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1.6 PreCalculusParent Functions
Graphing Techniques
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TransformationsVertical Translations Horizontal TranslationsGraph stays the same, but moves up or down.
Graph stays the same, but moves left or right.
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TransformationsVertical Stretch Horizontal StretchWidth stays the same, but height increases.
Height stays the same, but width increases.
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TransformationsVertical Compression Horizontal CompressionWidth stays the same, but height decreases.
Height stays the same, but width decreases.
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TransformationsReflection Over the x-axisGraph “flips” up-side down.
Reflection Over the y-axisGraph “flips” side-ways.
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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!
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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!
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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|
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Graphing Techniquesf(x) = x2 – 4 (down 4)
x
y
1. Graph f(x) = x2.
2. Shift all of the points down 4 units.
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Graphing Techniquesf(x) = (x – 3)3
(right 3)
x
y
1. Graph f(x) = x3.
2. Shift all of the points right 3 units.
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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.
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Graphing Techniquesf(x) = -x3
(reflect over x-axis)
x
y
1. Graph f(x) = x3.
2.Reflect all points over the x-axis.