1.6 PreCalculusParent Functions

Graphing Techniques

Transformations

Vertical Translations Horizontal Translations

Graph stays the same, but moves up or down.

Graph stays the same, but moves left or right.

Transformations

Vertical Stretch Horizontal StretchWidth stays the same, but height increases.

Height stays the same, but width increases.

Transformations

Vertical Compression Horizontal CompressionWidth stays the same, but height decreases.

Height stays the same, but width decreases.

Transformations

Reflection Over the x-axis

Graph “flips” up-side down.

Reflection Over the y-axis

Graph “flips” side-ways.

Quadraticf(x) = x2

Abs Valuef(x) = |x|

Square Rt. f(x) =

Translate Up

Translate Down

Translate Left

Translate Right

x

g(x) = x2 + A g(x) = |x| + A xg(x) = + A

g(x) = x2 − A

g(x) = (x + A)2

g(x) = (x − A)2

g(x) = |x| − A

g(x) = |x + A|

g(x) = |x − A|

xg(x) = − A

Ax +g(x) =

Axg(x) =

Assume that A is a positive, real number!

Quadraticf(x) = x2

Abs Valuef(x) = |x|

Square Rt. f(x) =

Vertical Stretch

Vertical Compression

Horizontal Stretch

Horizontal Compression

x

2xA

1=g(x) |x|

A

1=g(x) x

A

1=g(x)

g(x) = | 1 A x | g(x) = ( 1

A x ) 2

x=g(x) A1

Ax=g(x)

g(x) = Ax2

g(x) = (Ax)2

g(x) = A|x|

g(x) = |Ax|

xg(x) = A

Assume that A is a positive, real number!

Quadraticf(x) = x2

Abs Valuef(x) = |x|

Square Rt. f(x) =

Reflection over x-axis

Reflection over y-axis

x

Assume that A is a positive, real number!

g(x) = −x2 g(x) = −|x| xg(x) = −

x-=g(x)g(x) = (-x)2 g(x) = |-x|

Rational FunctionsTranslate

UpStretch

Translate Down

Compression

Translate Left

Reflection over x-axis

Translate Right

Reflection over y-axis

1( )f x

x

1( )g x A

x

1( )g x A

x

1( )g x

x A

1

( )g xx A

( )A

g xx

1( )g x

Ax

1( )g x

x

1( )g x

x

Identify each transformation from the parent graph f(x) = x2.

g(x) = x2 + 5 g(x) = x2 – 2

g(x) = (x + 1)2 g(x) = (x – 3)2

up 5 down 2

left 1 right 3

g(x) = −x2 g(x) = (-x)2reflection over x-axis

reflection over y-axis

2x2

1=g(x)

g(x) = ( 1 2 x )

2

g(x) = 2x2

g(x) = (2x)2

vertical stretchfactor of 2

vertical comp.factor of ½

Horiz. stretchFactor of 2

Horiz. Comp.Factor of ½

Identify each transformation from the parent graph f(x) = x2.

g(x) = -2x2 + 5

g(x) = -(x + 1)2

g(x) = (x – 3)2 − 2

up 5

down 2

left 1

right 3

reflection over x-axis

vertical stretchfactor of 2

reflection over x-axis

g(x) = (-2x)2 Horiz. Comp.Factor of ½

reflection over y-axis

Identify each transformation from the parent graph f(x) = |x|.

g(x) = |x| + 3 g(x) = |x| – 10

g(x) = |x + 5| g(x) = |x – 2|

up 3 down 10

left 5 right 2

g(x) = −|x| g(x) = |-x|reflection over x-axis

reflection over y-axis

|x|2

1=g(x)

g(x) = | 1 2 x |

g(x) = 2|x|

g(x) = |2x|

vertical stretchfactor of 2

vertical comp.factor of ½

Horiz. stretchFactor of 2

Horiz. Comp.Factor of ½

Identify each transformation from the parent graph f(x) = |x|.

g(x) = 5|x| − 4

g(x) = -|x| + 3

g(x) = 2|x – 5| - 3

down 4

down 3

up 3

right 5

vertical stretchfactor of 5

reflection over x-axis

g(x) = |-3x| Horiz. Comp.Factor of ⅓

reflection over y-axis

vertical stretchfactor of 2

Identify each transformation from the parent graph

xf(x) =

xg(x) = + 3xg(x) = − 2

2x +g(x) = 4xg(x) =

x2

1=g(x)

x=g(x) 21 2x=g(x)

xg(x) = 2

down 2 up 3

left 2 right 4

vertical stretchfactor of 2

vertical comp.factor of ½

horiz. stretchfactor of 2

horiz. Comp.factor of ½

x-=g(x) x-=g(x)reflection overx-axis

reflection overy-axis

Identify each transformation from the parent graph

xf(x) =

1+4+x2g(x) =

5-x2

1-=g(x)

up 1

right 5

vertical stretchfactor of 2

vertical comp.factor of ½

down 4 horiz. Comp.factor of ⅓

reflection overx-axis

reflection overy-axis

43x=g(x)

left 4

Find the function that is finally graphed after the following three transformations are

applied to the graph of y = |x|.

1. Shift left 2 units.

2. Shift up 3 units.

3. Reflect about the y-axis.

2y x

2 3y x

2 3y x

Find the function that is finally graphed after the following three transformations are

applied to the graph of

1. Shift down 5 units.

2. Shift right 2 units.

3. Reflect about the x-axis.

5y x

y = x

2 5y x

2 5y x

Graphing Techniquesf(x) = x2 – 4 (down 4)

x

y

1. Graph f(x) = x2.

2. Shift all of the points down 4 units.

Graphing Techniquesf(x) = (x – 3)3

(right 3)

x

y

1. Graph f(x) = x3.

2. Shift all of the points right 3 units.

Graphing Techniquesf(x) = |x - 2| + 3

(right 2, up 3)

x

y

1. Graph f(x) = |x|.

2. Shift all of the points right 2 and up 3.

Graphing Techniquesf(x) = -x3

(reflect over x-axis)

x

y

1. Graph f(x) = x3.

2.Reflect all points over the x-axis.