1.5 Combining Functions and Shifting Graphs

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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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Ch. 1: Preliminaies 1.5 Combining Functions and Shifting Graphs

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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



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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


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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Solution : :

+

=

:

= . = ( )( )

= : +: :

Solution : < < :

/:

/:

: =

:

/: <

/:


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



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h li i i bi i i hif i d li h


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Solution

Solution

Solution


Ch P li i i C bi i F i Shif i d S li G h


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Solution


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


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1.5 Combining Functions and Shifting Graphs