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

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.

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

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  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 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 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 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 ) = x2 - 4, find ƒ ( x ) + 5 for the given values. ** change is outside, so add 5 to y 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 ) = x2 - 4, find ƒ ( x ) + 5 for the given values. ** change is outside, so add 5 to y 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 ) = x2 - 4, find ƒ ( x ) + 5 for the given values. ** notice that x stays the same ƒ(x) ƒ(x) + 5

  10. 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 =

  11. 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 =

  12. 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 =

  13. 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 =

  14. 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 =

  15. 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 =

  16. 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 =

  17. 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 =

  18. 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

  19. FUNCTIONS : Translations and Transformations Graphing 1. 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

  20. FUNCTIONS : Translations and Transformations Graphing 1. 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

  21. FUNCTIONS : Translations and Transformations Graphing 1. 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

  22. FUNCTIONS : Translations and Transformations Graphing 1. 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

  23. FUNCTIONS : Translations and Transformations Graphing 1. 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

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