# Wk4 Functions and Graphing Techniques Complete the attached document below about functions and graphing techniques Chapter 2 Functions and Their Graphs Cha

Wk4 Functions and Graphing Techniques Complete the attached document below about functions and graphing techniques Chapter 2 Functions and Their Graphs

Chapter 2: Functions and Their Graphs

Section 2.4: Library of Functions; Piecewise-defined Functions

In the real world, seldom can data be represented by just one function. More often, data changes

over time and must be represented by many different functions. When a function is defined by

different equations on different parts of its domain, it is called a piecewise-defined function.

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Author in Action: Graph Functions Listed in the Library of Functions (6:07)Video

Be able to fill in the information for all of these functions.

Before we begin looking at piecewise-defined functions, we must make sure we are familiar with

a lot of different functions, which we will call our “library of functions.” It’s important that you

know the properties of each of the key functions here. Being able to visualize basic functions

will be helpful in later topics such as nonlinear systems and graphing using transformations.

Exploration 1: Graph the Functions Listed in the Library of Functions

For each of the functions in our library, fill in the table of values, graph the function, then

complete the table of properties.

Identity Function: y = x

x

−2

−1

y

0

1

2

Properties

Domain:

Range:

x-intercept(s):

y-intercept:

Symmetry:

Interval the function is decreasing:

Interval the function is increasing:

Local maxima/Minima:

Absolute maxima/Minima:

Copyright © 2016 Pearson Education, Inc.

Sections 2.4 & 2.5

Square Function: y = x 2

x

−2

−1

y

0

1

2

Properties

Domain:

Range:

x-intercept(s):

y-intercept:

Symmetry:

Interval the function is decreasing:

Interval the function is increasing:

Local maxima/Minima:

Absolute maxima/Minima:

Cube Function: y = x 3

x

−2

−1

y

0

1

2

Properties

Domain:

Range:

x-intercept(s):

y-intercept:

Symmetry:

Interval the function is decreasing:

Interval the function is increasing:

Local maxima/Minima:

Absolute maxima/Minima:

Copyright © 2016 Pearson Education, Inc.

Chapter 2 Functions and Their Graphs

Square Root Function: y = x

x

−4

−1

y

0

1

4

Properties

Domain:

Range:

x-intercept(s):

y-intercept:

Symmetry:

Interval the function is decreasing:

Interval the function is increasing:

Local maxima/Minima:

Absolute maxima/Minima:

Cube Root Function: y = 3 x

x

−8

y

−1

0

1

8

Properties

Domain:

Range:

x-intercept(s):

y-intercept:

Symmetry:

Interval the function is decreasing:

Interval the function is increasing:

Local maxima/Minima:

Absolute maxima/Minima:

Copyright © 2016 Pearson Education, Inc.

Sections 2.4 & 2.5

Reciprocal Function: y =

1

x

x

−2

−1

y

0

1

2

Properties

Domain:

Range:

x-intercept(s):

y-intercept:

Symmetry:

Interval the function is decreasing:

Interval the function is increasing:

Local maxima/Minima:

Absolute maxima/Minima:

Absolute Value Function: y = x

x

−2

−1

y

0

1

2

Copyright © 2016 Pearson Education, Inc.

Chapter 2 Functions and Their Graphs

Properties

Domain:

Range:

x-intercept(s):

y-intercept:

Symmetry:

Interval the function is decreasing:

Interval the function is increasing:

Local maxima/Minima:

Absolute maxima/Minima:

Some other important functions to know:

The Constant Function: f ( x) = b where b is a real number.

These graphs will always be a ___________________ line

with a y – intercept of ________.

Copyright © 2016 Pearson Education, Inc.

Sections 2.4 & 2.5

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Author in Action: Graph Piecewise-defined Functions (10:00)Video

Now that we have our “library of functions,” let’s look at piecewise – defined functions.

Post 1

Definition: When a function is defined by different __________ on different parts of its

________, it is called a piecewise-defined function.

Example 1*: Graph Piecewise – Defined Functions

The function f is defined as

x2

if x 0

f ( x ) = 2

if x = 0

x + 2 if x 0

(a) Find f ( −2) , f ( 0) , and f ( 3)

(b) Determine the domain of f.

(c) Graph f.

(d) Use the graph to find the range of f.

(e) Is f continuous on its domain?

Copyright © 2016 Pearson Education, Inc.

Chapter 2 Functions and Their Graphs

Section 2.5: Graphing Techniques: Transformations

In the last section we looked at our “library of functions” and made sure we could recognize any

of them. In most applications however, we are often asked to graph or recognize a function that

is “almost” like one that we already know. In this section, we will look at some of these

functions and develop techniques for graphing them. Collectively, these techniques are referred

to as transformations.

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Author in Action: Graph Functions Using Vertical and Horizontal Shifts (9:51)Video

Post 2

Exploration 1: Graph Functions Using Vertical Shifts

(a) Sketch the graph of y = x 2

(b) On the same set of axes sketch the graphs

y = x2 + 2 .

of

(c) What is the connection between the

graphs?

(d) Test your connection by predicting the graph of y = x 2 − 3 and then graphing it.

If we were asked to write the transformations y = x 2 + 2 and y = x 2 − 3 in terms of the original

function, we could say that y = x 2 + 2 = f ( x) + 2 and y = x 2 − 3 = f ( x) − 3 .

Copyright © 2016 Pearson Education, Inc.

Sections 2.4 & 2.5

Summary of Vertical Shifts

• If a positive real number k is added to the output of a function, y = f ( x) , the graph of

the new function, y = f ( x) + k , is the graph of f shifted _______________ k units.

•

If a positive real number k is subtracted from the output of a function, y = f ( x) , the

graph of the new function, y = f ( x) − k is the graph of f shifted __________ k units.

Post 3

Exploration 2*: Graph Functions Using Horizontal Shifts

(e) Sketch the graph of y = x 2

(f) On the same set of axes sketch the graphs

= ( − 2)2 .

of

(g) What is the connection between the

graphs?

(h) Test your connection by predicting the graph of = ( + 2)2 and then graphing it.

If we were asked to write the transformations = ( − 3)2 and = ( + 3)2 in terms of the

original function, we could say that = ( − 2)2 = ( − 2) and

= ( + 3)2 = ( + 3).

Summary of Horizontal Shifts

Copyright © 2016 Pearson Education, Inc.

Chapter 2 Functions and Their Graphs

•

If the argument x of a function f is replaced by x − h, h 0 , the graph of the new function

y = f ( x − h) is the graph of f___________________________ h units.

•

If the argument x of a function f is replaced by x + h, h 0 , the graph of the new function

y = f ( x + h) is the graph of f___________________________ h units.

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Author in Action: Functions Compressions and Stretches Part I (10:22)Video

Post 4

Exploration 3*: Graph Functions Using Vertical Stretches and Compressions

1

On the same set of axes, sketch the graphs of ( ) = | |, = 2 ( ), and y = f ( x )

2

Summary of Vertical Stretches and Compressions

The graph of y = af ( x) when a 0 is a transformation of the graph y = f ( x) .

Copyright © 2016 Pearson Education, Inc.

Sections 2.4 & 2.5

•

If a 1 , then the graph of the original function is ______________ vertically by a factor

of a units.

•

If 0 a 1, then the graph of the original function is ______________ vertically by a

factor of a units.

Post 5

Exploration 4: Graph Functions Using Horizontal Stretches and Compressions

On the same set of axes, sketch the graphs of ( ) = √ , = (2 ).

Copyright © 2016 Pearson Education, Inc.

Chapter 2 Functions and Their Graphs

Summary of Horizontal Stretches and Compressions:

The graph of y = f (ax) when a 0 is a transformation of the graph y = f ( x) .

• If a 1 , then the graph of the original function is ____________ horizontally by a factor

1

of

units.

a

•

If 0 a 1, then the graph of the original function is ____________ horizontally by a

1

factor of

units.

a

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Author in Action: Functions Compressions and Stretches Part II (6:00)Video

This shows how to combine a horizontal compression and a vertical stretch.

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Author in Action: Graph Functions Using Reflections about the x-Axis and the y-Axis

(4:40)Video

This shows how to combine horizontal and vertical reflections.

Summary of Horizontal and Vertical Reflections:

The graph of y = − f ( x ) is a transformation of the graph y = f ( x) . It reflects the graph of the

original function over the __________ axis.

The graph of y = f ( − x ) is a transformation of the graph y = f ( x) . It reflects the graph of

the original function over the __________ axis.

Copyright © 2016 Pearson Education, Inc.

Sections 2.4 & 2.5

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Author in Action: Summary of Transformations (13:56)

Know the table:

Summary of Transformations*

To Graph:

Draw the Graph of f and :

Vertical Shifts

Functional Change to f ( x ) :

y = f ( x) + k , k 0

y = f ( x) − k, k 0

Horizontal Shifts

y = f ( x + h) , h 0

y = f ( x − h) , h 0

Compressing or Stretching

y = af ( x ) , a 0

y = f ( ax ) , a 0

Reflection about the x – axis

y = − f ( x)

Reflection about the y – axis

y = f ( −x)

Copyright © 2016 Pearson Education, Inc.

Chapter 2 Functions and Their Graphs

Be able to follow the example.

Example 1*: Summary of Transformations

Use the graph of f ( x) = x to obtain the graph of g ( x ) = −2 x + 1 − 3 .

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