Algebra-2 Graphical Transformations. Parent Function: The simplest function in a family of functions...

Algebra-2Algebra-2

Graphical TransformationsGraphical Transformations

Graphical TransformationsGraphical TransformationsParent FunctionParent Function: The : The simplestsimplest function in a family function in a familyof functions (lines, parabolas, cubic functions, etc.)of functions (lines, parabolas, cubic functions, etc.)

xy

Graphical TransformationsGraphical TransformationsTransformationTransformation: an adjustment made to the parent function : an adjustment made to the parent function that results in a that results in a change to the graphchange to the graph of the parent function. of the parent function.

Changes could include:Changes could include:

shiftingshifting (“translating”) the graph (“translating”) the graph upup or or downdown,,

““translating”translating” the graph the graph leftleft or or rightright

vertical stretchingvertical stretching or or shrinkingshrinking(making the graph (making the graph more steep more steep or or less steepless steep) )

ReflectingReflecting across x-axis across x-axis

y = xy = x

CompareCompare the two lines. the two lines.

addingadding 1 to the parent function 1 to the parent functiontranslatestranslates the graph up by 1. the graph up by 1.

y = x + 1y = x + 1

Each point moves Each point moves upup by by oneone..

y = xy = x

CompareCompare the two lines. the two lines.

Each point moves Each point moves upup by by twotwo..

y = x + 2y = x + 2 addingadding 2 to the parent function 2 to the parent functiontranslatestranslates the graph up by 2. the graph up by 2.

Your Turn:y = xy = x11. Graph the parent function:. Graph the parent function:

22. .

33. Explain how . Explain how y = x – 3 transforms the parent function.transforms the parent function.

Graph the line:Graph the line: y = x – 1 y = x – 1

y = xy = x

11

CompareCompare the two lines. the two lines.

y = 2xy = 2x

11

11

The The most simplemost simple line of all. line of all.y = xy = x

CompareCompare the two lines. the two lines.The coefficient of ‘x’ The coefficient of ‘x’ makes the graph makes the graph steepersteeper. . (vertically stretches the graph)(vertically stretches the graph)

y = 2xy = 2x

22

11

Your Turn:Your Turn:

xy2

1

11

4. 4. Graph the following two lines on the sameGraph the following two lines on the same x-y axis.x-y axis.

xy

22

5. 5. Explain how the following equationExplain how the following equation transforms the parent function:transforms the parent function: y = 3x - 2.y = 3x - 2.

Graphical TransformationsGraphical Transformations

y = xy = x

Graph the lines.Graph the lines.

xy

Reflection across x-axis.Reflection across x-axis.

How does the graph of How does the graph of compare with the graph of ?compare with the graph of ?

Graphical TransformationsGraphical TransformationsTransformationTransformation: an adjustment made to the parent function : an adjustment made to the parent function that results in a that results in a change to the graphchange to the graph of the parent function. of the parent function.

xy

kaxy )1(

Reflection Reflection across x-axisacross x-axis

translating translating upup or or downdown

more/less more/less steepsteep

32 xy

xy

Twice as steep, Twice as steep, translated up 2translated up 2

Your turn:Your turn: xy

translated up 6translated up 6

6. 6. Explain how the following equation transforms the parent function:Explain how the following equation transforms the parent function:

25 xy

6xy

7. 7. Explain how the following equation transforms the parent function:Explain how the following equation transforms the parent function: 5 times as steep, 5 times as steep,

translated down 2translated down 2

8. 8. Explain how the following equation transforms the parent function:Explain how the following equation transforms the parent function:

43 xy Reflected across x-axis, Reflected across x-axis, 3 times as steep, 3 times as steep, translated up 4translated up 4

Translations to the left or rightTranslations to the left or right

y = xy = x

xy

Reflection across x-axis.Reflection across x-axis.

2xy

22

Describe the transformationDescribe the transformation

xy

-2-2

What if ‘x’ is replaced by (x + 3) What if ‘x’ is replaced by (x + 3)

Translations to the left or rightTranslations to the left or right

translated down 2translated down 2

2)3( xy

translated left 3translated left 3

1xy

Graphical TransformationsGraphical TransformationsTransformationTransformation: an adjustment made to the parent function : an adjustment made to the parent function that results in a that results in a change to the graphchange to the graph of the parent function. of the parent function.

Changes could include:Changes could include:

shiftingshifting (“translating”) the graph (“translating”) the graph upup or or downdown,,

““translating”translating” the graph the graph leftleft or or rightright

vertical stretchingvertical stretching or or shrinkingshrinking(making the graph (making the graph more steep more steep or or less steepless steep) )

ReflectingReflecting across x-axis across x-axis

Graphical TransformationsGraphical TransformationsParent FunctionParent Function: The : The simplestsimplest function in a family function in a familyof functions (lines, parabolas, cubic functions, etc.)of functions (lines, parabolas, cubic functions, etc.)

2xy

CompareCompare the two parabolas. the two parabolas.

addingadding 2 to the parent function 2 to the parent functiontranslatestranslates the graph up by 2. the graph up by 2.

Each point moves Each point moves upup by by twotwo..

22 xy

Your Turn:9.9. Describe the transformation to the parent function: Describe the transformation to the parent function: 2xy

42 xy

10.10. Describe the transformation to the parent function: Describe the transformation to the parent function:2xy

52 xy

translated down 4translated down 4

translated up 5translated up 5

Graphical TransformationsGraphical TransformationsParent FunctionParent Function: The : The simplestsimplest function in a family function in a familyof functions (lines, parabolas, cubic functions, etc.)of functions (lines, parabolas, cubic functions, etc.)

2xy

CompareCompare the two parabolas. the two parabolas.

MultiplyingMultiplying the parent function the parent function by 3, makes it 3 times as by 3, makes it 3 times as steep.steep.

23xy

Graphical TransformationsGraphical TransformationsParent FunctionParent Function: The : The simplestsimplest function in a family function in a familyof functions (lines, parabolas, cubic functions, etc.)of functions (lines, parabolas, cubic functions, etc.)

2xy

CompareCompare the two parabolas. the two parabolas.

MultiplyingMultiplying the parent function the parent function by -1, reflects across the x-by -1, reflects across the x-axis.axis.

2xy

Your Turn:

11.11. Describe the transformation to the parent function: Describe the transformation to the parent function:2xy

22 xy

12.12. Describe the transformation to the parent function: Describe the transformation to the parent function: 2xy

63 2 xy

Reflected across x-axis Reflected across x-axis and translated up 2and translated up 2

3 times as steep and 3 times as steep and translated down 6translated down 6

Graphical TransformationsGraphical TransformationsParent FunctionParent Function: The : The simplestsimplest function in a family function in a familyof functions (lines, parabolas, cubic functions, etc.)of functions (lines, parabolas, cubic functions, etc.)

2xy

CompareCompare the two parabolas. the two parabolas.

ReplacingReplacing ‘x’ with ‘x – 1’ ‘x’ with ‘x – 1’ translates the parent function translates the parent function rightright by 1. by 1.

2)4( xy

Graphical TransformationsGraphical Transformations xy

kaxy )1(

Reflection Reflection across x-axisacross x-axis

translating translating upup or or downdown

more/less more/less steepsteep

khxay 2)()1(

Translates Translates left/rightleft/right

32 xy

Twice as steep, Twice as steep, translated up 2translated up 2

4)3(2 2 xy

Reflected across x-axis, Reflected across x-axis, twice as steep, twice as steep,

translated up 4, translated up 4, translated right 3translated right 3

Your Turn:

13.13. Describe the transformation to the parent function: Describe the transformation to the parent function: 2xy 3)5( 2 xy

14.14. Describe the transformation to the parent function: Describe the transformation to the parent function: 2xy 2)1(2 xy

translated up 3 translated up 3 translated left 5translated left 5

2 times as steep 2 times as steep translated right 1translated right 1

15.15. Describe the transformation to the parent function: Describe the transformation to the parent function: 2xy

4)3(2

1 2 xy Reflected across x-axis Reflected across x-axis 1/2 as steep 1/2 as steep

translated up 4 translated up 4 translated left 3translated left 3

Absolute Value FunctionAbsolute Value Function xxf )(

Enter the absolute value equation:Enter the absolute value equation:““22ndnd” + “catalog” (“0” button)” + “catalog” (“0” button) then “enter”then “enter”

Clear your “y-editor”Clear your “y-editor”

Now graph it:Now graph it:

Your turn: Your turn: 1. What are the coordinates of the vertex?1. What are the coordinates of the vertex? 2. What is the slope of the right side of the “vee”2. What is the slope of the right side of the “vee”

Your turn:Your turn:

What is the transformation to the parent function?What is the transformation to the parent function?

16. 16. Graph Graph 12 xy

17. 17. Clear from your “y-editor” Clear from your “y-editor” then graph: then graph:

12 xy12 xy

xy 1

What is the transformation to the parent function?What is the transformation to the parent function?

translated translated upup 1 1

translated translated rightright 1 1

Your turn:Your turn:18. 18. Clear from your “y-editor” Clear from your “y-editor” then graph: then graph:

12 xy

xy 22

What is the transformation to the parent function?What is the transformation to the parent function?

xy 1

Twice as Twice as steepsteep

Slope on right side is +2 Slope on right side is +2 slope on left side is -2slope on left side is -2

Your turn:Your turn:19. 19. Clear from your “y-editor” Clear from your “y-editor” then graph: then graph:

xy 22

What is the transformation to What is the transformation to the parent function?the parent function?

xy 1

xy 2

Reflected across x-axisReflected across x-axis

Slope on right side is -1 Slope on right side is -1 slope on left side is +1slope on left side is +1

xy kaxy )1(

Reflection Reflection across x-axisacross x-axis

translating translating upup or or downdown

more/less more/less steepsteep

khxay 2)()1(

Translates Translates left/rightleft/right

52/1 xy

Half as steep, Half as steep, translated down 5 translated down 5 reflected across reflected across

x-axisx-axis2)5(3 2 xy

Three times as steep, Three times as steep, translated down 2, translated down 2,

translated left 5translated left 5

khxaxf )1()(

325)( xxf

5 times as steep, 5 times as steep, translated up 3 translated up 3

translated right 2 translated right 2 reflected across reflected across

x-axisx-axis

What does adding or subtraction “k” do to the parent function?What does adding or subtraction “k” do to the parent function?

kxxf )(

hxxf )(

What does adding or subtraction “h” do to the parent function?What does adding or subtraction “h” do to the parent function?

What does multiplying by ‘a’ do to the parent function?What does multiplying by ‘a’ do to the parent function?

xaxf )(

Vertical shiftVertical shift

Horizontal shiftHorizontal shift

Vertical stretchVertical stretch

What does multiplying by (-1) do to the parent function?What does multiplying by (-1) do to the parent function?

xxf )( Reflection across x-axisReflection across x-axis

VocabularyVocabularyAbsolute Value FunctionAbsolute Value Function: A function of the form: : A function of the form:

khxaxf )(

SlopeSlope (h, k)(h, k)

Vertex (h, k)Vertex (h, k)

riserun

432)( xxf

ExamplesExamples

432)( xxf

khxaxf )(

Slope of right side = ?Slope of right side = ?

vertex = ?vertex = ?

32)( xxf Slope of right side = ?Slope of right side = ?

vertex = ?vertex = ?

Your Turn:Your Turn: khxaxf )1()(

20.20.

21.21.

22.22.

Describe the transformation to the parent function.Describe the transformation to the parent function.

574)( xxf

232)( xxf

6)( xxf

4 times as steep, translated: up 5, right 74 times as steep, translated: up 5, right 7

Reflected across x-axis, 2 times as steep, Reflected across x-axis, 2 times as steep, translated: up 2, left 3translated: up 2, left 3

Translated: down 6Translated: down 6

Your Turn:Your Turn:Describe the transformation to the parent function:Describe the transformation to the parent function:

2xy 23. 23. 6)5( 2 xy

24. 24. 34 xy

25. 25. 25.0192 xy

translated: up 6, right 5translated: up 6, right 5

Reflected across x-axis, 2 times as steep, Reflected across x-axis, 2 times as steep, translated: down 0.25, right 19translated: down 0.25, right 19

Reflected across x-axis, 4 times as steep, Reflected across x-axis, 4 times as steep, translated: up 3translated: up 3

Your turn:Your turn:

1)2(2 xfy

26. 26. The following graph is the function f(x). The following graph is the function f(x).

On the same axis graph the transformation: On the same axis graph the transformation:

Points: Points: (-2, -1)(-2, -1)(-1, -1)(-1, -1)( 0, 0)( 0, 0)( 1, 1)( 1, 1)( 2, 1)( 2, 1)( 3, 0)( 3, 0)