pr2015 functions and relations.docx

8/20/2019 PR2015 FUNCTIONS AND RELATIONS.docx

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[Typethe document title]

LESSON 1 Function Models RevisitedOne of the most important and controversial problems in Earth and space science todayis measuring, understanding, and predicting global arming! "here is deep concern thatthe average annual surface temperature on Earth has been increasing over the pastcentury and that this change ill have important conse#uences for industry, agriculture,and personal lifestyles!

Many scientists believe that the most li$ely variable contributing to the increase inorld temperature is greenhouse gases that reduce radiation of energy from Earth%ssurface into space! "he ne&t graph gives data on change in atmospheric greenhousegases over the past 1,''( years!

)tmospheric *oncentrations of *arbon +io&ide, 1'''-''(

Source. !earth/policy!org0ndicators0*O-0-''23data4!htm

"hin$ )bout "his Situation

"he challenge for atmospheric scientists is deciding ho current trends in greenhouse gas

orld temperature change should be pro5ected into the future! +ifferent pro5ections imply

corrective actions

a. n hat ays do you imagine that future global arming could change Earth%s a

your on life6b. 7ased on the data given in the graph on the previous page, hat strategy for pro5

global temperature ould ma$e most sense to you6c. 7ased on the data given in the graph above, hat strategy for pro5ecting change

carbon dio&ide ma$es most sense to you6

Functions and models Page 1

http://www.earth-policy.org/Indicators/CO2/2008_data3.htmhttp://www.earth-policy.org/Indicators/CO2/2008_data3.htm


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nvestigation 1. 8ariables Related by +ata in "ables

One ay to investigate particular relationships is to study a table of values containingtypical pairs of values, 9input, output:! *onsider the folloing situation!

;eople can be found of many shapes and si 1?

@eight 9cm: A1 2> BB 11- 1-( 14( 1>( 1A2 1?'

Source. "he Corld )lmanac and 7oo$ of Facts! N=. ;haros 7oo$s! 1BB-

1! Chat can you say about the relation beteen the age and the average height6

)ge and height are numerical or #uantitative variables, and e can say that heightdepends on age or height is a function of age! 7ecause height depends on age, itshould seem reasonable to thin$ of age as the input variable and height as the outputvariable in the relation! )ccording to the table, the average height of 1>/year/old

females is 1A2 centimeters! Mathematicians and scientists might use the symbolicshorthand form @91>: D 1A2 to convey this information!

Ce read @91>: D 1A2 as @ of 1> e#uals 1A2!

se of notation li$e this indicates a relation in hich one variable is a function of an/other! n this case, height is a function of age!

"here are many interesting #uestions that you can anser by e&amining data in thetable For e&ample.

-! @o much does the average height of females increase from birth to ageto6 @o does that increase compare to the increase beteen ages 1? and

-'6

"hese changes may be calculated by subtraction.@eight at - G @eight at ' D @9-: G @9l: D 2> G A1 D 44!@eight at -' G @eight at 1? D @9-': G @91?: D 1?A G 1?' D A!

"hus the average height increases by 44 centimeters in the first to years andonly by A centimeters in the four/year period beteen ages 1? and -'!

4! )t hat age does the average height of females reach 1'' cm6"he table shos that average height is BB cm at age > and 11- cm at age ?H sothe average height reaches 1'' cm for some age beteen > and ? years!



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>! +uring hich -/year period shon in the table does the average height offemales increase the most6 "he folloing table shos the height differences,and the largest one has been mar$ed!

)ge 9yrs: ' to - - to > > to ? ? to 2 2 to 1' 1' to 1- 1- to 1> 1> to

*hange in height

9cm:

2>/A1

a! Chen the greatest increase occurs6 Chy6

"he age and height data given in the table of Situation -!1 are averages! "he actualgroth pattern for any individual girl might be #uite different from those averages!*onsider the folloing data for Elda, ho found the information in health chec$uprecords that her mother had $ept

)ge 9years: ' - > ? 2 1' 1- 1> 1?

@eight 9cm: A? 2A B' 1'- 114 1-A 1A- 1?B 1(2

A! EldaIs height is a function + of her age!

a! )ccording to the table, hat as +anIs height at age >6 "hat is,complete the folloing statement!

b! + 9>: D3333333 c! )ccording to the table, at hat age as +anIs height 1'- cm6

"hat is, complete the folloing statement!d! + 9333333: D 1'-!e! Chat information is conveyed by + 912: D 1246f! +id +an gro more beteen the ages of ? and 2 or beteen the

ages of 1> and 1?6g! Cas +an shorter or taller than the average height for females at

age >6 Chat about at age 1'6 )ge 1-6h! n hich to/year interval did +an gro the fastest6 7y ho

much did she gro and ho does this compare ith the increase

in average height for females beteen the same ages6

Summari


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*hec$ your understanding1! "he table belo gives the average heights of )merican males at specific

ages

)ge 9years: ' - > ? 2 1' 1- 1> 1? 12

@eight 9cm: A- 2( 1'4 11? 1-( 142 1A' 1?4 1(> 1((

Source. "he Corld )lmanac and 7oo$ of Facts! 1BB-!

"he average height of males is a function M of age!

a! )t hat age is the average height of males 1A' cm6 b! se functional notation to rite the sentence, "he average height

of males at age 1> is 1?4 cm! @int. M9 : D

c! *omplete the folloing statement. M9-: Dd! Crite in ords the meaning of M92: D 1-(!e! n hich of the to/year periods shon by the table does the

average height increase the most6 Chat is that increase6

f! @o does the pattern of change in average height for malescompare to that for females6

On your on1! "he "alent Sho advisor found the folloing record of profits from

previous "alent Shos! Note. Negative profit means the sho lost money!

=ear 1B2' 1B21 1B2- 1B24 1B2> 1B2A 1B2? 1B2(

;rofits 9J: /--1 /1AA /12 -44 2>( 1242 1(A' -144

;rofit can be considered a function ; of the year!

a! n hich year did the "alent Sho ma$e the greatest profit6 b! "he amount of profit is a function of the year! Chat does ; 91B21:

D G1AA tell about "alent Sho profit6

c! n 1B2A the "alent Sho had a profit of J1242! sing functionalnotation, e&press this fact!d! n hich year did "alent Sho profit increase the most from the

previous year6 Crite the calculations needed to find each year/to/year change as follos. ;91B21: G ;91B2': D 9/1AA: G 9/--1: D ??!"hen use your calculator to find the results!

e! 7eteen hich consecutive years did the profit decrease6f! +escribe the overall trend in "alent Sho profits shon in the table

and ma$e a prediction about profit that could be e&pected from the1B22 sho!

-! "he folloing table gives the number of record albums sold in the nitedStates for some recent years!

=ear 1B(4 1B(A 1B(( 1B(B 1B21 1B24 1B2A 1B2(

)lbums solds

9millions:-B' 444 4>A 44' -2' -4' 1BA 1(2

a! Chat trend, or pattern of change, do you see in the data in the table6Offer a possible e&planation for this trend!



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b! "he number of albums sold, ), is a function of the year! Chatinformation does the folloing sentence, ritten in functionalnotation, give about album sales6 ) 91B21: D -2' million!

c! Crite the folloing statement in functional notation. 44' millionalbums ere sold in 1B(B!

d! n hich to/year period did the reported sales change the most6

Chat as that change6e! Chat estimate do you believe is most reasonable for album salesin 1B2B6

4! ) fast/food restaurant found that the number of orders of french fries that itsells depends on the price of an order of fries

;rice 9J: '!A' '!?' '!(' '!2' '!B' 1!''

Orders of fries

per day1''' 2A' ('' AA' >'' -A'

a! +escribe the relation beteen price and number of orders that yousee in the table and e&plain hy it is or is not reasonable to you!

b! Crite functional notation for each of the folloing! @int. Since thenumber of orders depends on the price, a reasonable ay to rite therelation might be N9price: D number of orders!

i! Chen the price is fifty cents, 1''' orders of fries are sold!ii! Only -A' orders are sold hen one dollar is charged

c! E&plain in ords the meaning of each of these symbolic statementsi! N9'!2': D AA'ii! N9'!?': D 2A'

d! f the manager ants to sell at least A'' orders each day,hat price 9s: should the restaurant charge6

e! f the restaurant chain ill only allo the price of fries to be beteen seventy and eighty cents an order, ho many orders

of fries ill be sold each day at this location6

>! Suppose that the manager of the restaurant in problem > decides to try a nelarge si


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nvestigation -. 8ariables Related by +ata in Kraphs

"he ays in hich to variables might be related are not alays shon clearly bytables of input/output data! ;atterns in the data may be lost amid all the specificnumbers! @oever, hen data are displayed in a graph, it is often much easier to seetrends and therefore to ma$e predictions

1! "he table displays ordered pairs of data 9age, height:! Each pair also can beconsidered as coordinates hich locate e&actly one point on the graph! For e&ample, the point ith coordinates 91?, 1?': represents the fact that at age1? the average height of females is 1?' centimeters!

-! t is customary to indicate values of the input variable on the hori


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) drought in the )frican veldt causes the death of many animals! For e&ample, atypical herd of ildebeest might suffer losses li$e those indicated in the folloing table

Length of

drought 9months:' 1 - 4 > A ? ( 2 B 1'

Number of

ildbeetsA'' >'' 4'' -A' -'' 1(A 1A' 1-A 1'' B' (A

1! ;lot the ordered pairs of data given in the table above

-! Kive the

coordinates for the points that anser the folloing #uestionsa! Chat as the population after three months of drought b! @o long had the drought been going on hen the herd reached a

population of 1-A6c! For hat length of time did the population remain above -''6

4! Chat pattern in the graph shos the drop in population6

Summari


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*hec$ your understanding

Chen N)S) sends a space shuttle on a mission, scientists andengineers monitor conditions on the shuttle ith many differentinstruments! For e&ample, heat sensors in the nose cone steadily sendtemperature readings to N)S) computers! "hese readings are atched

closely from several hours before liftoff until the shuttle lands safely!"he table belo gives a sample of readings, 9time, temperature:, for atypical mission! "emperature of the nose cone is a function of time inthe mission! Notice that time before liftoff is indicated by negativenumbers and that temperatures also ta$e on both positive and negativevalues

Mission time

9min:/-A /-' /1A /1' /A ' A 1' 1A -' -A 4' 4A >' >A A'

Nose cone

temperature

9*elsius:

/1' /1' /A /A ' 1' (A B' 11A 14' 1'A 2' -' /A/

4'/>'

1! ;lot the data pairs, 9time,temperature:, given in thetable! Notice that pointsshoing negative time 9time before liftoff: are located onthe left side of the verticala&is and that points shoingnegative temperature are belo the hori


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ON =OR OCNSE" )

1! List the coordinates of each of the labeled points as an ordered pair! Estimate herenecessary!

-! On graph paper, dra a pair of coordinate a&es! Label the a&es ithappropriate units, then plot the folloing ordered pairs! )t each point, ma$e aheavy dot and rite the coordinates

a! 9', ': b! 9/4, 2:c! 92, /4:d! 9/2, /4:e! 9/A, ':f! 9', /A:g! 9A, ':h! 9?, />:i! 94, 4: 5! 9(, (:$! 9/A, /A:

l! 9/(, /(:

1! ) certainfunction has the graphshon in the figure belo!

a! *opyand complete thefolloing table for thisfunction

& ' 1 - 4

y

4! List the coordinates of each labeled point as an ordered pair! Noticethat each hori


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>! ) hot/air balloon is launched at noon, and the balloonist records the altitude inmeters every 1A minutes! "he recorded data are given in the folloing table

"ime 9min: ' 1A 4' >A ?' (A B' 1'A 1-' 14A 1A' 1?A

)ltitude 9m: ' 1'' (A' 2(A B'' BA' B'' B-' A'' ?-A 4'' '

a! *hoose reasonable scales for the hori


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h! Chat as the change in altitude during the second hour6i! Chat as the ma&imum altitude6 5! )t hat time 9s: as the balloon rising most #uic$ly6$! Chen as the balloon falling most rapidly6

A! n each of the folloing situations you are given data relating to variables! neach case, choose reasonable scales for a&es on a graph and plot the given

data! "hen rite a sentence describing the pattern in the graph and hat itsays about the relation beteen the to variables!

"he mass of an average )merican young person, in $ilograms, is a function of the personIs age in years! @ere are some sample data

)ge 9years: ' - > ? 2 1' 1- 1>

Mass 9$g: 4 11 1A -' -? 41 42 >B

) nespaper delivery person%s ee$ly pay is a function of the number of papers delivered each day! @ere are some sample data!

;apers delivered A' 1'' 1A' -'' 4'' A''

;ay dollars (' 1-' 1(' --' 4-' A-'

)nimal populations tend to rise and fall in cycles! Suppose the folloing datashos ho s#uirrel populations in a central ;ennsylvania city varied from1B(A to 1B2>

=ear (A (? (( (2 (B 2' 21 2- 24

population (A' ('' A-' ?2' (4' ?A' AA' ?-A (2'

Set -

1! Find the values of & and y!a! 9&, -&: D 94, ?:

b! x+ x+ =c! 9A&/1, : D 9/11, >: d! 2 x 3 +4 = x+e! 94& 1, -y A: D 94, /-: f!

-! "he coordinate graph ma$es it possibleto study geometry figure by means of numbers because each point of the figurecan be located by a pair of numbers! "hefolloing graph shos a circle hosecenter is 9A, A:! Chat are the coordinatesof each point on the circle that is

identified by a letter!



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One of the most important and controversial problems in Earth and space science todayis measuring, understanding, and predicting global arming! "here is deep concern thatthe average annual surface temperature on Earth has been increasing over the pastcentury and that this change ill have important conse#uences for industry, agriculture,and personal lifestyles!

"he graph that follos shos the pattern of change in average orld temperature over

the past 1>( years! Chile the average global temperature has increased by less than adegree, this is still a large amount relative to historical data! "his recent temperatureincrease is four to five times faster than any other climate change in the pastmillennium!

)nnual +eviation from )verage Klobal "emperature,1B?11BB'

Source. "he Cashington ;ost, October 14, -''AH !cru!uea!ac!u$0cru0info0arming0


http://www.cru.uea.ac.uk/cru/info/warming/http://www.cru.uea.ac.uk/cru/info/warming/http://www.cru.uea.ac.uk/cru/info/warming/


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!a " i a # l e s a n d $ u n c t i o n s % 15

Many scientists believe that the most li$ely variable contributing to the increase in orld temperature is greenhousegases that reduce radiation of energy from Earth%s surface into space! "he ne&t graph gives data on change inatmospheric greenhouse gases over the past 1,''( years!

)tmospheric *oncentrations of *arbon +io&ide, 1'''-''(

Source. !earth/policy!org0ndicators0*O-0-''23data4!htm

n$ )bout "his Situation

challenge for atmospheric scientists is deciding ho current trends in greenhouse gas amounts and

ld temperature change should be pro5ected into the future! +ifferent pro5ections imply different

ective actions

n hat ays do you imagine that future global arming could change Earth%s atmosphere andyour on life67ased on the data given in the graph on the previous page, hat strategy for pro5ecting change in

global temperature ould ma$e most sense to you67ased on the data given in the graph above, hat strategy for pro5ecting change in atmospheric

carbon dio&ide ma$es most sense to you6

Kraphs are used so often because they tell a story more easily than it can be told ith ords or numbers! "he story in thefolloing graph is apparent at a glance. "he number of tropical storms in the )tlantic increases to a ma&imum around theend of )ugust, then falls off rapidly! "here is a moderate increase in early October, after hich the decreasing pattern

resumes


http://www.earth-policy.org/Indicators/CO2/2008_data3.htmhttp://www.earth-policy.org/Indicators/CO2/2008_data3.htmhttp://www.earth-policy.org/Indicators/CO2/2008_data3.htm


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"he folloing activity as$s you to reflect on several familiar real/orld situations in hich one #uantity is related to asecond! =ou ill be as$ed to thin$ about ho graphs describing these situations might appear! Loo$ for patterns andtrends as you analy


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ion 1. Carm/up activity

Note that the graphs in the activity 9as others you ill be as$ed to dra: are shon ithout numerical scales! "hey sho#ualities that capture the $ey features of the situation 9patterns and trends:, but do not sho e&act #uantities!

For each of the folloing si& scenarios, a conte&t and a figure shoing several graphs are given! )fter discussing theconte&t ith your partner or group, anser the folloing #uestions for each situation!

• E&amine each of the graphs in the figure! Chich graph best models the situation6

• Chat features made you choose that particular graph6 Chat features made you discount the other graphs6

• Chat are the to #uantities or variables in the situation6

1. Situation 1. "he height of a person over his or her lifetime!

2. Situation -. "he circumference of a circle as its radius changes!

3. Situation 4. "he height of a ball as it is thron into the air!

4. Situation >. "he amount of observable mold on a piece of bread sitting at room temperature from the time it is ba$ed to several months later!

5. Situation A. "he daily average lo temperature in degrees Fahrenheit in Fairban$s, )las$a from anuary 1 to+ecember 41!

?! Situation ?. "he temperature of a cold drin$ left in a arm room

Situation 1 Situation -



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Situation 4 Situation >

Situation A Situation ?

ion -. Relations and Functions

se this machine to anser the #uestions on the ne&t page!

+8+ 8ending Machine



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1. Suppose you inserted your money and pressed )l! Chat item

ould you receive6

2. Suppose you inserted your money and pressed *-! Chat item

ould you receive6

3. Suppose you inserted your money and pressed 74! Chat item

ould you receive6

4. f the machine ere filled properly, hat ould happen if you

pressed any of those same buttons again6

Each time you press a button, an input, you may receive a +8+, an output!

5. n the +8+ vending machine situation, does every input have an

output6 E&plain your response!

6. Each combination of input and output can be e&pressed as a

mapping ritten input output! For e&ample, 7- Ci


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) relation is a set of ordered pairs! "he list of ordered pairs that you rote in tem ?9b: is a relation!

Relations can have a variety of representations!

*onsider the relation P91, >:, 9-, 4:, 9?, A:Q shon here as a set of ordered pairs!

"his relation can also be represented in these ays!

7. =ou represented the vending machine situation using

mappings in tem ?! Other representations can also be used to illustrate ho the inputs and outputs of thevending machine are related!

a. *reate a table to illustrate ho the inputs and outputs of the vending machine are related!

b. n representing the vending machine inputs and outputs, hat decisions ould need to be made to

create the graph6

) function is a relation in hich each input is paired ith, at most, one output!

8. *ompare and contrast the +8+ 8ending Machine ith a function!

1. Suppose hen pressing button *1 button on the vending machine both Finding +reamo and Raiders of the

Mossed 7ar$ come out! @o does this vending machine resemble or not resemble a function6

1'! magine a machine here you input an age and the machine gives you the name of anyone ho is that age*ompare and contrast this machine ith a function! E&plain by using e&amples and create a representation ofthe situation!

11! *reate an e&ample of a situation 9math or real/life: that behaves li$e a function and another that does not behaveli$e a function! E&plain hy you chose each e&ample to fit the category!

a. 7ehaves li$e a function.



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b. +oes not behave li$e a function.

1-! dentify hether each list of ordered pairs represents a function! E&plain your ansers!

14! sing positive integers, rite to relations as a list of ordered pairs belo, one that is a function and one that isnot a function!

Function.

Not a function.

"he set of all inputs for a function is $non as the domain of the function! "he set of all outputs for a function is $nonas the range of the function!1>! *onsider a vending machine here inserting -A cents dispenses one pencil, inserting A' cents dispenses -

pencils, and so forth up to and including all 1' pencils in the vending machine!

a! Chat is the domain in this situation6

b! Chat is the range in this situation6

1A! For each function belo, identify the domain and range!



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1?! Each of the functions that you have seen has a finite number of ordered pairs! "here are functions that have aninfinite number of ordered pairs! +escribe any difficulties that may e&ist trying to represent a function ith aninfinite number of ordered pairs using the four representations of functions that have been described thus far!

1(! Sometimes, machine diagrams are used to represent functions! n the function machine belo, the inputs arelabeled & and the outputs are labeled y! "he function is represented by the e&pression -& A!

a. f & D ( is used as an input, hat is the output6

b. f & D G- is used as an input, hat is the output6

c. f & D 1is used as an input, hat is the output6

d. s there any limit to the number of input values that can be used ith this e&pression6 E&plain!

*onsider the function machine belo!

12! se the diagram to find the 9input, output: ordered pairs for the folloing values!



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a. & D /Ab. &D Ac. &D G1'

19. Ma$e a function machine for the e&pression 1' G A&! se it to

find ordered pairs for & D 4, & D G?, & D '!-A, and & D >!

*reating a function machine can be time consuming and a$ard! "he function represented by the diagram in tem 1(

can also be ritten algebraically as the e#uation y D -& A!

19. Evaluate each function for & D G-, & D A, & D >, and & D '!(A

For each &/value, find the corresponding y/value! ;lace the results in a table!

a. y D B G >&

b. y D

Chen referring to the functions in tem -', it can be confusing to distinguish among them since each begins ith y D!

Function notation can be used to help distinguish among different functions!

For instance, the function y D B G >& in tem -'9a: can be ritten.

-1! "o distinguish among different functions, it is possible to use different names! se the name h to rite thefunction from tem -'b using function notation!

Function notation is useful for evaluating functions for multiple input values! "o evaluate x = − x for

x= , you substitute - for the variable & and rite = − Simplifying the e&pression yields

= !

--! se function notation to evaluate f9&: shon above at & D A, & D G4, and & D '!A!

23. se the values for & and f9&: from tem --! +isplay the values using each representation!

a. list of ordered pairs

b. table of values



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c. mapping

d. graph

->! Evaluate each function for & D GA, & D /1'

a. x = x−

b. x = x−

c. x =− − x

;R)*"*E EER*SESCrite your ansers on noteboo$ paper! Sho your or$!

1. "he set P94, A:, 9/1, -:, 9-, -:, 9', /1:Q represents a function! dentify the domain and range of the function! "hendisplay the function using each representation!

a. a table

b. a mapping

c. a graph

2. E&plain hy each of the folloing is not a function!

3. Evaluate the functions for each domain value indicated!

a. x = x+ for & D GA, ', >

b. = − for t D G-, ', A, (

ON =OR OCN



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+etermine hich of the folloing relations is a function ! Kive the domain and range

1

-

4

>

A

n e&ercises ? 11, the d mains + and rules of some functions are given! Find the range of each function

6. :→ − x =

7. : x→ x− =

8. → − = −

9.

→ − = −10. G : x → x −4 x+4 D= 0 1 2 3

11. m : z →1− z D= −2 −1 0 2

Kive the domain of each function



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12. −

13. g ( x )=+

14. h ( x )= 2

15. = −

16. f ( x )=2

17. G ( x )= x−1 x +2

Let = − and x = − x ! Find the indicated values

18.

19.

20. 2 − 1

21. − −22. 5 −2 2

23. f 1

3

24. g 1

3

25. a+ − a26. a+1 − a

27.g x + −g x



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Si f ( x )=−1 , si x0

, find the indicated values

28.

29.

30. −

31.

32. −

M)"@EM)"*)L REFLE*"ON. Chich representation of a function do you feel is most useful6 Chy6 Chich one do

you feel is least useful6 Chy6ation 4. +omain and range of continuous functions

Roller coasters are scary and fun to ride! Cooden roller coasters sha$e and rattle as part of the thrill of the ride! 7elo isthe graph of the heights reached by the cars of the ooden roller coaster, "hunderball, over its first 1-A' feet of trac$!"he graph displays a function because each input value has one and only one output value! =ou can see this visuallyusing the vertical line test!

Study this graph to determine the domain and range!

"he domain gives all values of the independent variable. distance along the trac$ in feet! "hese values are graphedalong the hori


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"he range gives the values of the dependent variable. height above the ground in feet! "he values are graphed on thevertical or y/a&is!

"he range can be ritten in set notation as.

Pall real values of y. 1' y 11'Q

Read this notation as. the set of all real values of y, beteen 1' and 11', inclusive!

"he graph above shos data that are continuous! "he points in the graph are connected, indicating that domain and rangeare sets if real numbers ith no brea$s in beteen! ) graph of discrete data consists of individual points that are notconnected by a line or curve!

la! se set notation to rite the domain and range for the graph belo! +oes this graph appear to represent afunction6 ustify your anser! )re the data discrete or continuous6 Chy6

1 b! "he graph belo shos the relationship beteen t, the length of time of the bath 9from the time ater starts

running through the time the tub is drained: and d, the depth of the ater in the bath tub! "he graph representsfunction d 9bath ater depth:! Chat are the dependent and independent variables6 E&plain! se set notation torite the domain and range of function d! )re the data discrete or continuous and hy6



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E)M;LE 1. Kive the domain and range of the function f ( x )=− 2

graphed belo!

Step 1. Study the graph!

"he s$etch of this graph is a portion of the function represented by the e#uation

f ( x )=− 2

!

Step -. Loo$ for values for hich the domain causes the function to be undefined! Loo$ ho the graph behaves near & D -!

Solution. "he domain and range for f ( x )=− 2

can be ritten.

+omain. Pall real values of &. & ≠ -Q

Range. Pall real values of y. y 'Q

"R= "@ESE

a! Kive the domain and range of the function

= + −graphed belo



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b. Kive the domain and range for the e#uation y D -& / 1! E&plain hether this e#uation represents a function andho you determined this!

"echnology "ime

•Cor$ ith a partner to investigate the e#uations listed in the chart using graphing technology! Every e#uation

given here is a function!

•+etermine the domain and range for each function from the possibilities listed belo the chart!

•Select the appropriate domain from choices 1/? and record your anser in the +omain column! "hen select the

appropriate range from choices a/f and record the appropriate range in the Range column!

•Chen the chart is complete, compare your ansers ith those from another group!

Function +omain Range

=− +

= − +

= −= x+

= +

y=

;ossible domains ;ossible Ranges

1: all real numbers b: all real numbers

-: all real &, such that & ≠− a: all real y, such that y ≠ '



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4: all real &, such that & ≠ b: all real y, such that y >

>: all real &, such that & ≠ c: all real y, such that y '

A: all real &, such that & d: all real y, such that y 1

?: all real &, such that & e: all real y, such that y 4

ON =OR OCNCrite your ansers on noteboo$ paper! Sho your or$!

1! Kive the domain and range for the function graphed belo! E&plain hy this graph represents a function!

-! ) student calculates ho far aay a lightning stri$e is, based on hen the thunder is heard! "he student ma$esthe table belo using t$m0sec as the average speed of sound under rainy conditions! f the thunder is only heardhen the lightning stri$e is ithin 1A $m of the listener, hat are the domain and range for this model6 s thisrelation a function6 @o do you $no6

"ime until thunder is heard 9sec: 1 - 4 > A ?

+istance from lighting stri$e9$m: 1 1 1 -

3. Kive the domain and range of the function

x =− x−

>! "he graph belo shos five points that ma$e up the function h! Kive the domain and the range for the functionh!



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5. eff al$s at an average rate of 1-A yards per minute! Mar$Is house is located -''' yards from effIs house! "hegraph belo shos ho far eff still needs to al$ to reach Mar$Is house! Kive the domain and range for thismodel! s this model a function6 E&plain!



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?! *apital letters s$etched in the coordinate plane may or may not be functions! ;ic$ one letter that represents afunction and to that do not! se the vertical line test as part of the e&planation for your selections!

(! M)"@EM)"*)L REFLE*"ON. +escribe at least three different methods for determining if a relation is afunction! Chich method do you prefer and hy6



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ERFORM)N*E ")ST . Representations of functions

Chile on vacation, orge and ac$ie traveled to 7ryce *anyon National ;ar$ in tah! "hey ere impressed by thediffering elevations at the viepoints along the road! "he graph describes the elevations for several viepoints in termsof the time since they entered the par$!

E9t:

"ime 9min: after entering the par$

1. "he graph above represents a function E9t:! +escribe hy the graph represents a function! dentify the domain

and range of the function!

2. s this discrete or continuous data6 E&plain!

Chile at 7ryce *anyon National ;ar$, orge and ac$ie ent hi$ing on the nder the Rim trail to =ello *ree$! "hetable shos their progress don to =ello *ree$! "he grid is provided for optional use to help you anser the#uestions belo!

+escent

"ime 9min: Elevation 9ft:

' (B''

1' (A''-' (1''

4' ?(''



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3. Find the slope for the data in the table! nterpret the slope as a rate of change, including units!

4. On the descent, hat as the elevation 12 min after orge and ac$ie began6 ustify your anser!

5.On the descent, hen ere they at (''' ft6 ustify your anser!



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- Lineal Functions

n$ )bout "his Situation

"hin$ about the connections among graphs, data patterns, function rules, and problem

conditions for linear relationships!

a @o does 7arryIs daily pay change as the number of applications he collects increases6 @o is

that pattern of change shon in the graph6

b f the linear pattern shon by the graph holds for other 9number of applications, daily pay: pairs,

ho much ould you e&pect 7arry to earn for a day during hich he collects 5ust 1 application6

For a day he collects 14 applications6 For a day he collects -A applications6

b. Chat information from the graph might you use to rite a rule shoing ho to calculate daily pay for any number of applications6



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nvestigation 1. Carm/up activity

1. sing the a&es li$e the ones belo, s$etch a graph to illustrate

this situation

2. *ompare your graph ith those dran by your neighbors! "ry to come to some agreement over a correct

version

3. Crite don en e&planation of ho you arrived at your anser! n particular, anser the folloing #uestions

• Should the graph Uslope upards%or Uslope donards%6 Chy6

• Should the graph be straight line6 Chy6

• Should the graph meet the a&es6 f so, here6 f not, hy not6

ion -. Cor$ing under pressure

Chen a diver descends in a la$e or ocean, pressure is produced by the eight of the ater on the diver! )s a diver simsdeeper into the ater, the pressure on the diverIs body increases at a rate of about 1 atmosphere of pressure per 1' meters

of depth! "he table and graph belo represent the total pressure, y, on a diver given the depth, &, under ater in meters

a! Match each function rule ith its graph! E&plain ho you could ma$e the matches ithout calculations orgraphing tool help!

b! Chat do the numbers in the rules for +arrylIs and FeliciaIs account balances tell you about the values of their purchases and their monthly payments6



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1.Crite an e#uation describing the pressure e&erted on a diver hen under ater!

2.Chat is the slope of the e#uation of the line that you found6 Chat are the units of the slope6

1.Chat is the y/intercept of the line6

Slope/ntercept Form of a Linear E#uation

= +

here m is the slope of the line and b is the y/intercept!

2.dentify the slope and y/intercept of the line described by the e#uation y D G-& B!

5. *reate a table of values for the e#uation y D G-& B! "hen plot the

points and graph the line


& y

' 1

1 1!1

- 1!-

4 1!4

> 1!>

A 1!A

? 1!?

&nit 2 F. 'inen( Functions 1


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6. E&plain ho to find the slope from the table!

7. E&plain ho to find the y/intercept from the table!

8. E&plain ho to find the slope from the graph!

9. E&plain ho to find the y/intercept from the graph

E)M;LE 1. Crite the e#uation, in slope/intercept form, of the line that passes through the point 91, >: and has a slopeof G4!

Step 1. Find the y/intercept by substituting the coordinates of the point and the slope in the e#uation!

= +

=− +Substitute /4 for m, 1 for &, and >

for y

=− +

+ =− + +

="he y intercept is (

Step -. Substitute the slope and y/intercept into the slope/intercept form!

= +

=− x+



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Solution.

=− +

"R= "@ESE )

a.Crite the e#uation, in slope/intercept form, of the line ith a slope of >and a y/intercept of A!

a.Find the e#uation, in slope/intercept form, of the line that passes through

the point 9/4, (: and has a slope of G 4!

b. Crite the e#uation of the line shon in the graph at the right!

=ou get this form of the e#uation by solving the slope formula m=

y− y1

x− x

for − by multiplying both sides by x− x

"he variable y is the dependent variable, and & is the independent variable! =ou may use this form hen you $no a point on the line and the slope!

E)M;LE -. Crite the e#uation of a line ith a slope of that passes through the point 9-, A:!

Step . Substitute the given values into point/slope form!

− =m x− x

y−5= ( x−2)

Step -. Solve for y

y= ( x−2 )+5

Solution. y= ( x−2)+5

"R= "@ESE 7Find the e#uation of the line given a point and the slope


Point-Slope Form of a Linear Equation

− =m x− x

where m is the slope of the line and ( x ) is a point


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a. 9/-, (:, m=

b. 9?, /1:, m=−

c. 92?, 1-A:, =−

"he ton of San Simon charges its residents for trash pic$up and ater usage on the same bill! Each month the citycharges a flat fee for trash pic$/up and a J'!-A per gallon usage fee for ater! n anuary, one resident used >> gallonsof ater, and received a bill for J1?!

10. f & is the number of gallons of ater used during a month, and y

represents the bill amount in dollars, rite a point 9&1, y1:!

11.Chat does the J'!-A per gallon represent6

12.se point/slope form to rite an e#uation that represents the bill cost y in terms of the number of gallons ofater & used in a month!

13.Crite the e#uation in tem 1- in slope/intercept form! Chat does the y/intercept represent6

E)M;LE 4. Crite the e#uation of the line that passes through the points 9?, >: and 94, A:!

Step 1. Find the slope by substituting the to points into the slope formula!

m= y

2− y

1

x − x =

5−43−6

Substitute 9?, >: for 9 x and

94, A: for 9 x

m= −−

=−

=−

Step -. Substitute the slope and one of the points into point/slope form!

− =m x− x

y−5=−

( x−3) Substitute 94, A: for 9 x , and

− for m

y=−

( x−3 )+5Solve for y



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Solution. y=−

( x−3 )+5

"R= "@ESE *

Find the e#uation in point/slope form of the line shon in the graph!

Find the e#uation in point/slope form of the line ith a slope of A that

passes through the point 91, G4:!

Find the slope and a point on the line/

y=3− ( x+ 3 )

Find the e#uation of the line that passes through the points 9/-, 1: and

9-, 4:!

e! Crite the e#uation of the line from tem 9d: above in slope/intercept form

Standard Form of a Linear E#uation

)& 7y D *

here ) ', ) and 7 cannot both be


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*@E*T =OR N+ERS")N+NK

1. Crite the standard form of the e#uation of the line ith a slope of ( that passes through the point 91, -:!

2. Crite the e#uation 4& G -y D 1? in slope/intercept form!

3.Crite the e#uation y D G> ?9& 1: in slope/intercept form!

4. Crite the e#uation in standard form of the line that is represented by the data in the table!

1. )s a diver descends into fresh ater, e have determined that for every 1' meters of depth, the pressure on the

diver increases by one atmosphere! Find the e#uation of the line if e $no the diver is at a depth of -A metersand has a pressure of 4!A atmospheres!

1.M)"@EM)"*)L REFLE*"ON Rate yourself on a scale of 1 to A on your perceived understanding ofe#uations of lines for Slope/intercept, ;oint/slope, and Standard forms of a line! "he loer the rating the loerthe level of understanding! Chat can you do to bring your level of understanding to a higher level6



45/58


ion 4. *rossing the a&es. and y intercepts

Fran$ is reading @arper LeeIs "o Till a Moc$ingbird for his English class! "he boo$ is -2' pages long! @e estimates heill need seven hours of reading time to finish the boo$! +isplaying y on the vertical a&is as pages left to read and & onthe hori! +ivide each side of the e#uation in tem 4 by the constant term and simplify so that there ill be a 1 on one side

of the e#ual sign!

A! Chat do you recogni6

?! "he e#uation belo is a variable representation of the e#uation that you created in tem >! )s a historicalmathematician, you have the honor of naming this form of the line! Crite your choice of name for this form ande&plain your decision using mathematical vocabulary!

E)M;LE. "he graph of y D -& G ? belo shos the line crossing the &/a&is at 4 and the y/a&is at G?!

a! 8erify the intercepts based on the definition!

b! "hen rite the e#uation in the form shon in tem ?!



46/58


Step 1. se the definition of intercept!From the definition, the y/intercept is the y/coordinate here & D '!

= −

= − Substitute ' for &

=−

Crite as an ordered pair 9', G?:! "he y/intercept is G?!



47/58


From the definition, the &/intercept is the &/coordinate here y D '!

= −

= −Substitute ' for y

− = − −Solve for & by subtracting -& from both

sides

− =−Simplify by dividing both sides by /-

x=

Crite as an ordered pair 94, ':! "he &/intercept is 4!

Step -. se algebra to rite the e#uation in the form used in tem ?!

"o rite = − in the form x + y =1 , isolate the constant to t right of the e#ual sign! "hen divide through

n appropriate number to create the 1 to the right of the e#uals sign

= −

− x=− xSubtract -& from each of the e#uation

− + =−Simplify the ' pair 9-&/-&D':

− x

− +

y

− =

−

−

+ivide each term by /?

x+

y

− =1

Simplify

Solution. "he y intercept is /?, and the & intercept is 4! "he e#uation of the line is x

+ y

− =1



48/58


"R= "@ESE )

a. Crite the e#uation 4& G >y D -> in intercept form! 8erify the intercepts for this e#uation algebraically!

b. Crite the e#uation y=−

x−4 in intercept form!

c. Crite the intercept form of the e#uation that has a y/intercept of GA and an &/intercept of G>!

"he 7ooster *lub is planning to sell refreshments as a pro5ect at the upcoming Fall 7a


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a. f the e#uation of the function in part b above is f 9&: D 4& (, ould you consider your estimate of the


50/58


*hec$ your understandingCrite your ansers on noteboo$ paper or grid paper! Sho your or$!

1! Matt sells used boo$s on the nternet! "he cost of the ee$ly ebsite fee is J(!A', and he earns J1!A' on each

boo$ that he sells! Matt uses the linear e#uation = − to figure his ee$ly earnings here ye#uals earnings in dollars and & e#uals the number of boo$s that he sells!

a. Kraph this function! Set up your a&es as shon belo!b. Crite the e#uation in intercept form!c. Chat meaning do the &/ and y/intercepts have in the conte&t of this problem6



51/58


)ssume that each line crosses the &/ and y/a&es at integer values! Match lines c and d on the graphs ith the interceptform of the e#uations!

Chat are the


52/58


>! "he slope/intercept form of a line is y=−

x−6 ! Crite this e#uation in the intercept form! Kive the


53/58


nvestigation >. E#uation from data

@o fast can you and your classmates pass a te&tboo$ from one person to the ne&t until the boo$ has been relayedthrough each person in class6

1. Suppose your entire class lined up in a ro! Estimate the length of time you thin$ it ould ta$e to pass a boo$

from the first student in the ro to the last! )ssume that the boo$ starts on a table and the last person must place the boo$ on another table at the end of the ro!

Estimated time to pass the boo$.33333333333333333333333333

2. )s a class, e&periment ith the actual time it ta$es to pass the boo$ using small groups of students in your

class! se the table belo to record the times!

Number of Students ;assingthe 7oo$ 4 ? B 11 14 1A

"ime to ;ass the 7oo$9nearest tenth of a second:

3. 7ased on the data you recorded in the table above, ould you revise your estimated time from tem 16

E&plain the reasoning behind your anser!

Functions and models Page 53Unit 2 Linear Functions

12


54/58


4. Kraph the data in your table from tem - as a scatter plot on the coordinate grid

5. )re the data that you collected linear data6

a. E&plain your anser using the scatter plot!

b. E&plain your anser using the table of data

6.+escribe ho the time to pass the boo$ changes as the number of students increases!

7.Cor$ as a group to predict the number of seconds it ill ta$e to pass the boo$ through the hole class!

a. ;lace a trend line on the scatter plot in tem > in a position that your group feels best models the data!

"hen, mar$ to points on the line!

b. n the spaces provided belo, enter the coordinates of the to points identified in ;art 9a:!

;ont 1. 933333, 33333: ;oint -. 933333, 33333:

c! Chy does your group thin$ that this line gives the best position for modeling the scatter plot data6

8. se the coordinate pairs you recorded in tem (9b: to rite the e#uation for your trend line 9or linear model: of

the scatter plot!

9. E&plain hat the variables in the e#uation of your linear model represent!

10.Chat is the meaning of the slope in your linear model6

11.se your e#uation to predict ho long it ould ta$e to pass the boo$ through all the students in your class!

;redicted time to pass the boo$.333333333333333333333333333333

12.sing all of the students in your class, find the actual time it ta$es to pass the boo$!

)ctual time to pass the boo$.33333333333333333333333333333

13. @o do your estimate from tem 1 and your predicted time from tem

11 compare to the actual time that it too$ to pass the boo$ through the entire class6



55/58


14. Suppose that another class too$ 1 minute and >( seconds to pass the

boo$ through all of the students in the class! se your linear model to estimate the number of students in theclass!



56/58


*@E*T =OR N+ERS")N+NKCrite your ansers on noteboo$ paper or grid paper! Sho your or$!"he table shos the number of days absent and the grades for several students in Ms! ReynosoIs )lgebra 1 class! sethe table for tems 1/2!

+ays )bsent ' 4 ? 1 - - >

Krade 9percent:

B2 22 ?B 2B B' 2? ((

1. *reate a scatter plot of the data using days absent as the independent variable!

2. )re the data linear6 E&plain using the scatter plot and the table of data!

3. 7ased on the data, ho do grades change as the number of days absent increases6

4. +ra a trend line on your scatter plot! dentify to points on the trend line and rite an e#uation for the line

containing those to points!

1. Chat is the meaning of the & and y variables in the e#uation you rote6

2. Chat is the meaning of the slope and the y/intercept of the trend line you dre6

3. se your e#uation to predict the grade of a student absent for A days!

4. Si&ty percent is a passing grade in Ms! ReynosoIs class! se your e#uation to find ho many days a

student could be absent and still earn a passing grade!



57/58


5. Crite the e#uation of a line passing through the given pairs of points!

a. 9/-, A: and 9A, ?:

b. 9', ?: and 9/>, G2:

6. M)"@EM)"*)L REFLE*"ONS. Chat ould be the sign of the slope of a trend line on a scatter plot

that compares the sale prices of cars to ages of cars6 Chy do you thin$ so6



58/58


;ERFORM)N*E ")ST

im as serving as a finish/line 5udge for the Striders 1'T Run! @e as interested in finding out ho three of hisfriends ere doing out on the course! @e as able to learn the folloing information from racing officials at differentlocations along the course!Matuba is running at a strong, steady rate of 4-' m every minute after running the first 1>'' m in a time of Aminutes!Rodrigue< ran the first -''' m in ? minutes, before he settled into his steady pace passing the >>'' m mar$ at1> minutes)ccording to his calculations, +onovan feels he can e#ual his best running time of 4- minutes for 1',''' mover this course!)nser tems 1/4 belo based on the information im received about his three running friends! se & as the numberof minutes since the race began and y as the number of meters completed!

1! *reate three linear models for each runnerIs progress toard the finish line!-! E&plain the order in hich the three runners ill finish the race based on the models you formed using this

information!

4! sing the models you formed, in hat order ould the runners have passed the AT mar$ in the race6

For the linear models you created, find the folloing numeric ansers! E&plain the significance, if any, for each anserin the conte&t of the problem situation!

>! the domain of the linear model for +onovanA! the y/intercept for MatubaIs linear model?! the slope of the linear model for Rodrigue

pr2015 functions and relations.docx

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