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Ch. 1: Preliminaries 1.5 Combining Functions, Shifting and Scaling Graphs
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Sums, Differences, Products and Quotients
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Example Combining Functions Algebraically
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Composition Function
A function can be described as a "machine" or black box"
that for each input returns a corresponding output
A composite function g(f(x)) can be
visualized as the combination of
two "machines". The first takes
input x and outputs f(x). The
second takes f(x) and outputs
g(f(x)).
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Example Viewing a Function as a composite
Example Finding Formulas for composites
Solution
Ch. 1: Preliminaries 1.5 Combining Functions, Shifting and Scaling Graphs
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Example Shifting a Graph
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Example Shifting a Graph
Scaling and Reflecting a Graph of a Function
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Example Scaling and Reflecting a Graph
(a) Vertical = Multiply or divide the right hand by 3:
= stretches thegraph vertically by a factor 3
= (/) compresses the
graph vertically by a factor 3
(a) Horizontal
Multiply or divide every x by 3:
=
= compress the graph
horizontally by a factor 3
= (/) stretches
the graph horizontally
by a factor 3.
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(c) Reflection =
The graph of = is
a reflection ofy = across xais
The graph of = is
a reflection ofy = across yxais
Ch. 1: Preliminaries 1.5 Combining Functions, Shifting and Scaling Graphs
Type equation here.
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Example Combining Scaling and Reflection
Solution
(1) Substitute -2x for x in the right hand side to get the formula
(2) The formula is
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Ellipses
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In Figure (b) the line segment joining the points /, is
called major axis of the ellipse, the minor axis is the line segment
joining , .
The axes of the ellipse are reversed in Figure (c), the major axis isthe line segment joining the points , and the minor axis
is the line segment joining /, .
Major axis is the line segment having the longer length.
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Ch. 1: Preliminaries 1.5 Combining Functions, Shifting and Scaling Graphs
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Solution : :
+
=
:
= . = ( )( )
= : +: :
Solution : < < :
/:
/:
: =
:
/: <
/:
Ch. 1: Preliminaries 1.5 Combining Functions, Shifting and Scaling Graphs
Change the mistakes
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Solution
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Solution
Solution
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Solution Solution
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x y
20.0 0.0
15.0 0.1
10.0 0.1
5.0 0.1
0.0 0.5
-0.5 0.7
-1.0 1.0
-1.5 2.0
x y
-2.5 -2.0
-5.0 -0.3-10.0 -0.1
-15.0 -0.1
-20.0 -0.1
-25.0 0.0
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Solution
Ch. 1: Preliminaries 1.5 Combining Functions, Shifting and Scaling Graphs
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Ch. 1: Preliminaries 1.5 Combining Functions, Shifting and Scaling Graphs
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Ch. 1: Preliminaries 1.5 Combining Functions, Shifting and Scaling Graphs
h li i i bi i i hif i d li h
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Solution
Solution
Solution
Ch. 1: Preliminaries 1.5 Combining Functions, Shifting and Scaling Graphs
Ch P li i i C bi i F i Shif i d S li G h
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Solution
Ch. 1: Preliminaries 1.5 Combining Functions, Shifting and Scaling Graphs
Ch P li i i C bi i F i Shif i d S li G h
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Solution
( (
))
(
)
=
Solve 1 ---to---76
Ch. 1: Preliminaries 1.5 Combining Functions, Shifting and Scaling Graphs