Section 2.5 Transformation of Functions

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Section 2.5Transformation of Functions
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• Graphs of Common Functions

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Reciprocal Function
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• Vertical Shifts

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Vertical Shifts
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Example
Use the graph of f(x)x to obtain g(x)x-2
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• Horizontal Shifts

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Horizontal Shifts
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Example
Use the graph of f(x)x2 to obtain g(x)(x1)2
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Combining Horizontal and Vertical Shifts
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Example
Use the graph of f(x)x2 to obtain g(x)(x1)22
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• Reflections of Graphs

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Reflections about the x-axis
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Example
Use the graph of f(x)x3 to obtain the graph of
g(x) (-x)3.
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Example
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• Vertical Stretching and Shrinking

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Vertically Shrinking
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Vertically Stretching
Graph of f(x)x3
Graph of g(x)3x3
This is vertical stretching each y coordinate
is multiplied by 3 to stretch the graph.
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Example
Use the graph of f(x)x to graph g(x) 2x
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• Horizontal Stretching and Shrinking

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Horizontal Shrinking
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Horizontal Stretching
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Example
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• Sequences of Transformations

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• A function involving more than one transformation
  can be graphed by performing transformations in
  the following order
• Horizontal shifting
• Stretching or shrinking
• Reflecting
• Vertical shifting

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Summary of Transformations
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A Sequence of Transformations
Starting graph.
Move the graph to the left 3 units
Stretch the graph vertically by 2.
Shift down 1 unit.
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Example
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Example
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Example
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(a) (b) (c) (d)
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g(x)
Write the equation of the given graph g(x). The
original function was f(x) x2
(a) (b) (c) (d)
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g(x)
Write the equation of the given graph g(x). The
original function was f(x) x
(a) (b) (c) (d)