Chapter 2 Functions and Graphs Section 2 Elementary Functions: Graphs and Transformations

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

(𝑘𝑒𝑦𝑝𝑜𝑖𝑛𝑡𝑠 h𝑠 𝑜𝑤𝑛𝑖𝑛𝑟𝑒𝑑)

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Square Function

Domain: All reals (-, )Range: [0, ∞)

h(x) x2

(𝑘𝑒𝑦𝑝𝑜𝑖𝑛𝑡𝑠 h𝑠 𝑜𝑤𝑛𝑖𝑛𝑟𝑒𝑑)

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Cube Function

Domain: All reals (-, )Range: All reals (-, )

m(x) x3

(𝑘𝑒𝑦𝑝𝑜𝑖𝑛𝑡𝑠 h𝑠 𝑜𝑤𝑛𝑖𝑛𝑟𝑒𝑑)

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Square Root Function

Domain: [0, ∞)

Range: [0, ∞)

n(x) x

(𝑘𝑒𝑦𝑝𝑜𝑖𝑛𝑡𝑠 h𝑠 𝑜𝑤𝑛𝑖𝑛𝑟𝑒𝑑)

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Cube Root Function

Domain: All reals (-, ) Range: All reals (-, )

p(x) x3

(𝑘𝑒𝑦𝑝𝑜𝑖𝑛𝑡𝑠 h𝑠 𝑜𝑤𝑛𝑖𝑛𝑟𝑒𝑑)

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Absolute Value Function

Domain: All reals (-, ) Range: [0, ∞)

p(x) x

(𝑘𝑒𝑦𝑝𝑜𝑖𝑛𝑡𝑠 h𝑠 𝑜𝑤𝑛𝑖𝑛𝑟𝑒𝑑)

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Transformations

Types of transformations performed on graphs:• Vertical shift (translation)• Horizontal shift (translation)• Vertical stretch/shrink (dilation)• Horizontal stretch/shrink (dilation)• 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 • Shifts the graph of y = f(x) up h units

The graph of y = f(x) - h • Shifts the graph of y = f(x) down h units

Graph y = |x|, y = |x| + 4, and y = |x| – 5.

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Vertical Shift

State the domain and range of each function.

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Domain & Range

y = |x|• D: (-, ) R: [0, )

y = |x| + 4• D: (-, ) R: [4, )

y = |x| – 5 D: (-, ) R: [-5, )

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Horizontal Shift

The graph of y = f(x + h) • Shifts the graph of y = f(x) left h units

The graph of y = f(x - h) • Shifts the graph of y = f(x) right h units

Graph y = |x|, y = |x + 4|, and y = |x – 5|.

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Horizontal Shift

State the domain and range of each function.

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Domain & Range

y = |x|• D: (-, ) R: [0, )

y = |x+4|• 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

𝑦=2|𝑥|Vertical stretch

𝑦=0.5|𝑥|

State the domain and range of each function.

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Domain & Range

y = |x|• D: (-, ) R: [0, )

y = 2|x|• 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

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x

y

𝑦=√𝑥 (4,2)𝑦=√2 𝑥(2,2)

𝑦=√0.5 𝑥(8,2)

Horizontal shrink

Horizontal stretch

State the domain and range of each function.

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Domain & Range

• D: [0, ) R: [0, )

• D: [0, ) R: [0, )

D: [0, ) R: [0, )

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Reflections

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• The graph of y = -f(x) • Reflects the graph of y = f(x) over the x-axis.

• The graph of y = f(-x) • Reflects the graph of y = f(x) over the y-axis.

• Graph , and

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Reflections

Barnett/Ziegler/Byleen Business Calculus 12e

x

y

𝑦=√𝑥

𝑦=−√𝑥

𝑦=√−𝑥

Reflected over x-axis

Reflected over y-axis

State the domain and range of each function.

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Domain & Range

• D: [0, ) R: [0, )

• D: [0, ) R: (-, 0]

D: (-, 0] R: [0, )

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Multiple Transformations

It is common for a graph to have multiple transformations. It’s 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

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𝑦=¿ 𝑥∨¿ 𝑦=−∨𝑥+3∨¿

x

y

x

y

𝐷 : (− ∞ , ∞ )𝑅 :¿ 𝐷 : (− ∞ , ∞ )𝑅 :¿

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Example 2

Describe the transformations for : y = (x – 5)2 + 4

Barnett/Ziegler/Byleen Business Calculus 12e

𝑦=𝑥2 𝑦=(𝑥−5)2+4

𝑦=𝑥2, h𝑠 𝑖𝑓𝑡𝑒𝑑 h𝑟𝑖𝑔 𝑡 5 ,𝑢𝑝 4

𝐷 : (− ∞ , ∞ )𝑅 :¿ 𝐷 : (− ∞ , ∞ )𝑅 :¿

x

y

x

y

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Example 3

Describe the transformations for : y =

, shifted left 4, stretched vertically by 2, down 3

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𝑦=√𝑥 𝑦=2√𝑥+7 − 3

x

y

x

y

𝐷 : [ 0 , ∞ )𝑅 :¿ 𝐷 : [ − 4 , ∞ )𝑅 :¿

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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, –2) is not.

2if2

2if22)(

xx

xxxf

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Piecewise Practice

Barnett/Ziegler/Byleen Business Calculus 12e

h𝐺𝑟𝑎𝑝 𝑓 (𝑥 )={ −√𝑥 𝑥>4(𝑥+1)2𝑥<2

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x

y

h𝐺𝑟𝑎𝑝 𝑓 (𝑥 )={ −√𝑥 𝑥>4(𝑥+1)2𝑥<2

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