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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 What is the difference between and ? y = f ( x) + ay = f ( x) + ay = f ( x + a)y = f ( x + a) x f (x)f (x) + a+ a y x f (x)f (x) + a+ a y y = f ( x) + ay = f ( x) + a 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. Higher Maths 1 2 2 Transformations1

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Any change to the position of a graph is called a translation. Translation y = f ( x) + ay = f ( x) + a Slides the graph of vertically. f ( x)f ( x) a f ( x) + af ( x) + a f ( x)f ( x) y = f ( x + a)y = f ( x + a) Slides the graph of horizontally. f ( x)f ( x) a f ( x + a)f ( x + a) f ( x)f ( x) up down + a+ a – a– a left right + a+ a – a– a Higher Maths 1 2 2 Transformations2

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If the positive and negative coordinates of a graph are inverted, the graph will be reflected across either the x or y -axis. Reflection y = - f ( x ) Reflects the graph of f ( x ) vertically across the x -axis. - f ( x)- f ( x) f ( x)f ( x) y = f ( - x ) Reflects the graph of f ( x ) horizontally across the y -axis. f (- x)f (- x) f ( x)f ( x) Higher Maths 1 2 2 Transformations3

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If every coordinate is multiplied by the same value, the overall shape of the graph will be distorted. Distortion y = a f ( x)y = a f ( x) f ( x)f ( x) a f ( x)a f ( x) Changes the vertical size of the graph of f ( x ). a > 1 stretch a < 1 compress y = f (a x)y = f (a x) f ( x)f ( x) Changes the horizontal size of the graph of f ( x ). a < 1 stretch a > 1 compress f (a x)f (a x) Higher Maths 1 2 2 Transformations4

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The diagram below shows the graph of. Sketch. Sketching Composite Transformations Higher Maths 1 2 2 Transformations5 Example y = - 2 f ( x + 3 ) y = f ( x ) y = f ( x + 3 ) (3,0) ( - 2, - 2) ( - 5,4) (0, - 2) Remember to label all relevant coordinates. Important y = - 2 f ( x + 3 )

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6.7 Graphing Absolute Value Equations. Vertical Translations Below are the graphs of y = | x | and y = | x | + 2. Describe how the graphs are the same.

6.7 Graphing Absolute Value Equations. Vertical Translations Below are the graphs of y = | x | and y = | x | + 2. Describe how the graphs are the same.

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