section 2.5 transformation of functions
DESCRIPTION
Section 2.5 Transformation of Functions. Graphs of Common Functions. Reciprocal Function. Vertical Shifts. Vertical Shifts. Example. Use the graph of f(x)=|x| to obtain g(x)=|x|-2. Horizontal Shifts. Horizontal Shifts. Example. Use the graph of f(x)=x 2 to obtain g(x)=(x+1) 2. - PowerPoint PPT PresentationTRANSCRIPT
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Section 2.5Transformation of Functions
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Graphs of Common Functions
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x
y
Reciprocal Function
Domain: - ,0 0,
Range: - ,0 0,
Decreasing on - ,0 0,
Odd function
and
1( )f xx
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Vertical Shifts
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Vertical ShiftsLet be a function and be a positive real number. The graph of is the graph of shifted units
vertically upward. The graph of is the graph of shifted
f cy f x c y f x c
y f x c y f x c
units
vertically downward.
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Vertical Shifts
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Example
Use the graph of f(x)=|x| to obtain g(x)=|x|-2
x
y
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Horizontal Shifts
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Horizontal ShiftsLet be a function and a positive real number. The graph of is the graph of shifted
to the left units. The graph of is the graph of shifted
to the
f cy f x c y f x
cy f x c y f x
right units.c
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Horizontal Shifts
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Example
Use the graph of f(x)=x2 to obtain g(x)=(x+1)2
x
y
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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)=(x+1)2+2
x
y
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Reflections of Graphs
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Refection about the -AxisThe graph of is the graph of reflected
about the -axis.
xy f x y f x
x
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Reflections about the x-axis
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Reflection about the y-AxisThe graph of is the graph of reflected about - axis.
y f x y f xy
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Example
Use the graph of f(x)=x3 to obtain the graph of g(x)= (-x)3.
x
y
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Example
x
y
Use the graph of f(x)= x to graph g(x)=- x
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Vertical Stretching and Shrinking
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Vertically Shrinking
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Vertically Stretching
x
y
x
yGraph 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)= 2|x|
x
y
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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
x
y
Use the graph of f(x)= to obtain the
1graph of g(x)=3
x
x
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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:
1. Horizontal shifting
2. Stretching or shrinking
3. Reflecting
4. Vertical shifting
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Summary of Transformations
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A Sequence of Transformations
Move the graph to the left 3 units
Starting graph.
Stretch the graph vertically by 2.
Shift down 1 unit.
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Example
x
y
1Given the graph of f(x) below, graph ( 1).2f x
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Example
x
y
Given the graph of f(x) below, graph - ( 2) 1.f x
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Example
Given the graph of f(x) below, graph 2 ( ) 1.f x
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(a)
(b)
(c)
(d)
x
y
Use the graph of f(x)= x to graph g(x)= -x.The graph of g(x) will appear in which quadrant?
Quadrant IQuadrant IIQuadrant IIIQuadrant IV
( )f x x
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(a)
(b)
(c)
(d)
x
y
Write the equation of the given graph g(x). The original function was f(x) =x2
g(x)
2
2
2
2
( ) ( 4) 3
( ) ( 4) 3
( ) ( 4) 3
( ) ( 4) 3
g x x
g x x
g x x
g x x
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(a)
(b)
(c)
(d)
Write the equation of the given graph g(x). The original function was f(x) =|x|
g(x)
( ) 4
( ) 4
( ) 4
( ) 4
g x x
g x x
g x x
g x x
x
y