Chapter 2
Functions and Graphs
Section 2
Elementary Functions: Graphs and
Transformations
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Learning Objectives for Section 2.2
The student will become familiar with a beginning library of elementary functions.
The student will be able to transform functions using vertical and horizontal shifts.
The student will be able to transform functions using reflections, stretches, and shrinks.
The student will be able to graph piecewise-defined functions.
Elementary Functions; Graphs and Transformations
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Identity Function
Domain: All reals (-, )Range: All reals (-, )
f (x) x
(πππ¦πππππ‘π hπ ππ€ππππππ)
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Square Function
Domain: All reals (-, )Range: [0, β)
h(x) x2
(πππ¦πππππ‘π hπ ππ€ππππππ)
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Cube Function
Domain: All reals (-, )Range: All reals (-, )
m(x) x3
(πππ¦πππππ‘π hπ ππ€ππππππ)
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Square Root Function
Domain: [0, β)
Range: [0, β)
n(x) x
(πππ¦πππππ‘π hπ ππ€ππππππ)
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Cube Root Function
Domain: All reals (-, ) Range: All reals (-, )
p(x) x3
(πππ¦πππππ‘π hπ ππ€ππππππ)
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Absolute Value Function
Domain: All reals (-, ) Range: [0, β)
p(x) x
(πππ¦πππππ‘π hπ ππ€ππππππ)
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Transformations
Types of transformations performed on graphs:β’ Vertical shift (translation)β’ Horizontal shift (translation)β’ Vertical stretch/shrink (dilation)β’ Horizontal stretch/shrink (dilation)β’ Reflection
Each one can be determined by examining the equation of the graph.
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Vertical Shift
The graph of y = f(x) + h β’ Shifts the graph of y = f(x) up h units
The graph of y = f(x) - h β’ Shifts the graph of y = f(x) down h units
Graph y = |x|, y = |x| + 4, and y = |x| β 5.
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Vertical Shift
State the domain and range of each function.
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Domain & Range
y = |x|β’ D: (-, ) R: [0, )
y = |x| + 4β’ D: (-, ) R: [4, )
y = |x| β 5 D: (-, ) R: [-5, )
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Horizontal Shift
The graph of y = f(x + h) β’ Shifts the graph of y = f(x) left h units
The graph of y = f(x - h) β’ Shifts the graph of y = f(x) right h units
Graph y = |x|, y = |x + 4|, and y = |x β 5|.
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Horizontal Shift
State the domain and range of each function.
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Domain & Range
y = |x|β’ D: (-, ) R: [0, )
y = |x+4|β’ D: (-, ) R: [0, )
y = |x-5| D: (-, ) R: [0, )
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Vertical Stretching/Shrinking
The graph of y = Af(x) can be obtained from the graph ofy = f(x) by multiplying each y-coordinate of f(x) by A.
If A > 1, the result is a vertical stretch by a factor of A.
If 0 < A < 1, the result is a vertical shrink by a factor of A.
Graph y = |x|, y = 2|x|, and y = 0.5|x|
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Vertical Stretching/Shrinking
(2,2)
(2,1)
(2,4 )
Vertical shrink
π¦=2|π₯|Vertical stretch
π¦=0.5|π₯|
State the domain and range of each function.
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Domain & Range
y = |x|β’ D: (-, ) R: [0, )
y = 2|x|β’ D: (-, ) R: [0, )
y = 0.5|x| D: (-, ) R: [0, )
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Horizontal Stretching/Shrinking
The graph of y = f(cx) can be obtained from the graph ofy = f(x) by multiplying each x-coordinate by .
If c > 1, the result is a horizontal shrink by a factor of
If 0 < c < 1, the result is a horizontal stretch by a factor of
Graph , , and
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Horizontal Stretching/Shrinking
Barnett/Ziegler/Byleen Business Calculus 12e
x
y
π¦=βπ₯ (4,2)π¦=β2 π₯(2,2)
π¦=β0.5 π₯(8,2)
Horizontal shrink
Horizontal stretch
State the domain and range of each function.
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Domain & Range
β’ D: [0, ) R: [0, )
β’ D: [0, ) R: [0, )
D: [0, ) R: [0, )
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Reflections
Barnett/Ziegler/Byleen Business Calculus 12e
β’ The graph of y = -f(x) β’ Reflects the graph of y = f(x) over the x-axis.
β’ The graph of y = f(-x) β’ Reflects the graph of y = f(x) over the y-axis.
β’ Graph , and
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Reflections
Barnett/Ziegler/Byleen Business Calculus 12e
x
y
π¦=βπ₯
π¦=ββπ₯
π¦=ββπ₯
Reflected over x-axis
Reflected over y-axis
State the domain and range of each function.
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Domain & Range
β’ D: [0, ) R: [0, )
β’ D: [0, ) R: (-, 0]
D: (-, 0] R: [0, )
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Multiple Transformations
It is common for a graph to have multiple transformations. Itβs important to know what the parent looks like so you
can perform each transformation on it.
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Example 1
Describe the transformations for the function: y = -|x + 3|
y = |x| shifted left 3, reflected over x-axis
Barnett/Ziegler/Byleen Business Calculus 12e
π¦=ΒΏ π₯β¨ΒΏ π¦=ββ¨π₯+3β¨ΒΏ
x
y
x
y
π· : (β β , β )π :ΒΏ π· : (β β , β )π :ΒΏ
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Example 2
Describe the transformations for : y = (x β 5)2 + 4
Barnett/Ziegler/Byleen Business Calculus 12e
π¦=π₯2 π¦=(π₯β5)2+4
π¦=π₯2, hπ πππ‘ππ hπππ π‘ 5 ,π’π 4
π· : (β β , β )π :ΒΏ π· : (β β , β )π :ΒΏ
x
y
x
y
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Example 3
Describe the transformations for : y =
, shifted left 4, stretched vertically by 2, down 3
Barnett/Ziegler/Byleen Business Calculus 12e
π¦=βπ₯ π¦=2βπ₯+7 β 3
x
y
x
y
π· : [ 0 , β )π :ΒΏ π· : [ β 4 , β )π :ΒΏ
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Writing Equations of Functions
Barnett/Ziegler/Byleen Business Calculus 12e
Write the equation for each function described:1. shifted left 3 units, reflected over the x-axis and
shifted down 7 units.Answer:
2. stretched horizontally by a factor of 6, shifted up 2 units.
Answer: 3. shifted right 5 units, stretched vertically by a
factor of 4. Answer:
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Piecewise-Defined Functions
Functions whose definitions involve more than one rule for different parts of its domain are called piecewise-defined functions.
Graphing one of these functions involves graphing each rule over the appropriate portion of the domain.
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Example of a Piecewise-Defined Function
Graph the function
Notice that the point (2,0) is included but the point (2, β2) is not.
2if2
2if22)(
xx
xxxf
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Piecewise Practice
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hπΊπππ π (π₯ )={ ββπ₯ π₯>4(π₯+1)2π₯<2
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x
y
hπΊπππ π (π₯ )={ ββπ₯ π₯>4(π₯+1)2π₯<2
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