welcome to mth 151

Welcome to MTH 151

Dr. Kim

University of Wisconsin-La Crosse

Fall 2014

DKSE: Dr. Kim’s Stock ExchangeToday is Day 1. The price of Pringles is $1. a1 = 1.

1

MTH 151

We’ll talk about(review) exponential and logarithmic functionstrigonometry: ratios, functions, identities, applicationsvectorsparametric and polar descriptions of graphssequences and series

I have a class webpage set up:

http://websites.uwlax.edu/ekim/teaching/2014-Fall-151/

2

MTH 151

We’ll talk about(review) exponential and logarithmic functionstrigonometry: ratios, functions, identities, applicationsvectorsparametric and polar descriptions of graphssequences and series

I have a class webpage set up:

http://websites.uwlax.edu/ekim/teaching/2014-Fall-151/

2

Who am I?

Dr. Kimekim@uwlax.edu1018 Cowley Hall

Office hours:

Day of week Time LocationMondays 3:20pm-4:15pm My office (1018 Cowley Hall)Tuesdays 4:00pm-5:30pm My office (1018 Cowley Hall)Wednesdays 5:30pm-7:00pm MLC (251 Murphy Library)Thursdays 9:55am-11:55am MLC (251 Murphy Library)

... and by appointment!

3

Who am I?

Dr. Kimekim@uwlax.edu1018 Cowley Hall

Office hours:

Day of week Time LocationMondays 3:20pm-4:15pm My office (1018 Cowley Hall)Tuesdays 4:00pm-5:30pm My office (1018 Cowley Hall)Wednesdays 5:30pm-7:00pm MLC (251 Murphy Library)Thursdays 9:55am-11:55am MLC (251 Murphy Library)

... and by appointment!

3

Who are you

Class questionnaire

4

Components of your grade

Writing assignmentQuizzesExams (in-class)Final examWeBWorK

I log in at https://webwork.uwlax.edu/webwork2/MTH151_Kim

I sample login: gow.joe is username AND passwordI ww00 is a review (complete right away)I If you’re always on the deadline date, you’re always behind

in this class!

5

Components of your grade

Writing assignment

QuizzesExams (in-class)Final examWeBWorK

I log in at https://webwork.uwlax.edu/webwork2/MTH151_Kim

I sample login: gow.joe is username AND passwordI ww00 is a review (complete right away)I If you’re always on the deadline date, you’re always behind

in this class!

5

Components of your grade

Writing assignmentQuizzes

Exams (in-class)Final examWeBWorK

I log in at https://webwork.uwlax.edu/webwork2/MTH151_Kim

I sample login: gow.joe is username AND passwordI ww00 is a review (complete right away)I If you’re always on the deadline date, you’re always behind

in this class!

5

Components of your grade

Writing assignmentQuizzesExams (in-class)

Final examWeBWorK

I log in at https://webwork.uwlax.edu/webwork2/MTH151_Kim

I sample login: gow.joe is username AND passwordI ww00 is a review (complete right away)I If you’re always on the deadline date, you’re always behind

in this class!

5

Components of your grade

Writing assignmentQuizzesExams (in-class)Final exam

WeBWorKI log in at https://webwork.uwlax.edu/webwork2/MTH151_Kim

I sample login: gow.joe is username AND passwordI ww00 is a review (complete right away)I If you’re always on the deadline date, you’re always behind

in this class!

5

Components of your grade

Writing assignmentQuizzesExams (in-class)Final examWeBWorK

I log in at https://webwork.uwlax.edu/webwork2/MTH151_Kim

I sample login: gow.joe is username AND passwordI ww00 is a review (complete right away)I If you’re always on the deadline date, you’re always behind

in this class!

5

Reviewing content

ww00Function CarnivalPrerequisite verification exam (take-home)

6

Quiz/Exam calculator policy

... so I suggest not using calculators on homework either. Onlyuse to check.

7

Quiz/Exam calculator policy

... so I suggest not using calculators on homework either. Onlyuse to check.

7

Quiz/Exam calculator policy

... so I suggest not using calculators on homework either. Onlyuse to check.

7

Taking responsibility for your own education

The answers to the following questions:What does the next quiz cover?When is the next test?What’s covered on the next test?When are your office hours?Where is your office?

... are found in the syllabus.

8

Ask yourself “commentary questions”

As you solve a problem, ask questions of the form:What do I want to get?How do I get that to happen?

As soon as you finish a problem, ask yourself:What did I do here?What strategy should I learn? Is there a faster/slower wayto solve it?

The key reason for asking questions of the second kind is thatthey will end up helping you answer questions of the first kind inthe next section.

9

Ask yourself “commentary questions”

As you solve a problem, ask questions of the form:What do I want to get?How do I get that to happen?

As soon as you finish a problem, ask yourself:What did I do here?What strategy should I learn? Is there a faster/slower wayto solve it?

The key reason for asking questions of the second kind is thatthey will end up helping you answer questions of the first kind inthe next section.

9

Ask yourself “commentary questions”

As you solve a problem, ask questions of the form:What do I want to get?How do I get that to happen?

As soon as you finish a problem, ask yourself:What did I do here?What strategy should I learn? Is there a faster/slower wayto solve it?

The key reason for asking questions of the second kind is thatthey will end up helping you answer questions of the first kind inthe next section.

9

How to answer these “commentary questions”

When answering these questions, provide commentary infull sentences. As much as possible, use nouns (such asf (x) or “x to any power n”) as opposed to using pronouns(such as “it”).

Using the correct definitions (and keeping in mind the“parts of speech”) really helps!

10

How to answer these “commentary questions”

When answering these questions, provide commentary infull sentences. As much as possible, use nouns (such asf (x) or “x to any power n”) as opposed to using pronouns(such as “it”).Using the correct definitions (and keeping in mind the“parts of speech”) really helps!

10

Reading the textbook55 minutes goes by quickly:

New concepts/formulas/etc. to introduceExamplesMaking connections to previous and upcoming materialImparting strategy

My goal is to cover as much as possible, but there is no way toget to all of it:

Someone will prefer more examples and fewer concepts inclassSomeone else will prefer more concepts with fewerexamplesCompare the pace of this class to secondary trig.TWICE AS FAST

MoralBecause some material is strategically skipped in class, youmust review each section in the textbook on your own time.

11

Reading the textbook55 minutes goes by quickly:

New concepts/formulas/etc. to introduceExamplesMaking connections to previous and upcoming materialImparting strategy

My goal is to cover as much as possible, but there is no way toget to all of it:

Someone will prefer more examples and fewer concepts inclassSomeone else will prefer more concepts with fewerexamplesCompare the pace of this class to secondary trig.TWICE AS FAST

MoralBecause some material is strategically skipped in class, youmust review each section in the textbook on your own time.

11

Reading the textbook55 minutes goes by quickly:

New concepts/formulas/etc. to introduceExamplesMaking connections to previous and upcoming materialImparting strategy

My goal is to cover as much as possible, but there is no way toget to all of it:

Someone will prefer more examples and fewer concepts inclassSomeone else will prefer more concepts with fewerexamplesCompare the pace of this class to secondary trig.TWICE AS FAST

MoralBecause some material is strategically skipped in class, youmust review each section in the textbook on your own time.

11

Class calendar and textbook

Examine the class calendar

Do a “quick read” before classI If you are spending more than 10 seconds per page, you

are spending too much time!!

Come to classDo a thorough read of the book section

I Find each definition/theorem/formula (in agreen/blue/orange box)

I Read through the example problems and solutions.

Do the WebWork and textbook exercises.

12

Class calendar and textbook

Examine the class calendar

Do a “quick read” before classI If you are spending more than 10 seconds per page, you

are spending too much time!!

Come to classDo a thorough read of the book section

I Find each definition/theorem/formula (in agreen/blue/orange box)

I Read through the example problems and solutions.

Do the WebWork and textbook exercises.

12

Class calendar and textbook

Examine the class calendarDo a “quick read” before class

I If you are spending more than 10 seconds per page, youare spending too much time!!

Come to classDo a thorough read of the book section

I Find each definition/theorem/formula (in agreen/blue/orange box)

I Read through the example problems and solutions.

Do the WebWork and textbook exercises.

12

Class calendar and textbook

Examine the class calendarDo a “quick read” before class

I If you are spending more than 10 seconds per page, youare spending too much time!!

Come to classDo a thorough read of the book section

I Find each definition/theorem/formula (in agreen/blue/orange box)

I Read through the example problems and solutions.

Do the WebWork and textbook exercises.

12

Class calendar and textbook

Examine the class calendarDo a “quick read” before class

I If you are spending more than 10 seconds per page, youare spending too much time!!

Come to class

Do a thorough read of the book sectionI Find each definition/theorem/formula (in a

green/blue/orange box)I Read through the example problems and solutions.

Do the WebWork and textbook exercises.

12

Class calendar and textbook

Examine the class calendarDo a “quick read” before class

I If you are spending more than 10 seconds per page, youare spending too much time!!

Come to classDo a thorough read of the book section

I Find each definition/theorem/formula (in agreen/blue/orange box)

I Read through the example problems and solutions.

Do the WebWork and textbook exercises.

12

Class calendar and textbook

Examine the class calendarDo a “quick read” before class

I If you are spending more than 10 seconds per page, youare spending too much time!!

Come to classDo a thorough read of the book section

I Find each definition/theorem/formula (in agreen/blue/orange box)

I Read through the example problems and solutions.

Do the WebWork and textbook exercises.

12

Class calendar and textbook

Examine the class calendarDo a “quick read” before class

I If you are spending more than 10 seconds per page, youare spending too much time!!

Come to classDo a thorough read of the book section

I Find each definition/theorem/formula (in agreen/blue/orange box)

I Read through the example problems and solutions.

Do the WebWork and textbook exercises.

12

Class calendar and textbook

Examine the class calendarDo a “quick read” before class

I If you are spending more than 10 seconds per page, youare spending too much time!!

Come to classDo a thorough read of the book section

I Find each definition/theorem/formula (in agreen/blue/orange box)

I Read through the example problems and solutions.

Do the WebWork and textbook exercises.

12

Learning mathematics

Math is not a spectator sport!

There simply isn’t enough time to cover everything in class.To do well, you must supplement class time withreading the book.You must try the exercises and be willing to get a littlestuck! Do not delude yourself into thinking that youhave mastered the material simply because you mostlyfollow my explanations in class!As you learn a new technique, you should look at problemsand start formulating in your own words when the newtechnique will/won’t work.Each day’s class will likely rely on the previous day’smaterial. If you’ve merely finished the WebWork whichwas recently due, you’re a week behind and class willnot be as useful to you.

13

Learning mathematics

Math is not a spectator sport!There simply isn’t enough time to cover everything in class.To do well, you must supplement class time withreading the book.

You must try the exercises and be willing to get a littlestuck! Do not delude yourself into thinking that youhave mastered the material simply because you mostlyfollow my explanations in class!As you learn a new technique, you should look at problemsand start formulating in your own words when the newtechnique will/won’t work.Each day’s class will likely rely on the previous day’smaterial. If you’ve merely finished the WebWork whichwas recently due, you’re a week behind and class willnot be as useful to you.

13

Learning mathematics

Math is not a spectator sport!There simply isn’t enough time to cover everything in class.To do well, you must supplement class time withreading the book.You must try the exercises and be willing to get a littlestuck! Do not delude yourself into thinking that youhave mastered the material simply because you mostlyfollow my explanations in class!

As you learn a new technique, you should look at problemsand start formulating in your own words when the newtechnique will/won’t work.Each day’s class will likely rely on the previous day’smaterial. If you’ve merely finished the WebWork whichwas recently due, you’re a week behind and class willnot be as useful to you.

13

Learning mathematics

Math is not a spectator sport!There simply isn’t enough time to cover everything in class.To do well, you must supplement class time withreading the book.You must try the exercises and be willing to get a littlestuck! Do not delude yourself into thinking that youhave mastered the material simply because you mostlyfollow my explanations in class!As you learn a new technique, you should look at problemsand start formulating in your own words when the newtechnique will/won’t work.

Each day’s class will likely rely on the previous day’smaterial. If you’ve merely finished the WebWork whichwas recently due, you’re a week behind and class willnot be as useful to you.

13

Learning mathematics

Math is not a spectator sport!There simply isn’t enough time to cover everything in class.To do well, you must supplement class time withreading the book.You must try the exercises and be willing to get a littlestuck! Do not delude yourself into thinking that youhave mastered the material simply because you mostlyfollow my explanations in class!As you learn a new technique, you should look at problemsand start formulating in your own words when the newtechnique will/won’t work.Each day’s class will likely rely on the previous day’smaterial. If you’ve merely finished the WebWork whichwas recently due, you’re a week behind and class willnot be as useful to you.

13

Keep up to date on material

To be up to date, you must do exercises (textbook andWebWork) each day.If you want to be last minute, you’ll end up doing exerciseson WebWork each day (but you will be mentally about aweek behind the class)

Either way, you will be doing work for this class each day. If youhave to work on this class each day anyway, you might aswell be up-to-date!

14

Keep up to date on material

To be up to date, you must do exercises (textbook andWebWork) each day.If you want to be last minute, you’ll end up doing exerciseson WebWork each day (but you will be mentally about aweek behind the class)

Either way, you will be doing work for this class each day.

If youhave to work on this class each day anyway, you might aswell be up-to-date!

14

Keep up to date on material

To be up to date, you must do exercises (textbook andWebWork) each day.If you want to be last minute, you’ll end up doing exerciseson WebWork each day (but you will be mentally about aweek behind the class)

Either way, you will be doing work for this class each day. If youhave to work on this class each day anyway, you might aswell be up-to-date!

14

Expression vs. Equation vs. Function

In math, the termsexpressionequationfunction

mean three DIFFERENT things!

Key to successUnderstanding the difference between these three things iscrucial.

15

Expression vs. Equation vs. Function

In math, the termsexpressionequationfunction

mean three DIFFERENT things!

Key to successUnderstanding the difference between these three things iscrucial.

15

Expression

An expression is mathematical notation which represents anumeric quantity.

Examples:3

2× x− 5

x2 + 3x − 1√x

Reading expressionsTo remove ambiguity, the order of operations dictates how tointerpret an expression.

16

Expression

An expression is mathematical notation which represents anumeric quantity.

Examples:3

2× x− 5

x2 + 3x − 1√x

Reading expressionsTo remove ambiguity, the order of operations dictates how tointerpret an expression.

16

Equation

An equation is a statement (using an equals sign) whichasserts that two expressions are equal.

In other words...An equation says that the expression on the left of an equalssign is actually the same number/quantity as the expressionwritten to the right of the equals sign.

Examples:x + 2 = 6

24 + 2x = x2 − 1

17

Equation

An equation is a statement (using an equals sign) whichasserts that two expressions are equal.

In other words...An equation says that the expression on the left of an equalssign is actually the same number/quantity as the expressionwritten to the right of the equals sign.

Examples:x + 2 = 6

24 + 2x = x2 − 1

17

Equation

An equation is a statement (using an equals sign) whichasserts that two expressions are equal.

In other words...An equation says that the expression on the left of an equalssign is actually the same number/quantity as the expressionwritten to the right of the equals sign.

Examples:x + 2 = 6

24 + 2x = x2 − 1

17

Anatomy of an Equation

3x + 22− 2x = x +√

x + 12345

Expression vs. EquationAn equation is two expressions and an equals sign inbetween!

18

Anatomy of an Equation

3x + 22− 2x︸ ︷︷ ︸expression

= x +√

x + 12345︸ ︷︷ ︸expression

Expression vs. EquationAn equation is two expressions and an equals sign inbetween!

18

Anatomy of an Equation

3x + 22− 2x︸ ︷︷ ︸expression

= x +√

x + 12345︸ ︷︷ ︸expression︸ ︷︷ ︸

equation

Expression vs. EquationAn equation is two expressions and an equals sign inbetween!

18

Anatomy of an Equation

3x + 22− 2x︸ ︷︷ ︸expression

= x +√

x + 12345︸ ︷︷ ︸expression︸ ︷︷ ︸

equation

Expression vs. EquationAn equation is two expressions and an equals sign inbetween!

18

You try it!

Which of the following are NOT equations?

A (x− 3)(x + 5)

B 13 = 3x + 1

C x3 − 1 = 4

x

D 2x · 3x

E 1− 2 + 3− 4 + 5

F (3z)2 + 1 = 0

19

You try it!

Which of the following are NOT equations?

A (x− 3)(x + 5) ← not an equationB 13 = 3x + 1

C x3 − 1 = 4

x

D 2x · 3x ← not an equationE 1− 2 + 3− 4 + 5 ← not an equationF (3z)2 + 1 = 0

19

Solving equations

To solve an equation... generally apply the order of operations“backwards”

Example2x−1

5 + 4 = 7, which means (2× x− 1)÷ 5 + 4 = 7

subtract 4 on both sides2x−1

5 = 3 mult. by 5 on both sides2x− 1 = 15 add 1 on both sides2x = 16 div. by 2 on both sidesx = 8

20

Solving equations

To solve an equation... generally apply the order of operations“backwards”

Example2x−1

5 + 4 = 7

, which means (2× x− 1)÷ 5 + 4 = 7

subtract 4 on both sides2x−1

5 = 3 mult. by 5 on both sides2x− 1 = 15 add 1 on both sides2x = 16 div. by 2 on both sidesx = 8

20

Solving equations

To solve an equation... generally apply the order of operations“backwards”

Example2x−1

5 + 4 = 7, which means (2× x− 1)÷ 5 + 4 = 7

subtract 4 on both sides2x−1

5 = 3 mult. by 5 on both sides2x− 1 = 15 add 1 on both sides2x = 16 div. by 2 on both sidesx = 8

20

Solving equations

To solve an equation... generally apply the order of operations“backwards”

Example2x−1

5 + 4 = 7, which means (2× x− 1)÷ 5 + 4 = 7

subtract 4 on both sides

2x−15 = 3 mult. by 5 on both sides

2x− 1 = 15 add 1 on both sides2x = 16 div. by 2 on both sidesx = 8

20

Solving equations

To solve an equation... generally apply the order of operations“backwards”

Example2x−1

5 + 4 = 7, which means (2× x− 1)÷ 5 + 4 = 7

subtract 4 on both sides2x−1

5 = 3

mult. by 5 on both sides2x− 1 = 15 add 1 on both sides2x = 16 div. by 2 on both sidesx = 8

20

Solving equations

To solve an equation... generally apply the order of operations“backwards”

Example2x−1

5 + 4 = 7, which means (2× x− 1)÷ 5 + 4 = 7

subtract 4 on both sides2x−1

5 = 3 mult. by 5 on both sides

2x− 1 = 15 add 1 on both sides2x = 16 div. by 2 on both sidesx = 8

20

Solving equations

To solve an equation... generally apply the order of operations“backwards”

Example2x−1

5 + 4 = 7, which means (2× x− 1)÷ 5 + 4 = 7

subtract 4 on both sides2x−1

5 = 3 mult. by 5 on both sides2x− 1 = 15

add 1 on both sides2x = 16 div. by 2 on both sidesx = 8

20

Solving equations

To solve an equation... generally apply the order of operations“backwards”

Example2x−1

5 + 4 = 7, which means (2× x− 1)÷ 5 + 4 = 7

subtract 4 on both sides2x−1

5 = 3 mult. by 5 on both sides2x− 1 = 15 add 1 on both sides

2x = 16 div. by 2 on both sidesx = 8

20

Solving equations

To solve an equation... generally apply the order of operations“backwards”

Example2x−1

5 + 4 = 7, which means (2× x− 1)÷ 5 + 4 = 7

subtract 4 on both sides2x−1

5 = 3 mult. by 5 on both sides2x− 1 = 15 add 1 on both sides2x = 16

div. by 2 on both sidesx = 8

20

Solving equations

To solve an equation... generally apply the order of operations“backwards”

Example2x−1

5 + 4 = 7, which means (2× x− 1)÷ 5 + 4 = 7

subtract 4 on both sides2x−1

5 = 3 mult. by 5 on both sides2x− 1 = 15 add 1 on both sides2x = 16 div. by 2 on both sides

x = 8

20

Solving equations

To solve an equation... generally apply the order of operations“backwards”

Example2x−1

5 + 4 = 7, which means (2× x− 1)÷ 5 + 4 = 7

subtract 4 on both sides2x−1

5 = 3 mult. by 5 on both sides2x− 1 = 15 add 1 on both sides2x = 16 div. by 2 on both sidesx = 8

20

FunctionA function is a rule which consistently assigns to each input anoutput.

Arrow diagram: arrows from inputs to outputs

all the inputs all the outputsdomain range

1

2

3

4

A

B

C

D

Domain of this function is {1, 2, 3, 4}Range of this function is {A,B,D}

21

FunctionA function is a rule which consistently assigns to each input anoutput.

Arrow diagram: arrows from inputs to outputs

all the inputs all the outputsdomain range

1

2

3

4

A

B

C

D

Domain of this function is {1, 2, 3, 4}Range of this function is {A,B,D}

21

FunctionA function is a rule which consistently assigns to each input anoutput.

Arrow diagram: arrows from inputs to outputs

all the inputs all the outputs

domain range

1

2

3

4

A

B

C

D

Domain of this function is {1, 2, 3, 4}Range of this function is {A,B,D}

21

FunctionA function is a rule which consistently assigns to each input anoutput.

Arrow diagram: arrows from inputs to outputs

all the inputs all the outputsdomain range

1

2

3

4

A

B

C

D

Domain of this function is {1, 2, 3, 4}Range of this function is {A,B,D}

21

FunctionA function is a rule which consistently assigns to each input anoutput.

Arrow diagram: arrows from inputs to outputs

all the inputs all the outputsdomain range

1

2

3

4

A

B

C

D

Domain of this function is {1, 2, 3, 4}Range of this function is {A,B,D}

21

Functions by arrow diagramsSecond example?

inputsusually called x-values

outputsusually called y-values

1

2

3

4

A

B

C

D

Person representing input 1 did not know who to throw theball toThis arrow diagram does NOT represent a function

22

Functions by arrow diagramsSecond example?

inputsusually called x-values

outputsusually called y-values

1

2

3

4

A

B

C

D

Person representing input 1 did not know who to throw theball to

This arrow diagram does NOT represent a function

22

Functions by arrow diagramsSecond example?

inputsusually called x-values

outputsusually called y-values

1

2

3

4

A

B

C

D

Person representing input 1 did not know who to throw theball toThis arrow diagram does NOT represent a function

22

Function: by formula

Describe a function using an expression:

f (x) = 3x + 1

input is x

output

If input is x = 2, what is the output?I Output is y = 7. Also written f (2) = 7.

If input is x = −10, what is the output?I Output is y = −29. Also written f (−10) = −29.

If input is x = 13 , what is the output?

I Output is y = 2. Also written f ( 13 ) = 2.

23

Function: by formula

Describe a function using an expression:

f (x) = 3x + 1

input is x

output

If input is x = 2, what is the output?I Output is y = 7. Also written f (2) = 7.

If input is x = −10, what is the output?I Output is y = −29. Also written f (−10) = −29.

If input is x = 13 , what is the output?

I Output is y = 2. Also written f ( 13 ) = 2.

23

Function: by formula

Describe a function using an expression:

f (x) = 3x + 1

input is x

output

If input is x = 2, what is the output?I Output is y = 7. Also written f (2) = 7.

If input is x = −10, what is the output?I Output is y = −29. Also written f (−10) = −29.

If input is x = 13 , what is the output?

I Output is y = 2. Also written f ( 13 ) = 2.

23

Function: by formula

Describe a function using an expression:

f (x) = 3x + 1

input is x

output

If input is x = 2, what is the output?

I Output is y = 7. Also written f (2) = 7.If input is x = −10, what is the output?

I Output is y = −29. Also written f (−10) = −29.

If input is x = 13 , what is the output?

I Output is y = 2. Also written f ( 13 ) = 2.

23

Function: by formula

Describe a function using an expression:

f (x) = 3x + 1

input is x

output

If input is x = 2, what is the output?I Output is y = 7. Also written f (2) = 7.

If input is x = −10, what is the output?I Output is y = −29. Also written f (−10) = −29.

If input is x = 13 , what is the output?

I Output is y = 2. Also written f ( 13 ) = 2.

23

Function: by formula

Describe a function using an expression:

f (x) = 3x + 1

input is x

output

If input is x = 2, what is the output?I Output is y = 7. Also written f (2) = 7.

If input is x = −10, what is the output?

I Output is y = −29. Also written f (−10) = −29.

If input is x = 13 , what is the output?

I Output is y = 2. Also written f ( 13 ) = 2.

23

Function: by formula

Describe a function using an expression:

f (x) = 3x + 1

input is x

output

If input is x = 2, what is the output?I Output is y = 7. Also written f (2) = 7.

If input is x = −10, what is the output?I Output is y = −29. Also written f (−10) = −29.

If input is x = 13 , what is the output?

I Output is y = 2. Also written f ( 13 ) = 2.

23

Function: by formula

Describe a function using an expression:

f (x) = 3x + 1

input is x

output

If input is x = 2, what is the output?I Output is y = 7. Also written f (2) = 7.

If input is x = −10, what is the output?I Output is y = −29. Also written f (−10) = −29.

If input is x = 13 , what is the output?

I Output is y = 2. Also written f ( 13 ) = 2.

23

Function: by formula

Describe a function using an expression:

f (x) = 3x + 1

input is x

output

If input is x = 2, what is the output?I Output is y = 7. Also written f (2) = 7.

If input is x = −10, what is the output?I Output is y = −29. Also written f (−10) = −29.

If input is x = 13 , what is the output?

I Output is y = 2. Also written f ( 13 ) = 2.

23

Function: by formulaYou try it!

Let g be the function defined by g(θ) = 2θ.

1 What symbol represents the input?

I θ

2 What expression is the formula for the output?

I 2θ

3 If the input is 3, what is the output? Simplify.

I g(3) = 8

4 If the input is 4, what is the output? Simplify.

I g(4) = 16

5 If the input is 0, what is the output? Simplify.

I g(0) = 1

6 If the input is −1, what is the output? Simplify.

I g(−1) = 12

Discuss in groups of three or four.

24

Function: by formulaYou try it!

Let g be the function defined by g(θ) = 2θ.

1 What symbol represents the input?I θ

2 What expression is the formula for the output?I 2θ

3 If the input is 3, what is the output? Simplify.I g(3) = 8

4 If the input is 4, what is the output? Simplify.I g(4) = 16

5 If the input is 0, what is the output? Simplify.I g(0) = 1

6 If the input is −1, what is the output? Simplify.I g(−1) = 1

2

Discuss in groups of three or four.

24

Function: by graph

x

y

for every input on x-axis

record its output as height (y-coord)

f (x) = 3x + 1

Vertical Line TestEach vertical line touches the graph in at most one point.

25

Function: by graph

x

y

for every input on x-axis

record its output as height (y-coord)

f (x) = 3x + 1

Vertical Line TestEach vertical line touches the graph in at most one point.

25

Function: by graph

x

y

for every input on x-axis

record its output as height (y-coord)

f (x) = 3x + 1

Vertical Line TestEach vertical line touches the graph in at most one point.

25

Function: by graph

x

y

for every input on x-axis

record its output as height (y-coord)

f (x) = 3x + 1

Vertical Line TestEach vertical line touches the graph in at most one point.

25

Function: by graph

x

y

for every input on x-axis

record its output as height (y-coord)

f (x) = 3x + 1

Vertical Line TestEach vertical line touches the graph in at most one point.

25

Function is consistent: vertical line test

The rule for a function is consistent:

Each person representing input knew exactly who to throwthe ball toThe graph of the function satisfies VLT

NOT the graph of a functionFails the VLT

26

Function is consistent: vertical line test

The rule for a function is consistent:Each person representing input knew exactly who to throwthe ball to

The graph of the function satisfies VLT

NOT the graph of a functionFails the VLT

26

Function is consistent: vertical line test

The rule for a function is consistent:Each person representing input knew exactly who to throwthe ball toThe graph of the function satisfies VLT

NOT the graph of a functionFails the VLT

26

Function is consistent: vertical line test

The rule for a function is consistent:Each person representing input knew exactly who to throwthe ball toThe graph of the function satisfies VLT

NOT the graph of a functionFails the VLT

26

Function is consistent: vertical line test

The rule for a function is consistent:Each person representing input knew exactly who to throwthe ball toThe graph of the function satisfies VLT

NOT the graph of a functionFails the VLT

26

Function: by table

input outputU.S.A. Washington D.C.

Canada OttawaMexico Mexico City

United Kingdom LondonFrance Paris

Hungary BudapestThailand BangkokNamibia WindhoekJapan Tokyo

Australia Canberra

27

A graphing exercise

Task: Graph the function f (x) = |x + 2|.

First thoughtAhh! I have no idea what the graph looks like!

Second thoughtPick some random inputs. Write a table of input/output values.

The table you make with random inputs doesn’t completelyrepresent the function, but it’s representative data! Plot thepoints! Connect the dots!

Try it!

28

A graphing exercise

Task: Graph the function f (x) = |x + 2|.

First thoughtAhh! I have no idea what the graph looks like!

Second thoughtPick some random inputs. Write a table of input/output values.

The table you make with random inputs doesn’t completelyrepresent the function, but it’s representative data! Plot thepoints! Connect the dots!

Try it!

28

A graphing exercise

Task: Graph the function f (x) = |x + 2|.

First thoughtAhh! I have no idea what the graph looks like!

Second thoughtPick some random inputs. Write a table of input/output values.

The table you make with random inputs doesn’t completelyrepresent the function, but it’s representative data! Plot thepoints! Connect the dots!

Try it!

28

A graphing exercise

Task: Graph the function f (x) = |x + 2|.

First thoughtAhh! I have no idea what the graph looks like!

Second thoughtPick some random inputs. Write a table of input/output values.

The table you make with random inputs doesn’t completelyrepresent the function, but it’s representative data! Plot thepoints! Connect the dots!

Try it!

28

A graphing exercise

Task: Graph the function f (x) = |x + 2|.

First thoughtAhh! I have no idea what the graph looks like!

Second thoughtPick some random inputs. Write a table of input/output values.

The table you make with random inputs doesn’t completelyrepresent the function, but it’s representative data! Plot thepoints! Connect the dots!

Try it!

28

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)

0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0

|0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2|

= |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2|

= 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 2

1 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 2

1 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21

|1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2|

= |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3|

= 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 3

2 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 3

2 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32

|2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2|

= |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4|

= 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 4

3 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 4

3 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43

|3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2|

= |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5|

= 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5

−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5

−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5

−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1

| − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2|

= |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1|

= 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1

−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1

−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2

| − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2|

= |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0|

= 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0

−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0

−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3

| − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2|

= | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1|

= 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1

−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1

−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4

| − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2|

= | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2|

= 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

29

Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

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Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

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Graphing f (x) = |x + 2|Getting help by making a table first

x f (x)0 |0 + 2| = |2| = 21 |1 + 2| = |3| = 32 |2 + 2| = |4| = 43 |3 + 2| = |5| = 5−1 | − 1 + 2| = |1| = 1−2 | − 2 + 2| = |0| = 0−3 | − 3 + 2| = | − 1| = 1−4 | − 4 + 2| = | − 2| = 2

x

y

This table does not represent ALL inputs/outputs – just SOME.

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Why graphs of functions? Why functions?

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Why graphs of functions? Why functions?

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Summary

Expression: Notation representing a number

Equation: Assertion that two expressions are equal

Function: A rule assigning output number for each input

I Describe by arrow diagramI Describe by formulaI Describe by graphI Describe by table

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Summary

Expression: Notation representing a number

Equation: Assertion that two expressions are equal

Function: A rule assigning output number for each inputI Describe by arrow diagramI Describe by formulaI Describe by graphI Describe by table

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Fun activity

The main point of this class is to introduce new types offunctions and to use functions in new ways.Go to

student.desmos.com

and type in our class code.

Do this before tomorrow!

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