functions and relations - home - polk school district notes.pdf · function notes 1 december 15,...

Function Notes

1

December 15, 2014

Functions and Relations

Lesson objectives Teachers' notes

Objectives:1) Summarize the definition of a function both visually and in your own words.

2) Apply definition to sets of relations and graphs to determine if they are functions.

Function Notes

2

December 15, 2014

Teachers' notesLesson objectives

Subject:

Topic:

Grade(s):

Prior knowledge:

Crosscurricular link(s):

Algebra II

Functions and Relations

910

Type text here

Type text here

Lesson notes:

Focus Question:How can we determine if a given graph or set of relations is a function?

Function Notes

3

December 15, 2014

What is a function?It is a special kind of relation.

Real Life Examples:Age Example:Height exampleYour Example???

Function Notes

4

December 15, 2014

Consider these numbers...Consider these numbers...2 and 105 and 257 and ??7 and 35

Have you figured out the relation between these numbers?relation

2 and -105 and -257 and ??7 and -35

Function Notes

5

December 15, 2014

relationA relation in mathis some numbers that are connectedto other numbers by a pattern.

ordered pairThe connected numbers

make an ordered pair.

(2,10)

(2,-10)

(5,25)(5,-25)

(7,35)(7,-35)

Function Notes

6

December 15, 2014

x y2 -102 105 -255 257 -357 35

(2,10)(2,-10)

(5,25)(5,-25)

(7,35)(7,-35)

There are lots of ways to show a relation.

ListList TableTable MapMap GraphGraph EquationEquationy=±5x

Function Notes

7

December 15, 2014

1

0

-1

-2

2

-3

-2-1

0

1

In this relation, what is the output when the input is 1 ?

In this relation, what is the value of x when y=-2 ?

Function Notes

8

December 15, 2014


In this relation, what is the value of x when y=1 ?

1

2

3

4

1

2

0

-1

Function Notes

9

December 15, 2014

In this relation, what is the value of x when y=0 ?


{( , ), ( , ), ( , ), ( , ), ( , )}-1 0 5-3-5 243-1 0 4

Function Notes

10

December 15, 2014

In this relation, what is the value of x when y=-2 ?

0

1

2

3

4

5

-1

-2

-3

-4

-5

1 2 3 4 5-1-2-3-4-5


Function Notes

11

December 15, 2014

What makes the values in a table a function?

• With a table, the condition means that if there are two rows of a table with the same “input” then they must also have the same output.

Function Notes

12

December 15, 2014

1 2

1000 200050 100

5 102 4

7 14

in out

function machine

Function Notes

13

December 15, 2014

Domain

The domain contains the numbers or values called the inputs.

x y

1

42

2

63

84

50

100

Function Notes

14

December 15, 2014

Range

The range contains the numbers or values called the outputs.

x y

1

42

2

63

84

50

100

Function Notes

15

December 15, 2014

Function:

A special relation such that for all xvalues(input) there exists one and only one y value.

Function Notes

16

December 15, 2014

The “Pairing” of names and heights is a relation.

Pairs of names and heights are “ordered” in functions.

(Person, Height)

OnetoOne

Function Notes

17

December 15, 2014

(Height, Person)

Not a Function

Function Notes

18

December 15, 2014

Mapping:

Function Notes

19

December 15, 2014

NOTES TIME!

•Use notes section of binder•Examples have no errors•110% attention necessary•Teacher is the only one talking

Function Notes

20

December 15, 2014

Function:Well behaved relation.

Definition: A special relation such that for all values of x there exists one and only one yvalue.

OnetoOne

Function Notes

21

December 15, 2014

Function Notes

22

December 15, 2014

Set of relations:

1. {(2, 3), (4, 5), (5, 6)}

2. {(2, 3), (4, 3), (5, 6)}

3. {(2, 3), (2, 5), (5, 6)}

Yes

No

Yes

Function Notes

23

December 15, 2014

Graphs:

Vertical Line Test:

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0

1

2

3

4

5

6

90°

Function Notes

24

December 15, 2014

Function Notes

25

December 15, 2014

Function Notation:

Example:Linear Function:y = 2x 1

Function Notes

26

December 15, 2014

4

8

2

4

5

What's my rule?Click on the green arrow and drag over to the IN slot and then drag it down into the Function Machine. The answer will appear from the OUT slot.Try all three inputs and see if you can determine the rule. Tip my hat to see if you are correct.

NextNext

IN

OUT

x 2

Function Notes

27

December 15, 2014

Input Output

2

4

5

4

8

10

Now that you know the rule see if you can enter some other Inputs and then determine the Outputs.

Take the Input --- Apply the Rule --- Get the Output

NextNext

Function Notes

28

December 15, 2014

6

13

10

21

7

What's my rule?Click on the green arrow and drag over to the IN slot and then drag it down into the Function Machine. The answer will appear from the OUT slot.Try all three inputs and see if you can determine the rule. Tip my hat to see if you are correct.

IN

OUT

NextNext

x 2

Function Notes

29

December 15, 2014

Input Output

6

7

10

13

15

21

Now that you know the rule see if you can enter some other Inputs and then determine the Outputs.

Take the Input --- Apply the Rule --- Get the Output

NextNext

Function Notes

30

December 15, 2014

Pay close attention to

the summary on the next slides!

Function Notes

31

December 15, 2014

Relation: simply a set of ordered pairs.

List the domain and range for each set above.

Function Notes

32

December 15, 2014

A function is a rule that relates two quantities so that each input value (domain) corresponds to exactly one output value (range). "one-to-one"

Functions can often be looked at as input-output machines.

Let's practice with some function "machines".

Machine #2Machine #1

List the ordered pairs:

List the ordered pairs:

Is the relation a function?

http://www.amblesideprimary.com/ambleweb/mentalmaths/functionmachines.htmlhttp://www.littlefishsw.co.uk/card/functionmachine.html

Function Notes

33

December 15, 2014

Vertical Line TestThis test can be used to see if a graph represents a function.To be a function a vertical line can only touch the graph at one point.

Practice

http://www.shodor.org/interactivate/activities/VerticalLineTest/

Function Notes

34

December 15, 2014

Functions can also be illustrated with input-output tables or x-y tables.

Use the rule y = 2x to complete the table

Does the graph pass the vertical line test?

Function Notes

35

December 15, 2014

Use the rule y = 3x - 1to complete the table


Function Notes

36

December 15, 2014

Use the rule y = x2 + 3to complete the table

Graph the ordered pairs


Function Notes

37

December 15, 2014

Practice # 1

Practice # 2

Practice # 3

Function Notes

38

December 15, 2014

Practice #1

Function Notes

39

December 15, 2014

Quiz

Decide if each of the following represents a function.

1. {(1, 3), (2, 5), (3, 6), (4, 12)}2. {(0, 4), (0, 5), (1, 5)}3. {(5, 1), (3, 1), (-2, 1), (4, 1)}4. y = 2x + 1, if x = {1, 2, 3}5. y = x - 5, if x = {2, 4, 6}

Function Notes

40

December 15, 2014

What did I learn last class?Determine if the following is a relation or a funtion and give the domain and range

x y1 102 123 144 165 18

x y2 124 82 44 00 4

x y8 84 40 04 48 8

x y2 26 618 1854 54

162 162

Function Notes

41

December 15, 2014

Graph the relation and determine the domain and range.

{(2, 1), (3, 5), (5, 12)}

6 5 4 3 2 1 0 1 2 3 4 5 6

6

5

4

3

2

1

1

2

3

4

5

6

x

yDomain:

Range:

Function Notes

42

December 15, 2014

Guided Practice:

Ex#1f(x) = 3x 2 h(x) = 2x + 1 g(x) = 5x

f(1) =f(0) = h(2) =g(5) = g(1/5) =

Function Notes

43

December 15, 2014

Guided Practice:

Ex#2Create a table for each of the functions: f(x) = 3x 2 h(x) = 2x + 1

x 3x 2 f(x) x 2x + 1 f(x)

Function Notes

44

December 15, 2014

From the Sleep Inn, Sam jumped out of the window and walked 2 miles north and 4 miles east to the hideout spot. If the origin represents the Sleep Inn, graph the point that represents the hideout spot.

0 1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

x

y

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Nov 4-10:28 AM Recognize a Function Pre-Calculus 1.3FunctionsHigh SchoolClick here to change textClick here to change textPre-Calc page 125 lesson 1.3%#Standard: How to decide whether relations between two variables are functions.Be able to recognize a function in graph or equation form.%#Subject:Topic:Grade(s):Prior knowledge:Cross-curricular link(s):undefinedundefinedundefinedundefinedundefinedundefinedundefinedundefinedundefinedundefinedundefinedundefinedundefinedundefinedundefinedundefinedundefinedundefinedundefinedundefinedundefinedundefinedtrueundefinedundefinedundefined

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Jul 28-2:04 PM Every equation is a relation.1Only some relations are functions.2In a function, every input has one and only one output.3For Graphs: use the vertical line test .45falsetrue Is It a Function?

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SMART Notebook

AFunction Tables

Complete the function table for each equation.

Name: Date:

Copyright ©2010 WorksheetWorks.com

x y

-2

-1

5

9

y = 3x G 3( 1 )

x y

-6

1

2

5

y = x Q 2( 2 )

x y

-9

-7

3

4

y = x G 6( 3 )

x y

-3

-1

1

5

y = x Q 3( 4 )

x y

-7

-5

0

4

y = x G 2( 5 )

x y

-9

-6

0

2

y = x G 1( 6 )

x y

-9

-8

-4

1

y = 3x G 4( 7 )

x y

0

2

4

5

y = Gx Q 5( 8 )

x y

-6

-4

-1

3

y = x G 4( 9 )

x y

-2

-1

0

1

y = x Q 1(10)

x y

-9

-8

-4

8

y = x Q 4(11)

x y

-7

-5

0

6

y = 3x G 2(12)

AFunction Tables


A N S W E R K E Y


x y

-2

-1

5

9

y = 3x G 3( 1 )

x y

-6

1

2

5

y = x Q 2( 2 )

x y

-9

-7

3

4

y = x G 6( 3 )

x y

-3

-1

1

5

y = x Q 3( 4 )

x y

-7

-5

0

4

y = x G 2( 5 )

x y

-9

-6

0

2

y = x G 1( 6 )

x y

-9

-8

-4

1

y = 3x G 4( 7 )

x y

0

2

4

5

y = Gx Q 5( 8 )

x y

-6

-4

-1

3

y = x G 4( 9 )

x y

-2

-1

0

1

y = x Q 1(10)

x y

-9

-8

-4

8

y = x Q 4(11)

x y

-7

-5

0

6

y = 3x G 2(12)

-9

-6

12

24

-4

3

4

7

-15

-13

-3

-2

0

2

4

8

-9

-7

-2

2

-10

-7

-1

1

-31

-28

-16

-1

5

3

1

0

-10

-8

-5

-1

-1

0

1

2

-5

-4

0

12

-23

-17

-2

16

a a

a aThis is a genuine, customized

worksheet from

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http://www.WorksheetWorks.com/math/geometry/graphing/function-table.html

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SMART Notebook

AFunction Tables


Name: Date:


x y

-7

-3

-2

7

y = x Q 5( 1 )

x y

-7

-6

-3

5

y = x G 4( 2 )

x y

-9

3

8

9

y = x G 1( 3 )

x y

-8

-3

2

7

y =1-4x Q 4( 4 )

x y

-5

-1

2

9

y = x G 6( 5 )

x y

-5

-2

0

6

y = x G 5( 6 )

x y

-9

-2

5

8

y = x Q 2( 7 )

x y

-8

-3

4

6

y = G4x( 8 )

x y

-9

-7

-3

-2

y = G3x( 9 )

x y

-3

-2

6

7

y =14x Q 1(10)

x y

-8

-2

4

5

y = 3x(11)

x y

-9

-8

-2

4

y =14x G 5(12)

AFunction Tables


A N S W E R K E Y


x y

-7

-3

-2

7

y = x Q 5( 1 )

x y

-7

-6

-3

5

y = x G 4( 2 )

x y

-9

3

8

9

y = x G 1( 3 )

x y

-8

-3

2

7

y =1-4x Q 4( 4 )

x y

-5

-1

2

9

y = x G 6( 5 )

x y

-5

-2

0

6

y = x G 5( 6 )

x y

-9

-2

5

8

y = x Q 2( 7 )

x y

-8

-3

4

6

y = G4x( 8 )

x y

-9

-7

-3

-2

y = G3x( 9 )

x y

-3

-2

6

7

y =14x Q 1(10)

x y

-8

-2

4

5

y = 3x(11)

x y

-9

-8

-2

4

y =14x G 5(12)

-2

2

3

12

-11

-10

-7

1

-10

2

7

8

6

4 343 122 14

-11

-7

-4

3

-10

-7

-5

1

-7

0

7

10

32

12

-16

-24

27

21

9

6

1412

2 122 34

-24

-6

12

15

-7 14-7

-5 12-4

a a


worksheet from







•


•


•




SMART Notebook

AFunction Tables


Name: Date:


x y

-8

6

7

8

y = 3x Q 1( 1 )

x y

-6

-4

0

8

y = Gx Q 5( 2 )

x y

-5

1

8

9

y = G2x Q 4( 3 )

x y

-9

-7

4

6

y = G2x Q 2( 4 )

x y

4

5

8

9

y = x Q 1( 5 )

x y

-512

-1

112

5

y = 2x G 3( 6 )

x y

-9

-6

-4

-1

y = G4x Q 3( 7 )

x y

-9

-4

-1

7

y = x G 4( 8 )

x y

-9

-8

3

8

y = x G 6( 9 )

x y

-9

-8

-412

-212

y = 4x G 6(10)

x y

-9

-4

-3

1

y = G3x G 6(11)

x y

-4

1

3

7

y = x G 3(12)

AFunction Tables


A N S W E R K E Y


x y

-8

6

7

8

y = 3x Q 1( 1 )

x y

-6

-4

0

8

y = Gx Q 5( 2 )

x y

-5

1

8

9

y = G2x Q 4( 3 )

x y

-9

-7

4

6

y = G2x Q 2( 4 )

x y

4

5

8

9

y = x Q 1( 5 )

x y

-512

-1

112

5

y = 2x G 3( 6 )

x y

-9

-6

-4

-1

y = G4x Q 3( 7 )

x y

-9

-4

-1

7

y = x G 4( 8 )

x y

-9

-8

3

8

y = x G 6( 9 )

x y

-9

-8

-412

-212

y = 4x G 6(10)

x y

-9

-4

-3

1

y = G3x G 6(11)

x y

-4

1

3

7

y = x G 3(12)

-23

19

22

25

11

9

5

-3

14

2

-12

-14

20

16

-6

-10

5

6

9

10

-14

-5

0

7

39

27

19

7

-13

-8

-5

3

-15

-14

-3

2

-42

-38

-24

-16

21

6

3

-9

-7

-2

0

4

a a


worksheet from







•


•


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SMART Notebook

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