1.6 PreCalculusParent Functions
Graphing Techniques
TransformationsVertical Translations Horizontal TranslationsGraph stays the same, but moves up or down.
Graph stays the same, but moves left or right.
TransformationsVertical Stretch Horizontal StretchWidth stays the same, but height increases.
Height stays the same, but width increases.
TransformationsVertical Compression Horizontal CompressionWidth stays the same, but height decreases.
Height stays the same, but width decreases.
TransformationsReflection Over the x-axisGraph “flips” up-side down.
Reflection Over the y-axisGraph “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 CompressionHorizontal
StretchHorizontal
Compression
x
2xA1
=g(x) |x|A1
=g(x) xA1
=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 xx
1( )g x Ax
1( )g x Ax
1( )g xx A
1( )g xx A
( ) Ag xx
1( )g xAx
1( )g xx
1( )g xx
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
2x21
=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|21
=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 graphxf(x) =
xg(x) = + 3xg(x) = − 2
2x +g(x) = 4xg(x) =
x21
=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 graphxf(x) =
1+4+x2g(x) =
5-x21
-=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.