

shrinks…

Stretches and Shrinks

xy f

c

Let c be a positive real number. Then the following trans-formations result in stretches or shrinks of the graph ofy = f(x):

a stretch by a factor of c if c > 1a shrink by a factor of c if c < 1

Horizontal Stretches or Shrinks

y c f x a stretch by a factor of c if c > 1a shrink by a factor of c if c < 1

Vertical Stretches or Shrinks

Let C be the curve defined by y = f(x) = x – 16x. Findequations for the following non-rigid transformations of C :

1

1

13

1. C is a vertical stretch of C by a factor of 312

2. C is a horizontal shrink of C by a factor of 1/213

2 3y f x 33 16x x 33 48x x

3 1/ 2

xy f

2f x 3

2 16 2x x 38 32x x

Let’s verify our algebraic work graphically…

Whiteboard problems…

Describe how the graph of can be transformed to the given equation.

reflect across x-axis

shift right 5

reflect across y-axis

reflect across y-axis

shift right 3

Describe how the graph of can be transformed to the given equation.

vertical stretch of 2

horiz. shrink of ½

horiz. stretch of 5

vertical shrink of 0.3

y x

y x

5y x

y x

3y x

3y x

32y x

3(2 )y x

3(0.2 )y x

30.3y x

More whiteboard problems…

Describe how to transform the graph of f into the graph of g.

right 6

reflect across x-axis, left 4

Describe the translation of to

Reflected across x-axis

Vertical stretch factor of 2

Shift left 4

Shift up 1

Homework: p. 139-140 1-23 odd

( ) 2

( ) 4

f x x

g x x

2

2

( ) ( 1)

( ) ( 3)

f x x

g x x

( )f x x

( ) 2 4 1g x x