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Published byAlfred Melton Modified over 9 years ago
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Graphical Transformations!!! Sec. 1.5a is amazing!!!
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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)
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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
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Translations Let c be a positive real number. Then the following trans- formations 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
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Each figure shows the graph of the original square root function, along with a translation function. Write an equation for each translation.
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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)
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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)
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Find an equation for the reflection of the given function across each axis: Across the x-axis: Across the y-axis: Let’s support our algebraic work graphically…
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On to vertical and horizontal stretches and shrinks…
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Stretches and Shrinks Let c be a positive real number. Then the following trans- formations result in stretches or shrinks of the graph of y = f(x): a stretch by a factor of cif c > 1 a shrink by a factor of cif c < 1 Horizontal Stretches or Shrinks a stretch by a factor of cif c > 1 a shrink by a factor of cif c < 1 Vertical Stretches or Shrinks
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Let C be the curve defined by y = f(x) = x – 16x. Find equations for the following non-rigid transformations of C : 1 1 1 3 1. C is a vertical stretch of C by a factor of 3 12 2. C is a horizontal shrink of C by a factor of 1/2 13 Let’s verify our algebraic work graphically…
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Whiteboard problems… Describe how the graph of can be transformed to the given equation. reflect across x-axis shift right 5 reflect across y-axis shift right 3 Describe how the graph of can be transformed to the given equation. vertical stretch of 2 horiz. shrink of ½ horiz. stretch of 5 vertical shrink of 0.3
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More whiteboard problems… Describe how to transform the graph of f into the graph of g. right 6 reflect across x-axis, left 4 Describe the translation of to Reflected across x-axis Vertical stretch factor of 2 Shift left 4 Shift up 1 Homework: p. 139-140 1-23 odd
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