1

Learning Objectives for Section 2.2

You will become familiar with some elementary functions.

You will be able to transform functions using vertical and horizontal shifts.

You will be able to transform functions using stretches and shrinks.

You will be able to graph piecewise-defined functions.

Elementary Functions; Graphs and Transformations

2

Six Graphs of Common Functions

___________________ Function ____________________ Function

Domain: ________________ Domain: ____________________

Range: ________________ Range: _____________________

1) ( )f x c 2) ( )f x x

3

•

Six Graphs of Common Functions (continued)

______________ Function _________________ Function

Domain: _________ Domain: _________

Range: _________ Range: _________

4) ( )f x x3) ( )f x x

4

Six Graphs of Common Functions (continued)

_______________ Function ____________ Function

Domain: _________ Domain: _________

Range: _________ Range: _________

25) ( )f x x 36) ( )f x x

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Vertical and Horizontal Translations

TRANSLATIONS, also called shifts, are simple transformations of the graph of a function whereby each point of the graph is translated (shifted) a certain number of units vertically and/or horizontally.

The shape of the graph remains the same.

Vertical translations are shifts upward or downward.

Horizontal translations are shifts to the right or to the left.

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Vertical and Horizontal Translations

ExampleGraph the function on your calculator and describe the transformation that the graph of must undergo to obtain the graph of h(x).

( ) 5

( ) 5

( ) 5

( ) 5

h x x

h x x

h x x

h x x

( )f x x

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Vertical and Horizontal Translations

Vertical and Horizontal Translations of the Graph of y = f(x)

For h, k positive real numbers:

1. Vertical shift k units UPWARD:

2. Vertical shift k units DOWNWARD:

3. Horizontal shift h units to the RIGHT:

4. Horizontal shift h units to the LEFT:

( ) ( )h x f x k

( ) ( )h x f x k

( ) ( )h x f x h

( ) ( )h x f x h

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Transformations of f(x) = x2

Given

22 4y x

Describe each transformation of the graph of y1.

23 4y x

2

4 4y x

2

5 4y x

21y x

9

Vertical and Horizontal Translations (continued)

Take a look*

Name the transformations that f(x) = x2 must undergo to obtain the graph of g(x) = (x + 3)2 – 5

f(x)

g(x)

Solution:

You can obtain the graph of g by translating f 3 units ___________

and 5 units ____________________.

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Reflecting Graphs

A REFLECTION is a mirror image of the graph in a certain line.

Reflection in the x-axis: h(x) = − f(x)

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Parent Function

f(x)= x

f(x) = x

Solution:

You can obtain the graph of g by reflecting f in the x-axis.

Reflecting Graphs (continued)

Take a look*

Sketch the graph of and . f(x)= x g(x)= - x

g(x)

f(x)

Reflection in x-axis

g(x)= - x

12

First name the parent function (one of the six common graphs). Describe the transformations of each of the following graphs as compared to the graph of its parent function. Then sketch the graph of the transformed function.

a)

Parent Graph: _________________

Transformations:

4y x

Example

b)

Parent Graph: _________________

Transformations:

c)

Parent Graph: _________________

Transformations:

3 1y x

3 2y x

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Nonrigid Transformations

Translations and reflections are called rigid transformations because the basic shape of the graph is NOT changed.

Nonrigid transformations cause the original shape of the graph to change or become distorted.

We will look briefly at VERTICAL STRETCHES AND SHRINKS.

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A VERTICAL STRETCH causes the graph to become more elongated (skinnier)

A VERTICAL SHRINK causes the graph to become squattier (wider).

For y = f(x),

y = A f(x) is a vertical stretch if A > 1.

y = A f(x) is a vertical shrink if 0 < A < 1.

Example:

Vertical Stretch and Shrink

2

2

2g(x) = 4

1h(x) =

f(x

x

) =

x4

x

Notice that the vertex of the parabola does not change; the graph just becomes narrower or wider depending upon the value of A.

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First name the parent function (one of the six common graphs). Describe the transformations of each of the following graphs as compared to the graph of its parent function. Then sketch the graph of the transformed function.

a)

Parent Graph: _________________

Transformations:

14

3y x

Examples

b)

Parent Graph: _________________

Transformations:

c)

Parent Graph: _________________

Transformations:

35 1y x

2 3y x

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Summary ofGraph Transformations

See Handout:

Transformations of the graph of y = f(x)

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More Practice

See Handout:Extra Practice: Domain, Range, and Transformations of Common Function

20

Piecewise-Defined Functions

The absolute value of a real number x can be defined as

Notice that this function is defined by different rules for different parts of its domain. Functions whose definitions involve more than one rule are called piecewise-defined functions.

Graphing one of these functions involves graphing each rule over the appropriate portion of the domain.

if 0| |

if 0

x xx

x x

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Example of a Piecewise-Defined Function

Graph the function

2if2

2if22)(

xx

xxxf

x y