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A  transformation  is an algebraic change to a function which affects the  shape  or  position  of the graph. Introduction to Transformations NOTE x f   ( x ) +   a y x f   ( x ) +   a y y   =   f   (   x )   +   a y   =   f   (   x   +   a ) Higher Maths  1  2  2  Transformations UNIT OUTCOME SLIDE PART What is the difference between  and  ? y   =   f   (   x )   +   a y   =   f   (   x   +   a ) Making a change  after the function will affect only the Making a change  before  the function will affect only the y x -coordinate. -coordinate.
Any change to the position of a graph is called a  translation . Translation NOTE y   =   f   (   x )   +   a a f   (   x )   +   a f   (   x ) x y y   =   f   (   x   +   a ) Slides the graph of  horizontally.   f   (   x ) a f   (   x   +   a ) f   (   x ) x y up down +   a –   a left right +   a –   a Higher Maths  1  2  2  Transformations UNIT OUTCOME SLIDE PART Slides the graph of  vertically.   f   (   x ) CAREFUL!
If the positive and negative coordinates of a graph are inverted, the graph will be  reflected  across either the   x   or   y -axis. Reflection NOTE y   =  -   f   (   x ) Reflects the graph of  f   (   x ) vertically  across the  x -axis. -   f   (   x ) f   (   x ) x y y   =  f   ( -   x ) Reflects the graph of  f   (   x ) horizontally  across the  y -axis. f   ( -   x ) f   (   x ) x y Higher Maths  1  2  2  Transformations UNIT OUTCOME SLIDE PART
If every coordinate is multiplied by the  same value , the overall shape of the graph will be  distorted . Distortion NOTE y   =   a   f   (   x ) f   (   x ) x y a   f   (   x ) Changes the  vertical  size of the graph of  f   (   x ) . a  >  1   ‘ stretch’ a  <  1   ‘ compress’ y   =   f   ( a   x ) f   (   x ) x y Changes the  horizontal  size of the graph of  f   (   x ) . a  <  1   ‘ stretch’ a  >  1   ‘ compress’ f   ( a   x ) Higher Maths  1  2  2  Transformations UNIT OUTCOME SLIDE PART CAREFUL!
The diagram below shows the graph of  . Sketch  . Sketching Composite Transformations NOTE Higher Maths  1  2  2  Transformations UNIT OUTCOME SLIDE PART Example y   =   - 2   f   (   x   +   3 ) y   =   f   ( x ) y   =   f   (   x ) y   =   f   (   x   +   3 ) y   =   f   (   x   +   3 ) (3,0) ( - 2, - 2) ( - 5,4) (0, - 2) x y x y Remember to label all relevant coordinates. Important y   =   - 2   f   (   x   +   3 )

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Higher Maths 1.2.2 - Graphs and Transformations

  • 1. A transformation is an algebraic change to a function which affects the shape or position of the graph. Introduction to Transformations NOTE x f ( x ) + a y x f ( x ) + a y y = f ( x ) + a y = f ( x + a ) Higher Maths 1 2 2 Transformations UNIT OUTCOME SLIDE PART What is the difference between and ? y = f ( x ) + a y = f ( x + a ) Making a change after the function will affect only the Making a change before the function will affect only the y x -coordinate. -coordinate.
  • 2. Any change to the position of a graph is called a translation . Translation NOTE y = f ( x ) + a a f ( x ) + a f ( x ) x y y = f ( x + a ) Slides the graph of horizontally. f ( x ) a f ( x + a ) f ( x ) x y up down + a – a left right + a – a Higher Maths 1 2 2 Transformations UNIT OUTCOME SLIDE PART Slides the graph of vertically. f ( x ) CAREFUL!
  • 3. If the positive and negative coordinates of a graph are inverted, the graph will be reflected across either the x or y -axis. Reflection NOTE y = - f ( x ) Reflects the graph of f ( x ) vertically across the x -axis. - f ( x ) f ( x ) x y y = f ( - x ) Reflects the graph of f ( x ) horizontally across the y -axis. f ( - x ) f ( x ) x y Higher Maths 1 2 2 Transformations UNIT OUTCOME SLIDE PART
  • 4. If every coordinate is multiplied by the same value , the overall shape of the graph will be distorted . Distortion NOTE y = a f ( x ) f ( x ) x y a f ( x ) Changes the vertical size of the graph of f ( x ) . a > 1 ‘ stretch’ a < 1 ‘ compress’ y = f ( a x ) f ( x ) x y Changes the horizontal size of the graph of f ( x ) . a < 1 ‘ stretch’ a > 1 ‘ compress’ f ( a x ) Higher Maths 1 2 2 Transformations UNIT OUTCOME SLIDE PART CAREFUL!
  • 5. The diagram below shows the graph of . Sketch . Sketching Composite Transformations NOTE Higher Maths 1 2 2 Transformations UNIT OUTCOME SLIDE PART Example y = - 2 f ( x + 3 ) y = f ( x ) y = f ( x ) y = f ( x + 3 ) y = f ( x + 3 ) (3,0) ( - 2, - 2) ( - 5,4) (0, - 2) x y x y Remember to label all relevant coordinates. Important y = - 2 f ( x + 3 )