1. 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

2. 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

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.

4. 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 xy -3-7 02 15 517 xy -7 2 5 17 ƒ(x)ƒ(x - 3)

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 ) = 3x + 2, find ƒ ( x - 3 ) for the given values. ** since the change is inside parens, we will add 3 to all x’s xy -3-7 02 15 517 x+3y 0-7 32 45 817 ƒ(x)ƒ(x - 3)

6. 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 xy -3-7 02 15 517 x+3y 0-7 32 45 817 ƒ(x)ƒ(x - 3)

7. 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 ) = x 2 - 4, find ƒ ( x ) + 5 for the given values. ** change is outside, so add 5 to y xy -20 0-4 1-3 412 xy -2 0 1 4 ƒ(x)ƒ(x) + 5

8. 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 ) = x 2 - 4, find ƒ ( x ) + 5 for the given values. ** change is outside, so add 5 to y xy -20 0-4 1-3 412 xy+5 -25 01 12 417 ƒ(x)ƒ(x) + 5

9. 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 ) = x 2 - 4, find ƒ ( x ) + 5 for the given values. ** notice that x stays the same xy -20 0-4 1-3 412 xy+5 -25 01 12 417 ƒ(x)ƒ(x) + 5

10. FUNCTIONS : Translations and Transformations Example : Given the coordinate ( 3, -1 ), perform each translation or transformation. a)ƒ ( x + 2 ) = b)ƒ ( 2x ) = c)ƒ ( x ) – 5 = d)3 ƒ(x) = e)½ ƒ(x) = f)ƒ = g)ƒ ( x – 6 ) = h)ƒ ( x ) + 1 =

11. FUNCTIONS : Translations and Transformations Example : Given the coordinate ( 3, -1 ), perform each translation or transformation. a)ƒ ( x + 2 ) = ( 1, -1 )- inside, change x by subtracting 2 b)ƒ ( 2x ) = c)ƒ ( x ) – 5 = d)3 ƒ(x) = e)½ ƒ(x) = f)ƒ = g)ƒ ( x – 6 ) = h)ƒ ( x ) + 1 =

12. FUNCTIONS : Translations and Transformations Example : Given the coordinate ( 3, -1 ), perform each translation or transformation. a)ƒ ( x + 2 ) = ( 1, -1 )- inside, change x by subtracting 2 b)ƒ ( 2x ) = ( 3/2, -1 )- inside, change x by dividing by 2 c)ƒ ( x ) – 5 = d)3 ƒ(x) = e)½ ƒ(x) = f)ƒ = g)ƒ ( x – 6 ) = h)ƒ ( x ) + 1 =

13. FUNCTIONS : Translations and Transformations Example : Given the coordinate ( 3, -1 ), perform each translation or transformation. a)ƒ ( x + 2 ) = ( 1, -1 )- inside, change x by subtracting 2 b)ƒ ( 2x ) = ( 3/2, -1 )- inside, change x by dividing by 2 c)ƒ ( x ) – 5 = ( 3, - 6 )- outside, change y by subtracting 5 d)3 ƒ(x) = e)½ ƒ(x) = f)ƒ = g)ƒ ( x – 6 ) = h)ƒ ( x ) + 1 =

14. FUNCTIONS : Translations and Transformations Example : Given the coordinate ( 3, -1 ), perform each translation or transformation. a)ƒ ( x + 2 ) = ( 1, -1 )- inside, change x by subtracting 2 b)ƒ ( 2x ) = ( 3/2, -1 )- inside, change x by dividing by 2 c)ƒ ( x ) – 5 = ( 3, - 6 )- outside, change y by subtracting 5 d)3 ƒ(x) = ( 3, - 3 )- outside, change y by multiplying by 3 e)½ ƒ(x) = f)ƒ = g)ƒ ( x – 6 ) = h)ƒ ( x ) + 1 =

15. FUNCTIONS : Translations and Transformations Example : Given the coordinate ( 3, -1 ), perform each translation or transformation. a)ƒ ( x + 2 ) = ( 1, -1 )- inside, change x by subtracting 2 b)ƒ ( 2x ) = ( 3/2, -1 )- inside, change x by dividing by 2 c)ƒ ( x ) – 5 = ( 3, - 6 )- outside, change y by subtracting 5 d)3 ƒ(x) = ( 3, - 3 )- outside, change y by multiplying by 3 e)½ ƒ(x) = ( 3, -1/2 )- outside, change y by dividing by 2 f)ƒ = g)ƒ ( x – 6 ) = h)ƒ ( x ) + 1 =

16. FUNCTIONS : Translations and Transformations Example : Given the coordinate ( 3, -1 ), perform each translation or transformation. a)ƒ ( x + 2 ) = ( 1, -1 )- inside, change x by subtracting 2 b)ƒ ( 2x ) = ( 3/2, -1 )- inside, change x by dividing by 2 c)ƒ ( x ) – 5 = ( 3, - 6 )- outside, change y by subtracting 5 d)3 ƒ(x) = ( 3, - 3 )- outside, change y by multiplying by 3 e)½ ƒ(x) = ( 3, -1/2 )- outside, change y by dividing by 2 f)ƒ = ( 2, -1 )- inside, change x by multiplying by 2/3 g)ƒ ( x – 6 ) = h)ƒ ( x ) + 1 =

17. FUNCTIONS : Translations and Transformations Example : Given the coordinate ( 3, -1 ), perform each translation or transformation. a)ƒ ( x + 2 ) = ( 1, -1 )- inside, change x by subtracting 2 b)ƒ ( 2x ) = ( 3/2, -1 )- inside, change x by dividing by 2 c)ƒ ( x ) – 5 = ( 3, - 6 )- outside, change y by subtracting 5 d)3 ƒ(x) = ( 3, - 3 )- outside, change y by multiplying by 3 e)½ ƒ(x) = ( 3, -1/2 )- outside, change y by dividing by 2 f)ƒ = ( 2, -1 )- inside, change x by multiplying by 2/3 g)ƒ ( x – 6 ) = ( 9, -1 )- inside, change x by adding 6 h)ƒ ( x ) + 1 =

18. FUNCTIONS : Translations and Transformations Example : Given the coordinate ( 3, -1 ), perform each translation or transformation. a)ƒ ( x + 2 ) = ( 1, -1 )- inside, change x by subtracting 2 b)ƒ ( 2x ) = ( 3/2, -1 )- inside, change x by dividing by 2 c)ƒ ( x ) – 5 = ( 3, - 6 )- outside, change y by subtracting 5 d)3 ƒ(x) = ( 3, - 3 )- outside, change y by multiplying by 3 e)½ ƒ(x) = ( 3, -1/2 )- outside, change y by dividing by 2 f)ƒ = ( 2, -1 )- inside, change x by multiplying by 2/3 g)ƒ ( x – 6 ) = ( 9, -1 )- inside, change x by adding 6 h)ƒ ( x ) + 1 = ( 3, 0 )- outside, change y by adding 1

19. 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) = 3x 2 - 1

20. 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) = 3x 2 - 1 xf(x) -2 0 1 2 ƒ(x) = 3x 2 - 1

21. 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) = 3x 2 - 1 xf(x) -211 2 0 12 211 ƒ(x) = 3x 2 - 1

22. 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) = 3x 2 - 1 xf(x) -211 2 0 12 211 ƒ(x) = 3x 2 - 1 ƒ( x – 4 ) xf(x) 211 32 4 52 611 - inside, change x by adding 4

23. 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) = 3x 2 - 1 xf(x) -211 2 0 12 211 ƒ(x) = 3x 2 - 1 ƒ( x – 4 ) xf(x) 211 32 4 52 611