Transformations of functions. Vocabulary Transformation: moves the graph up or down, left or right,...

Transformations of Transformations of functionsfunctions

VocabularyVocabularyTransformationTransformation: moves the graph up or down, left or right, : moves the graph up or down, left or right, stretches or shrinks it, or flips it. stretches or shrinks it, or flips it.

VocabularyVocabularyTranslationTranslation: a transformation that shifts the graph : a transformation that shifts the graph

horizontallyhorizontally or or verticallyvertically

Your turn:Your turn:

Without deleting the previous equation, graph: Without deleting the previous equation, graph:

Graph the following on your calculator:Graph the following on your calculator:

21 xy

522 xy

1. 1. What is the difference between the two graphs?What is the difference between the two graphs?


Change the second equation to:Change the second equation to:

Keep the following in your calculator:Keep the following in your calculator:

21 xy

322 xy


Your turn:Your turn:3. 3. What does adding or subtraction “k” do to the function?What does adding or subtraction “k” do to the function?

kxxf 2)(

We call this a We call this a vertical translationvertical translation (or “shift”) (or “shift”)


Without deleting the previous equation, enter the following Without deleting the previous equation, enter the following (close the parentheses before you add ‘5’)(close the parentheses before you add ‘5’)

Enter the following on your calculator:Enter the following on your calculator:

““22ndnd” “0” (catalog) then “enter”” “0” (catalog) then “enter”

xy 1

52 xy



Change the second equation to:Change the second equation to:

Keep the following in your calculator:Keep the following in your calculator:

xy 1

32 xy


Your turn:Your turn:6. 6. What does adding or subtraction “k” do to the function?What does adding or subtraction “k” do to the function?

kxy

We call this a We call this a vertical translationvertical translation (or “shift”) (or “shift”)

Does this work for every Does this work for every function?function?

kxfy )(

Let’s try another function that we’ll learn about later.Let’s try another function that we’ll learn about later.

Without deleting the previous equation, graph the followingWithout deleting the previous equation, graph the following (make sure to close the parentheses before you add 5) (make sure to close the parentheses before you add 5)

Graph the following:Graph the following: )ln(1 xy

3)ln(2 xy

Vocabulary Vocabulary Parent FunctionParent Function: the simplest function in a family of functions.: the simplest function in a family of functions.

2)( xxf

Parent of the Parent of the parabola functionparabola function family: family:

xxf )(

Parent of the Parent of the linearlinear functionfunction family: family:

xxf )(

Parent of the Parent of the absolute value functionabsolute value function family: family:

Vocabulary Vocabulary Parent FunctionParent Function: the simplest function in a family of functions.: the simplest function in a family of functions.

2)( xxf xxf )(

These are the only three we have learned so far.These are the only three we have learned so far.

We will learn several more this year. In the next chapterWe will learn several more this year. In the next chapter will will learn the following:will will learn the following:

3)( xxf 4)( xxf nxxf )(

xxf )(


7. 7. What does adding a number to the parent functionWhat does adding a number to the parent function do to the graph of the parent function?do to the graph of the parent function?

kxf )(

Your turn:Your turn:Enter the following on your calculator:Enter the following on your calculator:

21 xy 2

2 )4( xy

9. 9. What is the difference between the What is the difference between the graphs of these two equations?graphs of these two equations?

Replace the previous with the following:Replace the previous with the following:

22 )3( xy2

1 xy

8. 8. What is the difference between the What is the difference between the graphs of these two equations?graphs of these two equations?

Your turn:Your turn:10. 10. Enter the following on your calculator:Enter the following on your calculator:

xy 1 42 xy

11. 11. Replace the previous with the following:Replace the previous with the following:

xy 1 32 xy

What is the difference between the What is the difference between the graphs of these two equations?graphs of these two equations?



12. 12. What does replacing ‘x’ with ‘x ± h’ do to the parent function?What does replacing ‘x’ with ‘x ± h’ do to the parent function?

)( hxf

13. 13. Without graphing, describe how the parent of the parabola Without graphing, describe how the parent of the parabola function family been transformed by replacing function family been transformed by replacing ‘ ‘x’ with ‘(x + 2)’ and then adding ‘3’ to the function?x’ with ‘(x + 2)’ and then adding ‘3’ to the function?

3)7( 2 xy2xy

Your turn:Your turn:14. 14. Enter the following on your calculator (make sure you putEnter the following on your calculator (make sure you put a ‘2’ into the “y-editor” before you enter the “abs( “ function).a ‘2’ into the “y-editor” before you enter the “abs( “ function).

xy 1xy 22

15. 15. Replace the previous with the following:Replace the previous with the following:

21 xy 2

2 3xy



Your turn:Your turn:16. 16. Enter the following on your calculator (make sure you putEnter the following on your calculator (make sure you put a ‘2’ into the “y-editor” before you enter the “abs( “ function).a ‘2’ into the “y-editor” before you enter the “abs( “ function).

xy 1xy

2

12

17. 17. Replace the previous with the following:Replace the previous with the following:2

1 xy 22 3

1xy




18. 18. What “transformation” occurs to the parent function whenWhat “transformation” occurs to the parent function when you multiply the parent function by a number greater than ‘1’?you multiply the parent function by a number greater than ‘1’?

)(2 xf

19. 19. What “transformation” occurs to the parent function whenWhat “transformation” occurs to the parent function when you multiply the parent function by a number less than ‘1’?you multiply the parent function by a number less than ‘1’?

)(4

1xf

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

SummarySummary

khxfay )(*

The “Transformation Equation”The “Transformation Equation”stretchstretch shiftshift

shiftshift

xaxf )( Vertical stretchVertical stretch

hxxf )( Horizontal shiftHorizontal shift

kxxf )( Vertical shiftVertical shift

Without graphing Without graphing predictpredict how the graph of the following how the graph of the following function is going to be different from the parent function.function is going to be different from the parent function.

542)( xxf

6)1(25.0)( 2 xxh

2132)( xxg

2)20(37)( xxs

Vertically stretched by a factor of 2, Vertically stretched by a factor of 2, shifted left 4 and up 5shifted left 4 and up 5

Vertically stretched by a factor of 2/3, Vertically stretched by a factor of 2/3, shifted right 1 and down 2shifted right 1 and down 2

Vertically stretched by a factor of 0.25, Vertically stretched by a factor of 0.25, shifted left 1 and down 6shifted left 1 and down 6

Vertically stretched by a factor of 37, Vertically stretched by a factor of 37, shifted right 20shifted right 20

2

1)( 2 xxt shifted down 1/2shifted down 1/2

Your turn:Your turn:Describe the transformation of the parent function.Describe the transformation of the parent function.

32)( xxf20. 20.

21. 21. 4)3(21)( 2 xxg

22. 22. 9.7)7.4(5.2)( 2 xxh

Your turn:Your turn:Without using your calculator, draw both the parent functionWithout using your calculator, draw both the parent function and an approximate graph of the following (on the same plot)and an approximate graph of the following (on the same plot)

32)( xxf23. 23.

24. 24. 4)3(21)( 2 xxg


25. 25. What is the difference between them?What is the difference between them?

Graph the following two equations on your calculator:Graph the following two equations on your calculator:

xy 1 xy 2

Graph the following two equations on your calculator:Graph the following two equations on your calculator:

21 xy 2

2 xy

26. 26. What is the difference between them?What is the difference between them?

Reflections across the ‘x’ Reflections across the ‘x’ axisaxis

(2, 3)(2, 3)

(2, -3)(2, -3)

Changing ‘y’ to ‘– y’ Changing ‘y’ to ‘– y’ reflects the pointreflects the pointacross the x-axis.across the x-axis.

y = f(x) y = f(x) y = -f(x) y = -f(x)Multiplying the function Multiplying the function by (-1) by (-1) reflects the function reflects the function across the x-axis.across the x-axis.


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

)( )()1( xfxf

Vocabulary:Vocabulary:ReflectionReflection: a mirror image of the graph across a boundary line.: a mirror image of the graph across a boundary line.

Reflections across the ‘x’ Reflections across the ‘x’ and ‘y’ axisand ‘y’ axis

(-2, 3)(-2, 3) (2, 3)(2, 3)Changing ‘x’ to ‘– x’ Changing ‘x’ to ‘– x’ reflects the pointreflects the pointacross the y-axis.across the y-axis.

y = f(x) y = f(x) y = f(-x) y = f(-x)Replacing and input valueReplacing and input value by (-1) times the imput valueby (-1) times the imput value reflects the function reflects the function across the y-axis.across the y-axis.

Your turn:Your turn:27. 27. What does replacing ‘x’ in the parent function with (-x) doWhat does replacing ‘x’ in the parent function with (-x) do to the parent function? to the parent function? )( )( xfxf

For the other 2 functions we’ve learned (square function andFor the other 2 functions we’ve learned (square function and absolute value function) a reflection across the y-axis looks absolute value function) a reflection across the y-axis looks just like the orginal function.just like the orginal function.

x)( xf(Linear function)(Linear function)

VocabularyVocabulary

ReflectionReflection: makes “mirror-image” of the graph : makes “mirror-image” of the graph across some line. across some line.

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

xxf )(

Comparing the General Comparing the General Equation to a specific Equation to a specific example.example.

432)( xxf

khxaxf )(

Slope = ?Slope = ?

vertex = ?vertex = ?

32)( xxf Slope = ?Slope = ?

vertex = ?vertex = ?

xy

Your Turn:Your Turn: khxaxf )(

28.28.

29.29.

30.30.

Find the Find the slopeslope and and vertexvertex of each graph. of each graph.

574)( xxf

532)( xxf

6)( xxf

More TransformationsMore Transformations

y = f(x + 3) – 4 y = f(x + 3) – 4

((00, 0), 0) x x -- 33 ((-3-3, 0), 0)(-3, (-3, 00)) y – 4y – 4 (-3, (-3, -4-4))

khxfay )(*stretchstretch shiftshift

shiftshift

f(x): (0, 0) (2, 3)f(x): (0, 0) (2, 3)

((22, 3), 3) 22 -- 33 ((-1-1, 0), 0)(-1, (-1, 33)) 3 – 43 – 4 (-1, (-1, -1-1))

f(x + 3) – 4: (-3, -4) (-1, -1) f(x + 3) – 4: (-3, -4) (-1, -1)

More TransformationsMore Transformations

y = 5 * f(x + 3) – 4 y = 5 * f(x + 3) – 4

((22, 4), 4) x x -- 33 ((-1-1, 4), 4)

((-1-1, , 44)) 55 * y * y ((-1-1, , 2020))

((-1-1, , 88)) y y – 4– 4 ((-1-1, , 1616))

khxfay )(*stretchstretch shiftshift

shiftshift

f(x): (2, 4)f(x): (2, 4)

Left 3Left 3

Vertical stretch Vertical stretch factor of 5factor of 5

Down 4Down 4

5 *f(x + 3) – 4: (-1, 16) 5 *f(x + 3) – 4: (-1, 16)

Your Turn:Your Turn:

Graph: Graph: f(x+2) – 1

31. 31. Given: f(x) is made up of a segment withGiven: f(x) is made up of a segment with end points (2, 3) and (-4, 7)end points (2, 3) and (-4, 7)

Your Turn:Your Turn:

32. 32. Graph without using your calculatorGraph without using your calculator::

212)( xxf33. 33. Graph without using your calculatorGraph without using your calculator::

32)( xxf

Effect of ‘h’ and ‘k’ on the Effect of ‘h’ and ‘k’ on the Absolute Value FunctionAbsolute Value Function

xy h = 0, k = 0

11 xy

h = 1, k = 1

1

1

Effect of ‘a’ on the Absolute Effect of ‘a’ on the Absolute Value FunctionValue Function

xy 4

xy

xy4

1

xy

Sign changeSign changecauses a “Reflection”causes a “Reflection”

As the value of ‘a’ changes, the shape of the graph changes.As the value of ‘a’ changes, the shape of the graph changes.

Effect of ‘h’ and ‘k’ on the Effect of ‘h’ and ‘k’ on the Absolute Value FunctionAbsolute Value Function

xy xy

12 xy

234

1 xy

(0, 0)(0, 0)

(0, 0)(0, 0)

(2, 1)(2, 1)

(-3, -2)(-3, -2)

Transformations of functions. Vocabulary Transformation: moves the graph up or down, left or right,...

Documents

Transcript of Transformations of functions. Vocabulary Transformation: moves the graph up or down, left or right,...