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FUNCTIONS : Translations and Transformations Translation – the “shifting” of a function

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FUNCTIONS : Translations and Transformations

Translation – the “shifting” of a function

Transformation – the “stretching” or “shrinking” of a function

Shifts the function left

Shifts the function right

Shifts the function up

Shifts the function down

Stretches / shrinks the function vertically

Stretches / shrinks the function horizontally

FUNCTIONS : Translations and Transformations

The easiest way to describe these is to just show you an example with a few rules :

Rule #1 – if the “change” is outside parentheses, you change the y

coordinate by the exact operation that is given.

Rule #2 – if the “change” is inside parentheses, you change the x

coordinate by the opposite operation that is given

FUNCTIONS : Translations and Transformations

Rule #1 – if the “change” is outside parentheses, you change the y

coordinate by the exact operation that is given.

Rule #2 – if the “change” is inside parentheses, you change the x

coordinate by the opposite operation that is given

EXAMPLE : ƒ( x ) = 3x + 2, find ƒ ( x - 3 ) for the given values.

FUNCTIONS : Translations and Transformations

Rule #1 – if the “change” is outside parentheses, you change the y

coordinate by the exact operation that is given.

Rule #2 – if the “change” is inside parentheses, you change the x

coordinate by the opposite operation that is given

EXAMPLE : ƒ( x ) = 3x + 2, find ƒ ( x - 3 ) for the given values.

** since the change is inside parens, we will add 3 to all x’s

ƒ(x)

ƒ(x - 3)

FUNCTIONS : Translations and Transformations

Rule #1 – if the “change” is outside parentheses, you change the y

coordinate by the exact operation that is given.

Rule #2 – if the “change” is inside parentheses, you change the x

coordinate by the opposite operation that is given

EXAMPLE : ƒ( x ) = 3x + 2, find ƒ ( x - 3 ) for the given values.

** since the change is inside parens, we will add 3 to all x’s

ƒ(x)

ƒ(x - 3)

FUNCTIONS : Translations and Transformations

Rule #1 – if the “change” is outside parentheses, you change the y

coordinate by the exact operation that is given.

Rule #2 – if the “change” is inside parentheses, you change the x

coordinate by the opposite operation that is given

EXAMPLE : ƒ( x ) = 3x + 2, find ƒ ( x - 3 ) for the given values.

** notice that y stays the same

ƒ(x)

ƒ(x - 3)

FUNCTIONS : Translations and Transformations

Rule #1 – if the “change” is outside parentheses, you change the y

coordinate by the exact operation that is given.

Rule #2 – if the “change” is inside parentheses, you change the x

coordinate by the opposite operation that is given

EXAMPLE : ƒ( x ) = x2 - 4, find ƒ ( x ) + 5 for the given values.

** change is outside, so add 5 to y

ƒ(x)

ƒ(x) + 5

FUNCTIONS : Translations and Transformations

Rule #1 – if the “change” is outside parentheses, you change the y

coordinate by the exact operation that is given.

Rule #2 – if the “change” is inside parentheses, you change the x

coordinate by the opposite operation that is given

EXAMPLE : ƒ( x ) = x2 - 4, find ƒ ( x ) + 5 for the given values.

** change is outside, so add 5 to y

ƒ(x)

ƒ(x) + 5

FUNCTIONS : Translations and Transformations

Rule #1 – if the “change” is outside parentheses, you change the y

coordinate by the exact operation that is given.

Rule #2 – if the “change” is inside parentheses, you change the x

coordinate by the opposite operation that is given

EXAMPLE : ƒ( x ) = x2 - 4, find ƒ ( x ) + 5 for the given values.

** notice that x stays the same

ƒ(x)

ƒ(x) + 5

FUNCTIONS : Translations and Transformations

- Example : Given the coordinate ( 3 , -1 ), perform each translation
- or transformation.
- ƒ ( x + 2 ) =
- ƒ ( 2x ) =
- ƒ ( x ) – 5 =
- 3 ƒ(x) =
- ½ ƒ(x) =
- ƒ =
- ƒ ( x – 6 ) =
- ƒ ( x ) + 1 =

FUNCTIONS : Translations and Transformations

- Example : Given the coordinate ( 3 , -1 ), perform each translation
- or transformation.
- ƒ ( x + 2 ) = ( 1 , -1 )- inside, change x by subtracting 2
- ƒ ( 2x ) =
- ƒ ( x ) – 5 =
- 3 ƒ(x) =
- ½ ƒ(x) =
- ƒ =
- ƒ ( x – 6 ) =
- ƒ ( x ) + 1 =

FUNCTIONS : Translations and Transformations

- Example : Given the coordinate ( 3 , -1 ), perform each translation
- or transformation.
- ƒ ( x + 2 ) = ( 1 , -1 )- inside, change x by subtracting 2
- ƒ ( 2x ) = ( 3/2 , -1 )- inside, change x by dividing by2
- ƒ ( x ) – 5 =
- 3 ƒ(x) =
- ½ ƒ(x) =
- ƒ =
- ƒ ( x – 6 ) =
- ƒ ( x ) + 1 =

FUNCTIONS : Translations and Transformations

- Example : Given the coordinate ( 3 , -1 ), perform each translation
- or transformation.
- ƒ ( x + 2 ) = ( 1 , -1 )- inside, change x by subtracting 2
- ƒ ( 2x ) = ( 3/2 , -1 )- inside, change x by dividing by 2
- ƒ ( x ) – 5 = ( 3 , - 6 )- outside, change y by subtracting 5
- 3 ƒ(x) =
- ½ ƒ(x) =
- ƒ =
- ƒ ( x – 6 ) =
- ƒ ( x ) + 1 =

FUNCTIONS : Translations and Transformations

- Example : Given the coordinate ( 3 , -1 ), perform each translation
- or transformation.
- ƒ ( x + 2 ) = ( 1 , -1 )- inside, change x by subtracting 2
- ƒ ( 2x ) = ( 3/2 , -1 )- inside, change x by dividing by 2
- ƒ ( x ) – 5 = ( 3 , - 6 )- outside, change y by subtracting 5
- 3 ƒ(x) = ( 3 , - 3 )- outside, change y by multiplying by3
- ½ ƒ(x) =
- ƒ =
- ƒ ( x – 6 ) =
- ƒ ( x ) + 1 =

FUNCTIONS : Translations and Transformations

- Example : Given the coordinate ( 3 , -1 ), perform each translation
- or transformation.
- ƒ ( x + 2 ) = ( 1 , -1 )- inside, change x by subtracting 2
- ƒ ( 2x ) = ( 3/2 , -1 )- inside, change x by dividing by 2
- ƒ ( x ) – 5 = ( 3 , - 6 )- outside, change y by subtracting 5
- 3 ƒ(x) = ( 3 , - 3 )- outside, change y by multiplying by 3
- ½ ƒ(x) = ( 3 , -1/2 )- outside, change y by dividing by 2
- ƒ =
- ƒ ( x – 6 ) =
- ƒ ( x ) + 1 =

FUNCTIONS : Translations and Transformations

- Example : Given the coordinate ( 3 , -1 ), perform each translation
- or transformation.
- ƒ ( x + 2 ) = ( 1 , -1 )- inside, change x by subtracting 2
- ƒ ( 2x ) = ( 3/2 , -1 )- inside, change x by dividing by 2
- ƒ ( x ) – 5 = ( 3 , - 6 )- outside, change y by subtracting 5
- 3 ƒ(x) = ( 3 , - 3 )- outside, change y by multiplying by 3
- ½ ƒ(x) = ( 3 , -1/2 )- outside, change y by dividing by 2
- ƒ = ( 2 , -1 )- inside, change x by multiplying by 2/3
- ƒ ( x – 6 ) =
- ƒ ( x ) + 1 =

FUNCTIONS : Translations and Transformations

- Example : Given the coordinate ( 3 , -1 ), perform each translation
- or transformation.
- ƒ ( x + 2 ) = ( 1 , -1 )- inside, change x by subtracting 2
- ƒ ( 2x ) = ( 3/2 , -1 )- inside, change x by dividing by 2
- ƒ ( x ) – 5 = ( 3 , - 6 )- outside, change y by subtracting 5
- 3 ƒ(x) = ( 3 , - 3 )- outside, change y by multiplying by 3
- ½ ƒ(x) = ( 3 , -1/2 )- outside, change y by dividing by 2
- ƒ = ( 2 , -1 )- inside, change x by multiplying by 2/3
- ƒ ( x – 6 ) = ( 9 , -1 )- inside, change x by adding 6
- ƒ ( x ) + 1 =

FUNCTIONS : Translations and Transformations

- Example : Given the coordinate ( 3 , -1 ), perform each translation
- or transformation.
- ƒ ( x + 2 ) = ( 1 , -1 )- inside, change x by subtracting 2
- ƒ ( 2x ) = ( 3/2 , -1 )- inside, change x by dividing by 2
- ƒ ( x ) – 5 = ( 3 , - 6 )- outside, change y by subtracting 5
- 3 ƒ(x) = ( 3 , - 3 )- outside, change y by multiplying by 3
- ½ ƒ(x) = ( 3 , -1/2 )- outside, change y by dividing by 2
- ƒ = ( 2 , -1 )- inside, change x by multiplying by 2/3
- ƒ ( x – 6 ) = ( 9 , -1 )- inside, change x by adding 6
- ƒ ( x ) + 1 = ( 3 , 0 )- outside, change y by adding 1

FUNCTIONS : Translations and Transformations

Graphing1. Find coordinates for the original function

by picking some x’s

2. Create an x/y table with the “change”

EXAMPLE : Graph ƒ ( x – 4 ) if ƒ(x) = 3x2 - 1

FUNCTIONS : Translations and Transformations

Graphing1. Find coordinates for the original function

by picking some x’s

2. Create an x/y table with the “change”

EXAMPLE : Graph ƒ ( x – 4 ) if ƒ(x) = 3x2 - 1

ƒ(x) = 3x2 - 1

FUNCTIONS : Translations and Transformations

Graphing1. Find coordinates for the original function

by picking some x’s

2. Create an x/y table with the “change”

EXAMPLE : Graph ƒ ( x – 4 ) if ƒ(x) = 3x2 - 1

ƒ(x) = 3x2 - 1

FUNCTIONS : Translations and Transformations

Graphing1. Find coordinates for the original function

by picking some x’s

2. Create an x/y table with the “change”

EXAMPLE : Graph ƒ ( x – 4 ) if ƒ(x) = 3x2 - 1

ƒ( x – 4 )

ƒ(x) = 3x2 - 1

- inside, change x by adding 4

FUNCTIONS : Translations and Transformations

Graphing1. Find coordinates for the original function

by picking some x’s

2. Create an x/y table with the “change”

3. Graph the new coordinate set

EXAMPLE : Graph ƒ ( x – 4 ) if ƒ(x) = 3x2 - 1

ƒ( x – 4 )

ƒ(x) = 3x2 - 1