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Graphical Transformations!!! Sec. 1.5a is amazing!!! New Definitions Transformations – functions that map real numbers to real numbers Rigid Transformations – leave the size and shape of a graph unchanged (includes translations and reflections) Non-rigid Transformations – generally distort the shape of a graph (includes stretches and shrinks) New Definitions Rigid Transformations Vertical Translation – of the graph of y = f(x) is a shift of the graph up or down in the coordinate plane Horizontal Translation – a shift of the graph to the left or the right Translations Let c be a positive real number. Then the following transformations result in translations of the graph of y = f(x): Horizontal Translations y = f(x – c) a translation to the right by c units y = f(x + c) a translation to the left by c units Vertical Translations y = f(x) + c a translation up by c units y = f(x) – c a translation down by c units Each figure shows the graph of the original square root function, along with a translation function. Write an equation for each translation. y x5 y x 4 y x 1 New Definitions Rigid Transformations Points (x, y) and (x, –y) are reflections of each other across the x-axis. Points (x, y) and (–x, y) are reflections of each other across the y-axis. (–x, y) (x, y) (x, –y) Reflections The following transformations result in the reflections of the graph of y = f(x): Across the x-axis y = – f(x) Across the y-axis y = f(–x) Find an equation for the reflection of the given function across each axis: 5x 9 f x 2 x 3 Across the x-axis: Across the y-axis: 5x 9 9 5x y f x 2 2 x 3 x 3 5x 9 5 x 9 y f x 2 2 x 3 x 3 Let’s support our algebraic work graphically… On to vertical and horizontal stretches and shrinks… Stretches and Shrinks Let c be a positive real number. Then the following transformations result in stretches or shrinks of the graph of y = f(x): Horizontal Stretches or Shrinks x y f c a stretch by a factor of c a shrink by a factor of c if c > 1 if c < 1 Vertical Stretches or Shrinks y c f x a stretch by a factor of c a shrink by a factor of c if c > 1 if c < 1 3 Let C 1 be the curve defined by y 1 = f(x) = x – 16x. Find equations for the following non-rigid transformations of C 1 : 1. C 2 is a vertical stretch of C 1 by a factor of 3 y2 3 f x 3 x 16 x 3x3 48 x 3 2. C 3 is a horizontal shrink of C1 by a factor of 1/2 x 3 y3 f f 2 x 2 x 16 2 x 1/ 2 8 x 32 x 3 Let’s verify our algebraic work graphically… Whiteboard problems… Describe how the graph of y x can be transformed to the given equation. 3 y x Describe how the graph of can be transformed to the given equation. y x reflect across x-axis y 2 x3 vertical stretch of 2 y x5 shift right 5 y (2 x)3 horiz. shrink of ½ y x reflect across y-axis y (0.2 x)3 horiz. stretch of 5 y 3 x reflect across y-axis shift right 3 y 0.3x 3 vertical shrink of 0.3 More whiteboard problems… Describe how to transform the graph of f into the graph of g. f ( x) x 2 g ( x) ( x 3)2 g ( x) 2 x 4 1 right 6 g ( x) x 4 f ( x) ( x 1)2 Describe the translation of f ( x) x to reflect across x-axis, left 4 Reflected across x-axis Vertical stretch factor of 2 Shift left 4 Shift up 1 Homework: p. 139-140 1-23 odd