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Higher Maths 1.2.2 - Graphs and Transformations

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Higher Maths 1.2.2 - Graphs and Transformations

  1. 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. 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. 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. 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. 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 )

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