CSCOPE Algebra 1 Unit 1

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

By the Sea (pp. 1 of 2)
The graphing calculator can be used to plot points. Points are entered under the statistics function STAT! of the calculator. Points are then plotted using the STAT PL"T. These s#ills $ill be used o%er and o%er in future concepts $hen entering data and finding functions. 1. Turn off the a&es to the graphing $indo$. This is found under format. '. Set the ()*+"( %alues as , -'0. '0. 1! and / -10. 10. 1!. 1. Under the STAT23+)T #e4s. enter the follo$ing points in L1 & %alue! and L' 4 %alue!. 5e sure to enter them in the order gi%en. left to right. -6. 6! -1.10! 1. 10! 7. 6! 10. 8! 11. -1! -1. -9! -11. -1! -1'. 8! -6. 6! /our lists $ill loo# li#e the picture belo$ for these ordered pairs. /ou $ill need to scroll do$n to see the last three pairs on the screen. Use this same model for entering 4our lists in :7 and :9 belo$.

8. ;o to STAT PL"T and turn on the first plot. Under T4pe use the second graph. This dra$s and connects the points. Under ,list put L1. Under /list put L'. Under Mar# use the last small point. 0. Press ;<APH. (hat do 4ou see= (hat are 4ou ma#ing= +ra$ a >uic# s#etch of the present graph.

7. Under the STAT23+)T #e4s. enter the follo$ing points in L1 x %alue! and L8 y %alue!. 5e sure to enter them in the order gi%en. left to right. -1. -9! 10. 8! 7. 6! -1. -9! 1. 10! -1. 10! -1. -9! -6. 6! -1'. 8! -1. -9! ?. ;o to STAT PL"T and turn on the second plot. Under T4pe use the second graph. This dra$s and connects the points. Under ,list put L1. Under /list put L8. Under Mar# use the last small point. 6. Press ;<APH. (hat do 4ou see= Are 4ou getting a better idea 4et= +ra$ a >uic# s#etch of the present graph.

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

By the Sea (pp. 2 of 2)
9. Under the STAT23+)T #e4s. enter the follo$ing points in L0 & %alue! and L7 4 %alue!. 5e sure to enter them in the order gi%en. left to right. 0. -7! '. -10! -8. -10! -?. -7! 10. ;o to STAT PL"T and turn on the third plot. Under T4pe use the second graph. This dra$s and connects the points. Under ,list put L0. Under /list put L7. Under Mar# use the last small point. 11. Press ;<APH. +oesn@t that ma#e 4ou $ish 4ou $ere at the beach right no$= +ra$ a s#etch of the final graph.

1'. +e%elop a design of 4our o$n. <emember. points must connect in orderA /ou must use all si& lists and three scatter plots. S#etch 4our design on grid paper. labeling all points. Test and %erif4 results using the graphing calculator.

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Four Quadrant Grid

Set A Points:

Set 5 Points:

Set B Points:

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Miles and Intersections (pp. 1 of 3)
Mathematics. especiall4 algebra. is used to describe and interpret ho$ >uantities are related. )n this lesson 4ou $ill e&plore ho$ one >uantit4 might affect another. and ho$ their relationship is reflected in the graphs and tables that represent them. Ho$ man4 intersections are there on 4our route from home to school= )s there a relationship bet$een the distance in miles from school and the number of intersections= 1. Thin# about the route from 4our home to school. a. 3stimate the distance in miles from home to school. <ound 4our estimate to the nearest half-mile. b. (rite the number of intersections 4ou go through. c. Bollect the data from 4our class and enter it in the table belo$. Distance ( iles) !u "er of Intersections Distance ( iles) !u "er of Intersections

d. +escribe an4 patterns 4ou obser%e in the table. )f there is no pattern. sa4 so.

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Miles and Intersections (pp. 2 of 3)
'. Breate a scatterplot of the class data. ;raph the distance along the x-a&is and the number of intersections along the y-a&is. y

x

1. +escribe the scatterplot. (hat patterns do 4ou obser%e= )f there is no pattern. sa4 so.

8. Predict the number of intersections there $ill be if 4ou li%e ? miles from school. 3&plain ho$ 4ou made 4our prediction.

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Miles and Intersections (pp. 3 of 3)
0. Ho$ does the number of intersections change as the distance from home to school increases=

7. Ban the distance tra%eled be used to reliabl4 predict the number of intersections= (h4 or $h4 not=

?. Ho$ do the table and scatterplot support 4our ans$er to >uestion 7=

6. Bompare 4our prediction from >uestion 8 $ith the rest of the class. (ere 4our predictions the same or different=

9. Ho$ does 4our ans$er to >uestion 7 help e&plain 4our ans$er to >uestion 6 abo%e=

10. Although there is no pattern in the table or scatterplot that represents the number of intersections. the distance and the number of intersections are ###############. Ho$ does the table and scatterplot sho$ a ############### bet$een the distance and the number of intersections=

$eep your %or& fro

this e'ploration(

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Miles and Intersections Data )ollection !u "er of Intersections !u "er of Intersections

Distance ( iles)

Distance ( iles)

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Frayer Model*+elation Definition )n o$n $ords! )haracteristics

Word:
,'a ples from o$n life!

Relation

!on-e'a ples from o$n life!

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Burnin. )alories (p. 1 of 3)
)n Miles and Intersections. 4ou learned that t$o >uantities such as distance, number of intersections! is a relation as long as it forms a set of ordered pairs. /ou also learned that $hen there is no pattern in the data. it is difficult to ma#e predictions. (hat if there is a pattern= (hat if changing one >uantit4 changes the other >uantit4 in a predictable $a4= )n this e&ploration. 4ou $ill in%estigate another relation $ith special characteristics. The food 4ou eat pro%ides 4our bod4 the energ4 it needs to maintain bod4 functions such as temperature regulation. blood circulation. bone gro$th and muscle repair. )t also pro%ides the energ4 needed to be ph4sicall4 acti%e. (hen 4ou engage in ph4sical acti%it4. 4ou burn kilocalories. A #ilocalorie #cal! is the amount of energ4 re>uired to raise the temperature of one liter 1 L! of $ater one degree Belsius 1° B!. A dietar4 Balorie $ith a capital C! is e>ual to one #ilocalorie. 1. )magine 4ou are going to the g4m after school. /ou ha%e had e&tra snac#s during the da4. +oes the number of minutes 4ou need to e&ercise depend on the number of #ilocalories 4ou $ant to $or# off= 3&plain $h4 or $h4 not. The table belo$ sho$s the number of #ilocalories burned in one minute b4 a person $eighing 00 #g for se%eral different acti%ities. /cti0ity 5as#etball Bard pla4ing +ancing Cootball Dumping rope Pla4ing the piano Painting (al#ing !u "er of &ilocalories "urned per minute "y a 12-&. person 7.9 1.'0 1.?0 7.7 6.1 ' 1.? 8

'. Use proportions to calculate the number of minutes it $ould ta#e to burn off 00 #ilocalories $hile pla4ing bas#etball. (hat shortcut could be used to calculate the number of minutes it ta#es to burn off a gi%en number of #ilocalories=

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Burnin. )alories (p. 2 of 3)
1. Bhoose one acti%it4. <ecord the name of 4our acti%it4. Balculate the number of minutes re>uired to burn 100 #cal. '00 #cal. 100 #cal. 800 #cal. and 000 #cal rounded to the nearest tenth of a minute. /cti0ity ,ner.y Burned (&cal) ( in) 100 '00 100 800 000

8. Breate a scatterplot for the acti%it4 belo$. ;raph the number of #ilocalories along the x-a&is and time in minutes along the y-a&is. +ra$ a smooth line through the points on the scatterplot.

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Burnin. )alories (p. 3 of 3)
0. +escribe an4 patterns in the table and scatterplot.

7. Use the table and graph to predict the number of minutes it $ould ta#e to burn 600 #cal for the acti%it4 4ou chose. Ho$ do 4our results compare $ith others in the class that selected the same acti%it4=

?. Ho$ does the number of minutes change as the number of #ilocalories increases= Ho$ do the table and scatterplot support 4our ans$er=

6. Ho$ are the scatterplots created in Miles and Intersections and Burnin. )alories the same= Ho$ are the4 different=

9. )f 4ou $ere able to ma#e a prediction of the number of minutes it $ould ta#e to burn 600 #cal. ho$ is this situation different from that in Miles and Intersections $here 4ou $ere as#ed to predict the number of intersections if 4ou li%ed ? miles from school=

$eep your %or& fro

this e'ploration(

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

)artesian )oordinate Syste

(pp. 1 of 3)

Label the parts of the Bartesian Boordinates S4stem belo$ $ith the follo$ing: x-a&is. y-a&is. and Euadrant ). Euadrant )). Euadrant ))). Euadrant )F. origin. the coordinates of the origin. and $hen x is positi%e or negati%e. and $hen y is positi%e or negati%e in the ordered pair G. G!.

HHHH.HHHH!

HHHH.HHHH!

HHHHHHHHHH H

HHHH.HHHH!

HHHH.HHHH!

HHHH.HHHH!

The HHHHHHHHHHHHHHHHHHHHHHHHHHH is used to graph relationships bet$een >uantities. )t is composed of t$o number lines called the x-a&is and the y-a&is. These t$o number lines di%ide the plane into four >uadrants. • A HHHHHHHHHH or HHHHHHHHHHHHHHH is $ritten as x. y! or x. f x!! and can be located in an4 >uadrant or on the x-a&is or y-a&is. *"T3: Another $a4 to $rite y is f x!. { ( −'.1 ) .( 0.0 ) .( '. −0 ) } Set of ordered pairs • HHHHHHHHHH can be graphed as a point or a set of points. Cor the set of ordered pairs. in $hich >uadrant $ould each point be located= -' . 1! HHHHHHHHHH 0. 0! HHHHHHHHHHHHHHHHHHHHHHHHHHHH '. -0! HHHHHHHHHH 0. 1! HHHHHHHHHHHHHHHHHHHHHHHHHHHH The HHHHHHHHHH of the relationship is the set of permissible x %alues. The notation for domain is +:I-'. 0. 'J. +omains can be continuous or discrete. o HHHHHHHHHH data are indi%idual points that $ould not be connected $hen graphed because not all rational %alues define the domain. The set of points under the relation abo%e is discrete and +: I-'. 0. 'J. +iscrete data represented b4 the graph of a function are connected $ith a bro#en line on the graph.! o HHHHHHHHHH data are an infinite number of points that are connected $hen graphed because all real %alues can be defined in the domain. Bontinuous data is connected $ith a solid line on a graph. and domain is $ritten using ine>ualit4 notation such as: +: I-0 K x K ?J or +: Ix ≥ 0J The HHHHHHHHHH of the relationship is the set of permissible y %alues. The notation for range is $ritten using ine>ualit4 notation such as: <: I-0 K y K ?J or <: Iy ≥ 0J!
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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

)artesian )oordinate Syste


(pp. 2 of 3)

<elations in $hich each element of the domain is paired $ith e&actl4 one element of the range are called HHHHHHHHHHHHHHHH. o )f a set of data is a function. HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH o )f a set of data is a function. HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH Cunction Analog4: Bonsider the domain to be a set of people on a bus. Thin# of each bus stop along the $a4 as the range. The LfunctionM of the bus is to deli%er people to their respecti%e destinations. )t is possible for t$o or more people to get off at one bus stop y!. ho$e%er. it is not possible for the same person x! to get off at t$o different bus stops. A person x! is associated $ith onl4 one bus stop y!.

• •

)f the y %alue increases as the x %alue increases. the function is HHHHHHHHHHHHHHHH. "n the graph an increasing function $ill go up from left to right. )f the y %alue decreases as the x %alue increases. the function is HHHHHHHHHHHHHHHH. "n the graph a decreasing function $ill go do$n from left to right.

)onnections 1. <emember that a relation bet$een t$o >uantities is a set of ordered pairs of the form x, y!. a. )n Miles and Intersections is there a relation bet$een miles to school and intersections crossed= 3&plain $h4 or $h4 not.

b. )n Burnin. )alories is there a relation bet$een #ilocalories burned and minutes= 3&plain $h4 or $h4 not.

c. Relation and dependence both describe ho$ t$o >uantities can be connected. Ho$ are the t$o ideas different= .

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

)artesian )oordinate Syste

(pp. 3 of 3)

'. The number of minutes depends on the number of #cal. (hen calculating the number of minutes. changing the number of #cal changed the ans$er. +id the number of intersections depend on the distance= (h4 or $h4 not=

1. )dentif4 the independent and dependent %ariables in the acti%it4 Burnin. )alories.

8. ;i%e another e&ample of a situation $here one >uantit4 depends on another. Example: The amount I earn depends on the number of hours I work. !

0. (hich of the t$o pre%ious acti%ities represents a functional relationship= 3&plain 4our reasoning.

7. +oes Burnin. )alories represent a continuous or discrete domain= 3&plain 4our reasoning .

?. (hat is the domain and range of the relation in%estigated in Burnin. )alories=

6. )s the relation in 5urning Balories increasing or decreasing= 3&plain 4our reasoning .

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Independent3Dependent Sentence Strips )onnections
Ans$er Bhart

(pp. 1 of 2)

Independent 4aria"le

Dependent 4aria"le

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Independent3Dependent Sentence Strips )onnections

(pp. 2 of 2)

<e$rite each of the follo$ing as a statement of one attribute depending on the otherNa %erbal statement of the relationships 4ou created $ith the sentence strips.

1. The >ualit4 of a music performance is related to the amount of practice.

'. The amount of perfume2cologne applied is related to the se%erit4 of an allergic reaction.

1. The effects of h4peracti%it4 are related to the amount of caffeine consumed.

8. The amount of stud4 time is related to a test grade.

0. The amount of li>uid pic#ed up relates to the absorbenc4 of different paper to$el brands.

7. The rate of plant gro$th is related to the color of light to $hich it is e&posed.

?. (rite one more cause and effect relationships. )dentif4 the independent and dependent %ariables.

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Independent3Dependent Sentence Strips )ards
But out the $ord strips. Use a glue stic# to attach independent %ariables on the left and the related dependent %ariable on the right in the ans$er chart on the Bonnections pages.

Eualit4 of Performance

Allergic reaction

Plant gro$th

Perfume2cologne

H4peracti%it4

Stud4 time

Test grade

Practice time

Li>uid pic#ed up

Baffeine consumed

Absorbenc4 of paper to$els

Bolor of light

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Relations and Dependency (pp. 1 of 4)

A mathematical relation e&presses a dependent relationship $here one >uantit4 depends in a s4stematic $a4 on another >uantit4. )n some cases there is a cause and effect relationship $here the cause is the independent %ariable and the effect is the dependent %ariable. 1. 3&ample: La#e Tra%is $ill rise ' feet if it rains 10 inches in the $atershed. )n other cases there is not a cause and effect relationship. but there can still be an independent2dependent relationship. )n this t4pe of relationship either can be the independent %ariable. $hich then forces the other to be dependent. '. 3&ample: Henr4 has an arm span of 78 inches and a height of 77 inches. Some are generaliOed algebraic relationships. 1. 3&ample: y = 'x + 1 is a function and e&presses a dependenc4 relationship.
Input Independent Do ain 5utput Dependent +an.e

&

4

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

+elations and Dependency (pp. 2 of 6)
• • • The %alue of HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH. The %ariable x is called the HHHHHHHHHHHHHHHor HHHHHHHHHHHHHHHHHHHHHHHHHHH. The set of permissible %alues for the independent %ariable is called the HHHHHHHHHHHHHHHH. The %ariable y is called the HHHHHHHHHHHHHHHor HHHHHHHHHHHHHHHHHHHHHHHHHHH. The set of permissible %alues for the dependent %ariable is called the HHHHHHHHHHHHHHHH.

,'a ple

Independent

Dependent

Do ain (0alue)

+an.e (0alue)

1. La#e Tra%is $ill rise ' feet if it rains 10 inches in the $atershed. '. Henr4 has an arm span of 78 inches and a height of 77 inches. 1. y =' x +1 is a function and e&presses a dependenc4 relationship.

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

+elations and Dependency (pp. 3 of 6)
7ractice 7ro"le s 1. Sue Ann recei%ed a statement from her ban# listing the balance in her mone4 mar#et account for the past four 4ears. (hat is the independent >uantit4 in this table= ;i e 0 1 ' 1 8 Balance P1000 P1'80 P1190 P1800 P17''

'. ;arrett is in charge of ma#ing 1'0 corsages for homecoming. He decides to as# some of his classmates for help. The number of corsages each person can ma#e can be represented b4 1'0 the function f h! = $here h is the number of classmates that help ;arrett ma#e corsages. h +1 (hich is the dependent >uantit4 of this function=

1. The TM Tennis Team pla4ed a total of 17' matches last season. The number of matches the team lost. l. and the number of matches the team $on. w. are represented b4 the formula belo$. (hat >uantit4 does the dependent %ariable represent= l 8 192 : % 8. Pat hi#es at an a%erage rate of four miles per hour. The number of miles. m. she hi#es is %ie$ed as a function of the number of hours. h. she hi#es. (hat is the independent %ariable= 0. A ta&i dri%er charges an initial fee of P0.00 plus P0.00 per mile. (hat is the independent %ariable >uantit4 in this situation= 7. A long distance telephone compan4 charges P'.90 per month and P0.16 per minute for phone calls. (hat is the dependent %ariable >uantit4 in this situation=

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

?. A plumber charges fort4 dollars to ma#e a house call plus thirt4-fi%e an hour for labor. (hat are the independent and dependent %ariables=

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

+elations and Dependency (pp. 6 of 6)
6. The table belo$ represents the relationship bet$een the number of gallons of gas in a gas tan# and the number of miles that can be dri%en. (hich >uantit4 represents the dependent >uantit4 in this table= Gas in ;an& (.allons) 0 1 ' 1 8 0 Miles that )an Be Dri0en 0 '1 87 79 110 168

9. Larrissa ans$ered all t$ent4-fi%e >uestions on a multiple-choice histor4 e&am. Her score $as computed b4 multipl4ing the number of $rong ans$ers b4 four and then subtracting the number from one hundred. (hat >uantit4 represents the independent %ariable= 10. The cost for cop4ing a document is a function of the number of pages in the document. )n this situation. $hat is the dependent %ariable= 11. Bharles partiall4 filled a container $ith sand. The container $as shaped li#e a bo& and had dimensions ' feet long. 1.0 feet $ide. and 7 inches high. )f w represents the height of the sand in inches!. and the %olume V in cubic inches! of the sand is gi%en b4 the formula V8 3w. $hich >uantit4 is the independent %ariable= A. 5. B. +. The height of the container The %olume of the container The height of the sand in the container The %olume of the sand in the container

1'. )n the situation belo$. there are three functional relationships. )dentif4 at least one independent and dependent relationship. )n that relationship. tell $hich one is the independent %ariable and $hich one is the dependent %ariable. The monthl4 cost of electricit4 for a home is based on the number of #ilo$att-hours #$h! of electricit4 used. The number of #ilo$att hours used is based on the number of $atts of electricit4 each light bulb or appliance uses and the amount of time it is used.

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Facts /"out Functions (pp. 1 of <)
1. )dentif4 $hich relationships are functional and e&plain 4our reasoning. a. I 1.'!. 1.1!. -'.0!. 0.-1!J b. I 1.'!. 8.'!. -'.1!. 0.0!J

c. 1 -' 0 ' 1 0 -1

d. 1 8 -' 0 ' 1 0

e. -' -2 0 1 3 1 3 0 -1 -1 ' 2 1 3

f. -' 0 1 8 1 0 ' '

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Facts /"out Functions (pp. 2 of <)
g. h.

'. Cor a set of points. determine if it is a function and identif4 the domain and range. I -8. 0!. 0. 7!. 0. -?!. 1. 0!. 6. 9!. -'. -'!J

1. Cor a table of data. determine if it is a function. identif4 the independent and dependent %ariable. and state the domain and range. Seconds ;e perature (') (y) ' 0 8 -' 7 -8 6 -7 10 -6

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Facts /"out Functions (pp. 3 of <)
8. Cor gi%en graphs. determine if it is a function and identif4 the domain and range.
10 6 7 8 ' -10 -6 -7 -8 -' -' -8 -7 -6 -10 ' 8 7 6 10 -10 -6 -7 -8 -' -' -8 -7 -6 -10 10 6 7 8 ' ' 8 7 6 10

10 6 7 8 ' -10 -6 -7 -8 -' -' -8 -7 -6 -10 ' 8 7 6 10 -10 -6 -7 -8 -'

10 6 7 8 ' ' -' -8 -7 -6 -10 8 7 6 10

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Facts /"out Functions (pp. 6 of <)
0. Cor a relation 4 Q!. determine if it is a function. identif4 the independent and dependent %ariable. and state the domain and range. ;raph the relation. a. 4 Q '& R 1

b. 4 Q &' R '

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Facts /"out Functions (pp. 1 of <)
Practice Problems 1. Bompare and contrast the characteristics of relations and functions. Stud4 the statements belo$ that are about relations and functions. Place a chec# mar# in the appropriate bo&es if the statement is true for all relations. or if the statement is true for all functions. State ent )t can be discrete or continuous. )t has a domain and range. )t can be represented b4 ordered pairs in the form x, y!. )t matches e&actl4 one independent %alue $ith each dependent %alue. )t can be represented b4 a graph. )t can be represented b4 a table. /ttri"ute of /ll +elations /ttri"ute of /ll Functions

a. (hat conSecture can 4ou ma#e about relations and functions= Cill in each blan# $ith the $ord relation or function to ma#e a true statement. A HHHHHHHHHH is al$a4s a HHHHHHHHHH. but a HHHHHHHHHH is not al$a4s a HHHHHHHHHH. b. (hat is the defining characteristic of functions= )n other $ords. $hat ma#es a relation a function= '. +etermine if the mapping represents a function. a. b. -1 -1 0 ' 0 6 8 7 6 8 7 6 -1 -1 0 ' 0 6

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Facts /"out Functions (pp. 9 of <)
1. Plot the point -8. 0!. )dentif4 the domain and range.

8. )dentif4 the domain and range. +etermine if it is a function and tell $h4. a. Plot the set of points I 1. 0!. -'. 1!. 0. -7!. -1. 0!. -8. -'!. 0. 1!. 0. 1!. 1. 7!J. b.

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Facts /"out Functions (pp. < of <)
0. Ma#e a table of %alues and plot the relationship 4 Q '& G 1. )dentif4 the domain and range. +etermine if it is a function and tell $h4.

7. Ma#e a table of %alues and plot the relationship 4 Q & ' G 1. )dentif4 the domain and range. +etermine if it is a function and tell $h4.

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

;ic&et 7rices (pp. 1 of 6)
+oes the number of concert tic#ets sold depend on the price of the tic#et= )f one %ariable depends on another. can there be more than one dependent %ariable for each independent %ariable= )n this e&ploration. 4ou $ill further in%estigate the idea of dependence bet$een t$o >uantities and functional relationships. A popular band is scheduled to pla4 at the Starple& AmphitheaterT ho$e%er. the amphitheater@s management and the band cannot come to an agreement about the price of the tic#ets. 1. The amphitheater. $hich seats 00.000. has seen a drop of 100 tic#ets sold for each dollar increase in tic#et price. The number of tic#ets sold can be calculated using the formula t Q 00000 − 100p. $here t represents the number of tic#ets sold and p represents the price per tic#et in dollars. Bomplete the table. 7rice per tic&et in dollars (p) 0 10 '0 10 80 00 70 p 7rocess 00000 − 100 0! 00000 − 100 10! !u "er of tic&ets sold (t) 00.000 86.000

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

;ic&et 7rices (pp. 2 of 6)
'. Breate a scatterplot. ;raph the price in dollars along the x-a&is and the number of tic#ets sold along the y-a&is. Label and scale the a&es appropriatel4.

1. Ho$ did 4ou determine the number of tic#ets sold=

8. (hen 4ou calculated the number of tic#ets sold. ho$ man4 different ans$ers did 4ou get for each different tic#et price=

0. Ho$ is 4our ans$er to >uestion 8 reflected in the graph=

7. (hat are the independent and dependent %ariables in the problem situation=

?. +oes this situation represent discrete or continuous data= (h4=

6. (hat happens to the number of tic#ets sold as the price increases= Ho$ is this reflected in the table and graphs=

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Algebra 1 HS Mathematics Unit: 01 Lesson: 01

;ic&et 7rices (pp. 3 of 6)
9. Predict the price at $hich no tic#ets $ill be sold.

10. Ho$ did 4ou determine the solution to >uestion 9=

11. +oes the number of tic#ets sold depend on the price= (h4 or $h4 not=

1'. )s the price of the tic#ets and the number of tic#ets sold a relation= (h4 or $h4 not=

11. Bompare the three situations 4ou in%estigated: Miles and Intersections. Burnin. )alories. and ;ic&et 7rices. a. Ho$ are the tables the same or different=

b. Ho$ are the graphs the same or different=

c. Cor $hich situations $ere 4ou able to ma#e a prediction=

d. (hat seems to be the connection bet$een dependence and the abilit4 to ma#e a prediction=

09210210

Algebra 1 HS Mathematics Unit: 01 Lesson: 01

;ic&et 7rices (pp. 6 of 6)
18. 3&amine the table and scatterplot for one acti%it4 from Burnin. )alories. a. Ho$ man4 points are plotted for each #cal %alue=

b. )s there e&actl4 one dependent %alue minutes! matched $ith e&actl4 one independent %alue #cal!=

10. 3&amine the table and scatterplot for ;ic&et 7rices. a. Ho$ man4 points are plotted for each tic#et price %alue=

b. )s there e&actl4 one dependent %alue tic#ets sold! matched $ith e&actl4 one independent %alue tic#et price!=

17. The relations number of kcal, number of minutes ! and ticket price, number of tickets sold! are both special t4pes of relations. The4 are both functions. )n both functions. there are independent and dependent %ariables. and more importantl4. there is exactly one dependent %alue matched $ith each independent %alue. +o 4ou thin# all relations are functions= (h4 or $h4 not=

09210210

Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Function !otation (pp. 1 of 3)
Cunctions can be $ritten in t$o formats. y8 for at 4 Q 1& G 0 4 Q '&'G 1 x 4Q 1 4 Q '& f(') for at f &! Q 1& G 0 g &! Q '&' G 1 x h &! Q 1 S &! Q '&

The f &! format is called function notation. Cunction notation has t$o benefits o%er 4Q format. • • ;i%es different functions their specific LnameM. )n other $ords f &! denotes a specific rule. and g &! denotes a different rule. )t can be used to designate $hat %alue to e%aluate. )f it is $ritten as f '!. it means to find rule LfM and substitute in a '.  1 f ÷ 1

,'a ple f ( −' ) f &! Q 1& G 0 7ractice 1. g &! Q '&' G 1 h ( 0) x 1
0.?0!

f ( 0)

−f ( 0 )

f(

)

f ( !)

! ( −' )

! ( −1)

! ( 0)

! ( 0)

!(

)

'. h &! Q 1. S &! Q '&

h ( −1' )

h ( 1)

 1 h ÷ 1

h( f )

( −8 )

( 0)

(f )

8. m &! Q ? R 1&

m ( 8)

m ( −8 )

 1 m ÷ 1

−m ( 1 )

09210210

Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Function !otation (pp. 2 of 3)
T$o area distributors. 5ar#ing Lot ;rooming and Tid4 Pa$s. sell and deli%er the same #ind of shampoo for dogs and cats to area %eterinar4 clinics. The functions used b4 each distributor to calculate the cost to the clinics are gi%en belo$. 5ar#ing Lot ;rooming 4 Q 0& G 1 Tid4 Pa$s 4 Q 1& G '1

)f both dependent %ariables are $ritten as L yM. it is hard to distinguish $hich e>uation represents $hich distributor. To #eep trac# of se%eral functions it is sometimes necessar4 to distinguish them $ith a name. This is done b4 putting the functions in function or f &! notation. 5ar#ing Lot ;rooming b &! Q 0& G 1 Tid4 Pa$s p &! Q 1& G '1

0. (hat differences do 4ou obser%e in the cost functions $ritten in f &! notation= 7. (hat s4mbols are used to represent the dependent %ariable= ?. (rite an ordered pair for each distributor using the appropriate s4mbols. +o not use numbers. 6. Use the appropriate function notation to e%aluate the cost for 7 bottles. 9 bottles. and 10 bottles of shampoo for each distributor.

9. The follo$ing function notation $as gi%en for 5ar#ing Lot ;rooming: b 0! Q '6. a. (hat does the 0 represent= b. (hat does the '6 represent= 10. The follo$ing function notation $as gi%en for Tid4 Pa$s: p 10! Q 01. a. (hat does the 10 represent= b. (hat does the 01 represent=

09210210

Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Function !otation (pp. 3 of 3)
Cunction notation can also be used to find function %alues b4 appl4ing the graphing calculator. )nstead of naming the functions $ith %ariables. functions are named using 4 1. 4'. 41. and so on. The follo$ing steps are used to find function %alues in the graphing calculator.  Put function into /Q.  ;o to Home Screen and Blear.  ;o to Fars. /-%ars. Cunction. /1. /ou should get /1 on the Home Screen. 3nter '!. )t should gi%e the %alue of the function at ' on the Home Screen.  'nd 3ntr4 $ill bring it bac# up so 4ou can t4pe o%er the ' and find another %alue. 11. Use the graphing calculator to chec# 4our ans$ers on the pre%ious problems.

09210210

Algebra 1 HS Mathematics Unit: 01 Lesson: 01

Flyin. %ith Functions
On the back of the puzzle show the work for each problem using function notation. Verify results using the graphing calculator. After working the problems connect the ots in or er. f x! = x + 7 h x! = 1x + 8 ! x! = x ' − 1 ' p x! = − x " x! = 'x − 8 x + 1 1. f 8! 8. > 1! ?. g -8! 10. p -9! 11. h -8! 17. f -6! 19. > 7! ''. g '! 1 '0. p − ! ' '6. h -1.'! '. g 1! 0. p 7! 6. h 0! 11. f -11! 18. > 10! 1?. g -10! '0. p 99! '1. h 0! '7. f 0.8! '9. > 1 ! ' 1. h -1! 7. f -9! 9. > -1! 1'. g 1! 10. p 11! 16. h 1! '1. f 0! '8. > 1! 1 '?. g ! ' 10. p !!

09210210

Algebra 1 HS Mathematics Unit: 01 Lesson: 01

;a&e a =oo& at the Data (pp. 1 of 6)
The table belo$ sho$s the latitude and a%erage dail4 lo$ temperature for se%eral cities in *orth America and Ha$aii. )ity Miami. CL Honolulu. H) Houston. T, Philadelphia. PA 5urlington. FT Dac#son. MS Bhe4enne. (/ San +iego. BA =atitude (° !) '7 '1 10 80 88 1' 80 11 /0era.e Daily =o% ;e perature in >anuary (° F) 09 77 80 '1 6 11 10 89 7oints (=atitude? ;e p)

1. Breate a scatterplot of the data in the table. ;raph latitude along the x-a&is and a%erage temperature along the y-a&is.

'. +escribe an4 patterns in the data.

1. Ho$ do the table and scatterplot reflect the patterns in the data=

09210210

Algebra 1 HS Mathematics Unit: 01 Lesson: 01

;a&e a =oo& at the Data (pp. 2 of 6)
8. As the latitude increases. ho$ does the temperature change=

0. )s latitude and temperature a relation= 3&plain 4our response.

7. (hat is the domain and range of the relation=

?. +oes this relation represent a function=

Stud4 the diagram belo$ to determine the relationship bet$een perimeter and stages.

Stage 1 Perimeter Q 8

Stage ' Perimeter Q 6

Stage 1 Perimeter Q 1' Perimeter

6. Use the data from the diagram to fill in the table. Stage 1 ' 1 8 0 7 &
09210210

Process

Algebra 1 HS Mathematics Unit: 01 Lesson: 01

;a&e a =oo& at the Data (pp. 3 of 6)
9. Ma#e a scatterplot of the data on the grid belo$. Label and scale a&es o%er an appropriate domain and range.

10. (hat patterns do 4ou obser%e in the diagram= Ho$ are the4 represented in the table and on the scatterplot= 11. +oes the data represent a relation= 3&plain 4our reasoning. 1'. +oes the data represent a function= 3&plain 4our reasoning. 11. )dentif4 the independent and dependent %ariable. 18. )s the relationship continuous or discrete= 3&plain. 10. )s the relationship increasing or decreasing= 3&plain. 17. Cind f 18!. (hat does this represent in the problem situation= 1?. )s it possible in this problem situation to sa4 that f &! Q 60= 3&plain 4our reasoning.

09210210

Algebra 1 HS Mathematics Unit: 01 Lesson: 01

;a&e a =oo& at the Data (pp. 6 of 6)
The area of a rectangular pool $ith a perimeter of 600 feet is gi%en b4 the formula f &! Q 800& R &' $here & represents the length of the pool in feet and f &! represents the area of the pool in s>uare feet. 16. Breate a table. ' 0 00 100 100 '00 '00 100 100 800 f(') 19. Breate a graph.

'0. (hat patterns do 4ou obser%e in the table and on the scatterplot= '1. +oes the relation represent a function= 3&plain 4our reasoning. ''. )dentif4 the independent and dependent %ariable. '1. )s the relationship continuous or discrete= 3&plain. '8. )s the relationship increasing or decreasing= 3&plain. '0. Cind f 1?0!. (hat does this represent in the problem situation=

09210210

Algebra 1 HS Mathematics Unit: 01 Lesson: 01

/naly@in. +elations and Functions (1 of 6)
1. ;i%en the data set I -6. 8!. ?. 9!. -8. -7!. 1. -0!. 0. 1!. 1. 0!. ?. -0!. '. 8!. -'. 1!J a. Breate a table. ' y b. Breate a graph.

c. (hat patterns. if an4. do 4ou see in the data=

d. )s the data continuous or discrete= (hat are the domain and range of the data=

e. +oes the data represent a relation= 3&plain.

f. +oes the data represent a function= 3&plain.

g. Bould the representations of the data be used to ma#e predictions= 3&plain.

09210210

Algebra 1 HS Mathematics Unit: 01 Lesson: 01

/naly@in. +elations and Functions (2 of 6)
'. +uring a treadmill test the heart rate of the patient and the amount of o&4gen the patient consumes is measured. The table sho$s the heart rate and o&4gen consumption as the treadmill@s ele%ation $as increased. The o&4gen consumed can be calculated using the formula c &! Q 0.018& − 0.8?

$here c &! represents the o&4gen consumed and & represents the heart rate. a. Bomplete the table. Aeart +ate Beats per Minute 70 60 90 110 1'0 180 100 1?0 160 b. (hich is the independent >uantit4= (hich is the dependent >uantit4=

5'y.en )onsu ption =iters per Minute 0.88

c. Breate a scatterplot of the data. ;raph the heart rate along the &-a&is and the o&4gen consumption along the 4-a&is. +ra$ a smooth line through the scatterplot since in the real $orld situation partial beats can be read.

09210210

Algebra 1 HS Mathematics Unit: 01 Lesson: 01

/naly@in. +elations and Functions (3 of 6)
d. +oes the data represent a relation= 3&plain. e. +oes the data represent a function= 3&plain. f. )s the data continuous or discrete= 3&plain. g. )s the data increasing or decreasing= h. Ho$ can the function be described %erball4= i. Cind the %alue of c '10!. (hat does this represent in the problem situation=

1. Stud4 the diagram belo$ of one-inch s>uare tiles that are being used to determine the relationship bet$een side length and area.

Side length Q 1 in. Area Q 1 in.'

Side length Q ' in. Area Q 8 in.'

Side length Q HHHHH Area Q HHHHH

a. Breate a table. ' 1 ' 1 8 0 7 & 7rocess y

b. Breate a graph.

09210210

Algebra 1 HS Mathematics Unit: 01 Lesson: 01

/naly@in. +elations and Functions (6 of 6)
c. +oes the data represent a relation= 3&plain.

d. +oes the data represent a function= 3&plain.

e. (hat are the independent and dependent %ariables=

f. )s the data continuous or discrete= 3&plain.

g. )s the data increasing or decreasing= 3&plain 4our reasoning.

h. Ho$ can the function be described %erball4=

i. Cind the %alue of f '0!. (hat does this represent in the problem situation=

S. (hat is the %alue of & in f &! Q 188= 3&plain 4our reasoning.

#. )s f &! Q 00 possible in this problem situation= 3&plain.

09210210

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