1-5 Guided Notes TE - Parent Functions and Transformations

Name: _________________________________________________ Period: ___________ Date: ________________

Parent Functions and Transformations Guided Notes

A family of functions is a group of functions with graphs that display one or more similar characteristics.

The Parent Function is the simplest function with the defining characteristics of the family. Functions in the same family

are transformations of their parent functions.

Family - Constant Function

Family - Linear Function

Family - Quadratic Function

Graph

Graph

Graph

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?(?) = ?

Domain = (?¡Þ, ¡Þ)

Range = [?]

Rule

?(?) = ?

Domain= (?¡Þ, ¡Þ)

Range = (?¡Þ, ¡Þ)

Rule

Family - Cubic Function

Family - Square Root Function

Family - Reciprocal Function

Graph

Graph

Graph

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?(?) = ?

Domain= (?¡Þ, ¡Þ)

Range = (?¡Þ, ¡Þ)

Rule

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?(?) = ??

Domain= (?¡Þ, ¡Þ)

Range = [?, ¡Þ)

Rule

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?(?) = ¡Ì?

Domain= [?, ¡Þ)

Range = [?, ¡Þ)

Rule

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?

?(?) = ?

Domain= (?¡Þ, ?) ¡È (?, ¡Þ)

Range = (?¡Þ, ?) ¡È (?, ¡Þ)

Name: _________________________________________________ Period: ___________ Date: ________________

Parent Functions and Transformations Guided Notes

Family ¨C Absolut Value Function

Family - Greatest Integer Function

Graph

Graph

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?(?) = |?| = {

?? ?? ? < ?

? ?? ? ¡Ý ?

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?(?) = ???

Domain= (?¡Þ, ¡Þ)

Range ??? ???????

Rule

Domain= (?¡Þ, ¡Þ)

Range

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= [?, ¡Þ)

Transformations

Transformations

A change in the size or position of a figure or graph of the function is called a transformation.

Rigid transformations change only the position of the graph, leaving the size and shape unchanged.

Appearance in Function

?(?) ¡ú ?(?) + ?

?(?) ¡ú ?(?) ? ?

?(?) ¡ú ?(? ? ?)

?(?) ¡ú ?(? + ?)

?(?) ¡ú ??(?)

Vertical Translations

Horizontal Translations

Reflections in x-axes

?(?) ¡ú ?(??)

Reflections in y-axes

Transformation of Graph

? ????? ??

? ????? ????

? ????? ?????

? ????? ????

????????? ?? ??? ? ????

????????? ?? ??? ? ????

Transformation of

Point

(?, ?) ¡ú (?, ? + ?)

(?, ?) ¡ú (?, ? ? ?)

(?, ?) ¡ú (? + ?, ?)

(?, ?) ¡ú (? ? ?, ?)

(?, ?) ¡ú (?, ??)

(?, ?) ¡ú (??, ?)

Non rigid transformations distort the shape of the graph.

Appearance in Function

Transformation of Graph

Transformation of

Point

?(?) ¡ú ??(?) ? > ?

?(?) ¡ú ??(?) ? < ? < ?

???????? ??????????

?????????? ??????????

(?, ?) ¡ú (??, ?)

Vertical Dilations

Horizontal Dilations

?(?) ¡ú ?(??) ? > ?

?(?) ¡ú ?(??) ? < ? < ?

?????????? ????????????

???????? ????????????

?

(?, ?) ¡ú ( , ?)

?

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Name: _________________________________________________ Period: ___________ Date: ________________

Parent Functions and Transformations Guided Notes

Sample Problem 1: Identify the parent function and describe the transformations.

a.

?(?) = (? ? ?)?

Parent : ?(?) = ??

Transformation: Translation 1 unit right

b.

?(?) = ?? ? ?

Parent : ?(?) = ??

Transformation: Translation 5 units down

Parent : ?(?) = |?|

Transformation: Reflection in x-axis

Translation 4 units left

Parent : ?(?) = ??

?(?) = ??? + ?

d.

Transformation: Expand vertically by a factor of 3

Translation 7 units up

Sample Problem 2: Given the parent function and a description of the transformation, write the equation of the

transformed function ?(?) .

c.

?(?) = ?|? + ?|

a.

Quadratic - expanded horizontally by a factor of 2,

translated 7 units up.

b.

Cubic - reflected over the x axis and translated 9

units down.

c.

Absolute value - translated 3 units up, translated 8

units right.

d.

Reciprocal - translated 1 unit up.

?(?) =

? ?

? +?

?

?(?) = ??? ? ?

?(?) = |? ? ?| + ?

?(?) =

?

+?

?

Sample Problem 3: Use the graph of parent function to graph each function. Find the domain and the range of the

new function.

a.

?(?) = ?(? ? ?)? ? ?

?(?) = ?(? ? ?)? ? ?

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Parent function ?(?) = ?

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Transformation:

Expand vertically by a factor of 2

Translated 2 units down

Translated 3 units right

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? = (?¡Þ, ¡Þ)

? = (??, ¡Þ)

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Copyright ?

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Name: _________________________________________________ Period: ___________ Date: ________________

Parent Functions and Transformations Guided Notes

b.

?(?) = ¡Ì? ? ? + ?

?(?) = ¡Ì? ? ? + ?

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Parent function ?(?) = ¡Ì?

Transformation:

Translated 3 units up

Translated 5 units right

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? = [?. ¡Þ)

? = (?, ¡Þ)

c.

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?(?) = ?|? + ?| ? ?

?(?) = ?|? + ?| ? ?

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Parent function ?(?) = |?|

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Transformation:

Reflected in the x axis

Translated 1 unit down

Translated 4 units left

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? = (?¡Þ. ¡Þ)

? = (?¡Þ, ??]

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Transformations with Absolute Value

?(?) = |?(?)|

This transformation reflects any portion of the graph of ?(?) that is below the ? -axis so that it is above the ? -axis.

?(?) = ?(|?|)

This transformation results, in the portion of the graph of ?(?) that is to the left of the ?-axis, being replaced by a

reflection of the portion to the right of the ? -axis.

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Name: _________________________________________________ Period: ___________ Date: ________________

Parent Functions and Transformations Guided Notes

Sample Problem 4: Graph each function.

a.

?(?) = ?? ? ??

????? ?(?) = |?? ? ??|

?(?) = ?? ? ??

?(?) = |?? ? ??|

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?(?) =

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????? ?(?) =

?

|? ? ?|

?

???

?

= |???|

?(?) =

?(?)

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Copyright ?

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