Section2.5 Transformations Transformations Translations - - PowerPoint PPT Presentation

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Section2.5 Transformations Transformations Translations - - PowerPoint PPT Presentation

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Section2.5 Transformations Transformations Translations Horizontal Translations: Vertical Translations: The graph of f ( x c ) is f ( x ) shifted c units to the right. f ( x ) f ( x c ) Translations Horizontal Translations: Vertical

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SLIDE 1

Section2.5

Transformations

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SLIDE 2

Transformations

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SLIDE 3

Translations

Horizontal Translations: The graph of f (x − c) is f (x) shifted c units to the right.

f (x) f (x − c)

Vertical Translations:

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SLIDE 4

Translations

Horizontal Translations: The graph of f (x − c) is f (x) shifted c units to the right.

f (x) f (x − c)

The graph of f (x + c) is f (x) shifted c units to the left.

f (x) f (x + c)

Vertical Translations:

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SLIDE 5

Translations

Horizontal Translations: The graph of f (x − c) is f (x) shifted c units to the right.

f (x) f (x − c)

The graph of f (x + c) is f (x) shifted c units to the left.

f (x) f (x + c)

Vertical Translations: The graph of f (x) + c is f (x) shifted c units up.

f (x) f (x) + c

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SLIDE 6

Translations

Horizontal Translations: The graph of f (x − c) is f (x) shifted c units to the right.

f (x) f (x − c)

The graph of f (x + c) is f (x) shifted c units to the left.

f (x) f (x + c)

Vertical Translations: The graph of f (x) + c is f (x) shifted c units up.

f (x) f (x) + c

The graph of f (x) − c is f (x) shifted c units down.

f (x) f (x) − c

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SLIDE 7

Examples

  • 1. Graph y = x2 and y = (x +3)2.
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SLIDE 8

Examples

  • 1. Graph y = x2 and y = (x +3)2.

−4 −2 2 4 2 4 6 8 −6 −4 −2 2 4 2 4 6 8

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SLIDE 9

Examples

  • 1. Graph y = x2 and y = (x +3)2.

−4 −2 2 4 2 4 6 8 −6 −4 −2 2 4 2 4 6 8

  • 2. Graph y = √x and y = √x − 2
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SLIDE 10

Examples

  • 1. Graph y = x2 and y = (x +3)2.

−4 −2 2 4 2 4 6 8 −6 −4 −2 2 4 2 4 6 8

  • 2. Graph y = √x and y = √x − 2

−4 −2 2 4 6 8 0.5 1 1.5 2 2.5 3 −4 −2 2 4 6 8 −2 −1.5 −1 −0.5 0.5 1

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SLIDE 11

Stretching/Shrinking

Horizontal Stretch/Shrink: If c is positive, f (cx) is f (x) stretched or shrunk by a factor

  • f 1

c from/towards the y-axis.

f (x) f (cx)

f (x) f (cx)

Vertical Stretch/Shrink:

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SLIDE 12

Stretching/Shrinking

Horizontal Stretch/Shrink: If c is positive, f (cx) is f (x) stretched or shrunk by a factor

  • f 1

c from/towards the y-axis.

f (x) f (cx)

f (x) f (cx)

Vertical Stretch/Shrink: If c is positive, cf (x) is f (x) stretched or shrunk by a factor

  • f c from/towards the x-axis.

cf (x) f (x)

f (x) cf (x)

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SLIDE 13

Examples

  • 1. Graph y = x3 and y = ( 1

2x)3.

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SLIDE 14

Examples

  • 1. Graph y = x3 and y = ( 1

2x)3.

−4 −2 2 4 −2 2 −4 −2 2 4 −2 2

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SLIDE 15

Examples

  • 1. Graph y = x3 and y = ( 1

2x)3.

−4 −2 2 4 −2 2 −4 −2 2 4 −2 2

  • 2. Graph y = x3 and y = 2x3.
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SLIDE 16

Examples

  • 1. Graph y = x3 and y = ( 1

2x)3.

−4 −2 2 4 −2 2 −4 −2 2 4 −2 2

  • 2. Graph y = x3 and y = 2x3.

−4 −2 2 4 −2 2 −4 −2 2 4 −6 −4 −2 2 4 6

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SLIDE 17

Reflection

Horizontal Reflection: f (−x) is f (x) reflected across the y-axis. Vertical Reflection:

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SLIDE 18

Reflection

Horizontal Reflection: f (−x) is f (x) reflected across the y-axis. Vertical Reflection: −f (x) is f (x) reflected across the x-axis.

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SLIDE 19

Examples

Graph y = √x and y = √−x and y = −√x.

−2 2 4 6 8 10 0.5 1 1.5 2 2.5 3

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SLIDE 20

Examples

Graph y = √x and y = √−x and y = −√x.

−2 2 4 6 8 10 0.5 1 1.5 2 2.5 3 −10 −8 −6 −4 −2 2 0.5 1 1.5 2 2.5 3

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SLIDE 21

Examples

Graph y = √x and y = √−x and y = −√x.

−2 2 4 6 8 10 0.5 1 1.5 2 2.5 3 −10 −8 −6 −4 −2 2 0.5 1 1.5 2 2.5 3 −2 2 4 6 8 10 −3 −2.5 −2 −1.5 −1 −0.5

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SLIDE 22

“Starting” Functions

y = x2 y = x3 y = 1

x

y = √x y =

3

√x y = |x|

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SLIDE 23

Graphing Equations with Multiple Transformations

Let’s say we want to graph y = 1

2

√4 − x + 1

  • 1. Identify the “starting” function.

y = √x

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SLIDE 24

Graphing Equations with Multiple Transformations

Let’s say we want to graph y = 1

2

√4 − x + 1

  • 1. Identify the “starting” function.

y = √x

  • 2. Compare the “starting” function with the final function. Anything

that’s changed where the x is in the starting function will correspond to a horizontal transformation. Anything else will correspond to a vertical transformation. y =

  • x

vs y = 1 2

  • 4 − x + 1

Horizontal Vertical

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SLIDE 25

Graphing Equations with Multiple Transformations (continued)

  • 3. Graph the horizontal transformations, by going against the order of
  • perations.

−10 −5 5 10 0.5 1 1.5 2 2.5 3

y = √x

−10 −5 5 10 0.5 1 1.5 2 2.5 3

y = √x + 4 = √4 + x

−10 −5 5 10 0.5 1 1.5 2 2.5 3

y = √4 − x

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SLIDE 26

Graphing Equations with Multiple Transformations (continued)

  • 4. Graph the vertical transformations, by following the order of
  • perations.

−10 −5 5 10 0.5 1 1.5 2 2.5 3

y = 1

2

√4 − x

−10 −5 5 10 0.5 1 1.5 2 2.5 3

y = 1

2

√4 − x + 1

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SLIDE 27

Examples

  • 1. Graph y = (x − 1)2 − 3
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SLIDE 28

Examples

  • 1. Graph y = (x − 1)2 − 3

−4 −2 2 4 −2 2 4 6

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SLIDE 29

Examples

  • 1. Graph y = (x − 1)2 − 3

−4 −2 2 4 −2 2 4 6

  • 2. Graph y = −2√x + 2
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SLIDE 30

Examples

  • 1. Graph y = (x − 1)2 − 3

−4 −2 2 4 −2 2 4 6

  • 2. Graph y = −2√x + 2

−4 −2 2 4 6 8 10 −6 −4 −2