ALGEBRA 1 Introduction to Functions: Input & …...ALGEBRA 1 Introduction to Functions: Input &...

by Dr. José D. De León

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U3IntroductiontoFunctions:Input&OutputValues

Name:___________________________________ Date:________________Thesequestionswillgiveyoupracticetoimproveyourproficiencylevelforaportionofthefollowingstandard:MAFS.912.F-IF.1.1Understandthatafunctionfromoneset(calledthedomain)toanotherset(calledtherange)assignstoeachelementofthedomainexactlyoneelementoftherange.Ifisafunctionandisanelementofitsdomain,thendenotestheoutputofcorrespondingtotheinput.Thegraphofisthegraphoftheequation.MAFS.912.F-IF.1.2Usefunctionnotation,evaluatefunctionsforinputsintheirdomains,andinterpretstatementsthatusefunctionnotationintermsofacontext.

===========WORKTHEFOLLOWINGPROBLEMSWITHAPARTNER(IFNEEDED)===========1. Agiraffe’shungerleveldependsonthesizeofitslastmeal.

PartA:Whatistheindependentvariable?Whatisthedependentvariable?2. Youearn$20perhourdoinglandscapingwork.Yourtotalearningsdependontheamountofhoursyouspend

landscaping.

PartA:Whatistheindependentvariable?Whatisthedependentvariable?

PartB:Writeafunctiontorepresentthesituation.3. Cristianohasamessyhouse.ThefunctionthatrepresentsthetotalcostCristianospendshavinghishousecleanedis

! ( " ) = 125 " ,where " representsthenumberofvisits.PartA:DefinetheinputandoutputinthegivenscenarioanddeterminetheamountofmoneyCristianospendper

visit?PartB:WhatisthetotalcostforCristianoifhegetssevenvisitstocleanhishouse?PartC:HowmanyvisitsdidCristianogetifhepaidatotalof$500?


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4. Mr.Khansisbuyingstaplersforhisofficeduringtheback-to-schoolnotaxseason.Eachstaplercosts$16.99.

PartA:Whatdoeshisfinaltotalcostdependupon?PartB:Inthisscenario,whatistheinput?Whatistheoutput?PartC:Writeafunctiontorepresentthesituation.PartD:IfMr.Khansbuys15staplers,itwouldcosthim$254.85.Howwouldyouwritethisusingfunctionnotation?PartE:HowwouldthefunctionchangeifMr.Khansbuysthestaplesbeforeoraftertheback-to-schoolnotaxseason?Supposethattaxis6%.

5. Considerthefollowingincompletemappingdiagrams.

PartA:CompleteDiagramAsothatitisafunction.

PartB:IsitpossibletocompleteDiagramBsothatitisNOTafunction.Ifso,completethediagramtoshowarelation,butnotafunction.Ifnot,justifyyourreasoning.

PartC:IsitpossibletocompletethemappingdiagramforDiagramCsoitrepresentsafunction?Ifso,completethediagramtoshowafunction.Ifnot,justifyyourreasoning.

32-1

555

555

32-1

555

32-1

Diagram A Diagram B Diagram C


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6. Thecosttomanufacture # chairscanberepresentedbythefunction ! # = 36 # .Circlethepairofnumbers(onein

eachbox)thatcorrectlycompletesthestatementaboutthefunction.

If ! (63) = 2268,then chairscost$7. Whatisthedifferencebetweenarelationandafunction?8. Whichofthefollowingrelationsarenotfunctions?Selectallthatapply.

o (1, 3) , ( 3, 7) , ( 5, 11) , ( 7, 15) , (9, 19) o (1, 3) , ( 1, 7) , ( 5, 11) , ( 5, 15) , (9, 19) o (−2, 4) , ( −1, 1) , ( 0, 0) , ( 1, 1) , (2, 4) o (2, 4) , ( 1, 1) , ( 0, 0) , ( 1, −1) , ( 2, −4) o (6, 3) , ( 4, 1) , ( 2, 1) , ( 0, −1) , (−2, −3) o (1, 3) , ( 3, 7) , ( 3, 11) , ( 7, 15) , (9, 19) o (1, 3) , ( 3, 7) , ( 5, 11) , ( 9, 15) , (9, 19)

0663378

6.189.378.2,268.


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Name:___________________________________ Date:________________Thesequestionswillgiveyoupracticetoimproveyourproficiencylevelforaportionofthefollowingstandard:MAFS.912.F-IF.1.1Understandthatafunctionfromoneset(calledthedomain)toanotherset(calledtherange)assignstoeachelementofthedomainexactlyoneelementoftherange.Ifisafunctionandisanelementofitsdomain,thendenotestheoutputofcorrespondingtotheinput.Thegraphofisthegraphoftheequation.MAFS.912.F-IF.1.2Usefunctionnotation,evaluatefunctionsforinputsintheirdomains,andinterpretstatementsthatusefunctionnotationintermsofacontext.

===========WORKTHISPROBLEMINDEPENDENTLY===========1. TheHamptonsFamilyhasafishtankholding10,450mlofwater.Thewaterisleakingatarateof270mlper

minute.PartA:Definetheinputandoutputinthegivenscenario.

PartB:Writeafunctiontomodelthissituation.

PartC:Howmanylitersofwaterwillbeleftinthetankafter10minutes?

PartD:Howmanyminuteswillittakeforthetanktobeempty?

2. Considerthefollowingrelation: (1,12) , (3, 8) , (3, 11) , (6, 9) , (7, 11) .Whichorderedpaircouldberemovedsothattherelationisafunction?


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U3 Representing, Naming and Solving Functions

Name: ___________________________________ Date: ________________ These questions will give you practice to improve your proficiency level for a portion of the following standard: MAFS.912.F-‐IF.1.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If is a function and is an element of its domain, then denotes the output of corresponding to the input. The graph of is the graph of the equation. MAFS.912.F-‐IF.1.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. MAFS.912.F-‐IF.2.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

= = = = = = = = = = = WORK THE FOLLOWING PROBLEMS WITH A PARTNER (IF NEEDED) = = = = = = = = = = = 1. A ball is thrown into the air with an initial velocity of 22 meters per second. The quadratic function

ℎ(𝑡) = −4.9𝑡+ + 22𝑡 + 5.5 represents the height of the ball above the ground, in meters, with respect to time 𝑡, in seconds.

Part A: Determine ℎ(3) and explain what it represents.

The graph below represents the height of the ball with respect to time.

Part B: What is a reasonable domain for the function? Part C: What is a reasonable range for the function?


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2. On the moon, the time, in seconds, it takes for an object to fall a distance, 𝑑, in feet, is given by the function 𝑓 𝑑 = 1.11 𝑑.

Part A: Determine 𝑓(2) and explain what it represents.

Part B: The Imbrium Basin is the largest basin on the moon. A reasonable domain for the height above the lowest point in the basin is given by 𝑑 0 ≤ 𝑑 ≤ 3805774 . What does this tell you about the basin?

Part C: How long would it take a rock to drop from the rim to the bottom of the basin?

3. The function that represents the amount of caffeine, in milligrams, remaining in a body after drinking the two Mountain Dew sodas is given by 𝑓 𝑡 = 110 0.8855 6 where 𝑡 is time in hours. How much caffeine is left in a person’s body 18 hours after drinking two Mountain Dew sodas?

4. The function 𝑓 𝑡 = 98.6 − 𝑥 , represents a person’s variance from normal body temperature, where 𝑥 represents a person’s current body temperature in degrees Fahrenheit. Medical professionals say healthy individuals should have a variance of no more than 0.5℉. Part A: What can be said of an individual with a body temperature of 99.4℉? Justify your answer.

Part B: What can be said of an individual with a body temperature of 98.4℉? Justify your answer.

Part C: What is a reasonable domain that would indicate the individual is not healthy?

5. For some function f, the percentage of battery life remaining is dependent on the number of minutes that a cell

phone has been in use. Part A: Describe in words what 𝑓 22 represents.

Part B: Explain what the statement f (a) = 50 means.


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6. The Miami airport offers several different parking options for visitors. A new short term (no more than 24 hours)

parking garage charges $1 per hour (or any part thereof) for the first two hours and after that, $3 per hour (or any part thereof) not to exceed $20 per day. What is the domain of this function?

7. The math club decided to have a car wash to raise money for competition expenses. The graph below shows the

relationship between cars washed and earnings (in dollars). Describe the domain of the graphed function as precisely as possible.

8. John makes DVDs of his friend’s shows. He has realized that, because of his fixed costs, his average cost per DVD

depends on the number of DVDs he produces. The cost of producing 𝑥 DVDs is given by 𝐶 𝑥 = 2500 + 1.25𝑥. John wants to figure out how much to charge his friend for the DVDs. He’s not trying to make any money on the venture, but he wants to cover his costs. Suppose John made 100 DVDs. What is the domain and how would you determine the charge your friend should pay you to at least cover your costs?


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Name: ___________________________________ Date: ________________ These questions will give you practice to improve your proficiency level for a portion of the following standard: MAFS.912.F-‐IF.1.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If is a function and is an element of its domain, then denotes the output of corresponding to the input. The graph of is the graph of the equation. MAFS.912.F-‐IF.1.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. MAFS.912.F-‐IF.2.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

= = = = = = = = = = = WORK THIS PROBLEM INDEPENDENTLY = = = = = = = = = = = 1. The volume of a cylinder, 𝑉(ℎ), is given by the area of the base times the height, ℎ. The base of the cylinder shown

below is 5 inches. Part A: Write the function, 𝑉(ℎ), that represents the volume of the cylinder. Part B: Find 𝑉(3) and tell what it represents.

2. Use the graph (for example, by marking specific points) to illustrate the statements in 𝑎 – 𝑑 . If possible, label the

coordinates of any points you draw.

Part A: 𝑓 0 = 2

Part B: 𝑓 −3 = 𝑓 3 = 𝑓 9 = 0

Part C: 𝑓 2 = 𝑔 2

Part D: 𝑔 𝑥 > 𝑓 𝑥 𝑓𝑜𝑟 𝑥 > 2

3. Let 𝑓(𝑡) be the number of people, in millions, who own cell phones 𝑡 years after 1990. Explain the meaning of

𝑓 10 = 100.3 and discuss a reasonable domain for this function.

ℎinches


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U3 Adding, Subtracting and Multiplying Functions

Name: ___________________________________ Date: ________________ These questions will give you practice to improve your proficiency level for a portion of the following standards: MAFS.912.A-‐SSE.1.1 Interpret expressions that represent a quantity in terms of its context. Interpret parts of an expression, such as terms, factors, and coefficients. MAFS.912.A-‐SSE.1.2 Use the structure of an expression to identify ways to rewrite it. MAFS.912.A-‐APR.1.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

= = = = = = = = = = = WORK THE FOLLOWING PROBLEMS WITH A PARTNER (IF NEEDED) = = = = = = = = = = =

1. Jocelyn and Anna are comparing the size of their villages in the Clash of Clans app. The area of Jocelyn’s

village is represented by the polynomial, 2𝑤# + 10𝑤 + 12. The area of Anna’s village is represented by the polynomial, 3𝑤# + 4𝑤 − 5, where 𝑤 represents the width, in meters of their Town Hall. Part A: Find the expression that represents the additional area of Jocelyn’s village. Part B: Find the expressions that represents the combined total area of their villages.

2. Evaluate 𝑓(𝑥) ∙ 𝑔(𝑥) by modeling or by using the distributive property.

𝑓 𝑥 = (−3𝑥 + 2) and 𝑔 𝑥 = (2𝑥# − 5𝑥 − 1)

3. Jamie used the distributive property to find the product of 𝑠(𝑡) and ℎ(𝑡). His work was marked incorrect. Identify Jamie’s mistake. What advice would you give Jamie to avoid this mistake in the future.


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U3 Adding, Subtracting and Multiplying Functions


= = = = = = = = = = = WORK THIS PROBLEM INDEPENDENTLY = = = = = = = = = = =

4. Consider the triangle below.

5𝑚6 + 3𝑚# − 2𝑚 5𝑚6 + 3𝑚# − 2𝑚

8𝑚6 − 4𝑚# + 3𝑚

Part A: What is the perimeter of the triangle? Part B: If the height of the triangle is represented by the binomial 𝑚 + 40, what is the area of the triangle?

5. The polynomial, 5𝑣6 − 4𝑣# + 10𝑣 − 1 is subtracted from the polynomial, 7𝑣# + 2𝑣6 − 7𝑣. The difference can be expressed as 𝑎𝑣6 + 𝑏𝑣# + 𝑐𝑣 + 𝑑. Rudy said that the value of 𝑎 is 3 and that 𝑑 = 1. Is Rudy correct? Justify your answer.


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U3 Closure Property

Name: ___________________________________ Date: ________________ These questions will give you practice to improve your proficiency level for a portion of the following standards: MAFS.912.A-‐APR.1.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

= = = = = = = = = = = WORK THE FOLLOWING PROBLEMS WITH A PARTNER (IF NEEDED) = = = = = = = = = = =

1. For the following exercises determine if the closure property applies to the following statements by circling ‘True’ or

‘False’. Then provide an example of each statement.

Statement A Answer Choice Polynomials are closed under addition. True False Example:

Statement B Answer Choice Polynomials are closed under subtraction. True False Example:

Statement C Answer Choice Polynomials are closed under multiplication. True False Example:

Statement D Answer Choice Polynomials are closed under division. True False Example:

2. Check the boxes for the following sets that are closed under the given operations.

Set + − x ÷

{... −7, −6, −5, −4, −3} o o o o

{0, 1, 4, 9, 16, 25...} o o o o

{... −()*, −+

*, −,

*, −-

*, −*

*...} o o o o

{... −4, −3, -‐2, -‐1, 0, 1, 2, 3, 4 …} o o o o


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U3 Closure Property


= = = = = = = = = = = WORK THIS PROBLEM INDEPENDENTLY = = = = = = = = = = =

3. Consider the following standard:

MAFS.912.A-‐APR.1.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. What does it mean to say that polynomials form a system analogous to integers, as related to closure?

4. Choose from the following words and expressions to complete the statement below.

4𝑥0* + 6𝑦 − 4 1

27𝑥* 9𝑦 + 21𝑥 (5𝑦)* − 10𝑥2 + 15

integers

variables

whole numbers

coefficients rational numbers exponents

The product of 5𝑥- − 3𝑥2 + 2 and illustrates the closure property because the _________________ of the product are and the product is a polynomial.


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U3Real-WorldCombinationsandCompositionsofFunctions

Name:___________________________________ Date:________________Thesequestionswillgiveyoupracticetoimproveyourproficiencylevelforaportionofthefollowingstandards:MAFS.912.F-BF.1.1.b.c.Writeafunctionthatdescribesarelationshipbetweentwoquantities.b.Combinestandardfunctiontypesusingarithmeticoperations.Forexample,buildafunctionthatmodelsthetemperatureofacoolingbodybyaddingaconstantfunctiontoadecayingexponential,andrelatethesefunctionstothemodel.c.Composefunctions.Forexample,ifisthetemperatureintheatmosphereasafunctionofheight,andistheheightofaweatherballoonasafunctionoftime,thenisthetemperatureatthelocationoftheweatherballoonasafunctionoftime.

===========WORKTHEFOLLOWINGPROBLEMSWITHAPARTNER(IFNEEDED)===========

1. Findeachcombinationoffunctions.

a. If𝑓 𝑥 = 𝑥$ + 8𝑥 − 20and𝑔 𝑥 = 11 − 9𝑥,find𝑓 𝑥 + 𝑔(𝑥).

b. Ifℎ 𝑥 = 11𝑥 + 4and𝑝 𝑥 = 𝑥$ − 12,find𝑝 𝑥 − ℎ(𝑥).

2. Findeachcompositionoffunctions.

a. If𝑓 𝑥 = 𝑥$ + 8𝑥 − 20and𝑔 𝑥 = 6𝑥,find(𝑓 ∘ 𝑔) 𝑥 .

b. Ifℎ 𝑥 = 11𝑥 + 4and𝑝 𝑥 = 𝑥$ − 12,find𝑝 ℎ(𝑥) .


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3. Evaluateeachcompositevalue.

a. If𝑓 𝑥 = 𝑥$ + 5𝑥 − 6and𝑔 𝑥 = 2𝑥,find(𝑓 ∘ 𝑔) 2 .

b. Ifℎ 𝑥 = 4𝑥 + 3and𝑝 𝑥 = 𝑥$ − 8,find𝑝 ℎ(−1) .

4. ThestudentgovernmentassociationissellingrosesforValentine’sDaytoraisemoneyfortheirtriptothestate

convention.Thecostofeachroseis$1.50andthefloristchargesadeliveryfeeof$25.Theclassplanstoselltherosesfor$5.00each.

a. Definethevariable.

b. Writeacostfunction.

c. Writearevenuefunction.

d. Writeaprofitfunction.

5. Alocalcivicgroupissellingt-shirtstoraisefundsforRelaytoLife.Theyplantosell2500t-shirtsfor$10.They

considerraisingthet-shirtpriceinordertoearnmoreprofit.Foreach$1increase,theywillsell100fewert-shirts.Let𝑥representthenumberof$1increases.

a. Writeafunction,𝐶(𝑥),torepresentthecostofonet-shirtbasedonthenumberofincreases.

b. Writeafunction,𝑇(𝑥),torepresentthenumberoft-shirtssoldbasedonthenumberofincreases.

c. Writearevenuefunction,𝑅(𝑥),forthet-shirtsellthatcouldbeusedtomaximizerevenue.


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6. Annagetspaid$8.75/hourworkingasabaristaatCoffeeBreak.Herbosspaysher$9.00/hourforcreatingthe

weeklyadvertisementsigns.Sheworksatotalof25hourseachweek.

a. Let𝑥representthehoursthatAnnaworkseachweekasabarista.Writeafunctionℎ(𝑥)torepresenttheamountofmoneythatAnnaearnsworkingasabarista.

b. Writeafunction,𝑓(𝑥)torepresentthehoursAnnaworkscreatingthesigns.

c. Let𝑠representthenumberofhoursthatAnnaworkscreatingthesigns.Createafunction𝑔(𝑠)torepresentthemoneyAnnaearnscreatingthesigns.

d. Find𝑔 𝑓 𝑥 .Whatdoesthiscompositefunctionrepresent?

e. WhatfunctionscouldbecombinedtorepresentAnna’stotalearnings?CombinethefunctionstowriteanexpressionthatcanbeusedtorepresentAnna’stotalearnings,where𝑥representsthenumberofhourssheworksasabarista.

7. Oliviaworksatanelectronicsstore.Shereceivesaweeklysalaryof$425andispaida3%commissiononweekly

salesover$2000.Writeafunction𝑓(𝑥)torepresentOlivia’sweeklysales(𝑥)over$2000,afunction𝑔(𝑥)torepresentOlivia’scommission,andacompositefunction,(𝑔 ∘ 𝑓) 𝑥 torepresenttheamountofmoneyOliviaearnsoncommission.

8. Considerthefollowingstandards.

MAFS.912.F-BF.1.1b:Combinestandardfunctiontypesusingarithmeticoperation.Forexample,buildafunctionthatmodelsthetemperatureofacoolingbodybyaddingaconstantfunctiontoadecayingexponential,andrelatethesefunctionstothemodel.

MAFS.912.F-BF.1.1c:Composefunctions.Forexample,ifT(y)isthetemperatureintheatmosphereasafunctionofheight,andh(t)istheheightofaweatherballoonasafunctionoftime,thenT(h(t))isthetemperatureatthelocationoftheweatherballoonasafunctionoftime.Describethedifferenceincombiningfunctionsandcomposingfunctions.


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Name:___________________________________ Date:________________Thesequestionswillgiveyoupracticetoimproveyourproficiencylevelforaportionofthefollowingstandards:MAFS.912.F-BF.1.1.b.c.Writeafunctionthatdescribesarelationshipbetweentwoquantities.b.Combinestandardfunctiontypesusingarithmeticoperations.Forexample,buildafunctionthatmodelsthetemperatureofacoolingbodybyaddingaconstantfunctiontoadecayingexponential,andrelatethesefunctionstothemodel.c.Composefunctions.Forexample,ifisthetemperatureintheatmosphereasafunctionofheight,andistheheightofaweatherballoonasafunctionoftime,thenisthetemperatureatthelocationoftheweatherballoonasafunctionoftime.

===========WORKTHISPROBLEMINDEPENDENTLY===========

1. Considerthefollowingfunctionsthatrepresentcostandrevenuefortheband’ssalesofspiritmugs.

𝑓 𝑥 = 100 + 3𝑥𝑔 𝑥 = 8𝑥

a. Whichfunctionrepresentsthecostfunction?

b. Whichfunctionrepresentstherevenuefunction?

c. Howcanthefunctionsbecombinedtocreateaprofitfunction?

d. Writetheprofitfunction.2. Aresorthotelcharges12.5%intaxes.Additionally,theychargea$25resortfeeeachday.(Theresortfeeisnot

subjecttosalestax.)

Thefollowingfunctionsrepresentthesituation:𝑓 𝑎 = 1.125𝑎𝑔 𝑏 = 𝑏 + 25

a. Matchthefollowingtotheirdescription. 𝑎 A. Thecostofthehotelroomandsalestax,butwithouttheresortfee. 𝑏 B. Thecostofthehotelroomwithoutsalestaxortheresortfee. 𝑓(𝑎) C. Thecostofthehotelroom,withsalestaxandtheresortfee. 𝑔(𝑏) b. Is𝑔 𝑓 𝑎 = 𝑔(𝑏)?Justifyyouranswer.


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U3InterpretingKeyFeaturesofGraphsofFunction

Name:___________________________________ Date:________________Thesequestionswillgiveyoupracticetoimproveyourproficiencylevelforaportionofthefollowingstandard:MAFS.912.F-IF.1.1Understandthatafunctionfromoneset(calledthedomain)toanotherset(calledtherange)assignstoeachelementofthedomainexactlyoneelementoftherange.Ifisafunctionandisanelementofitsdomain,thendenotestheoutputofcorrespondingtotheinput.Thegraphofisthegraphoftheequation.MAFS.912.F-IF.2.5Relatethedomainofafunctiontoitsgraphand,whereapplicable,tothequantitativerelationshipitdescribes.MAFS.912.A-REI.4.10Understandthatthegraphofanequationintwovariablesisthesetofallitssolutionsplottedinthecoordinateplane,oftenformingacurve(whichcouldbealine).MAFS.912.F-IF.3.7Graphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandinsimplecasesandusingtechnologyinmorecomplicatedcases.


1. Thefollowinggraphfailstheverticallinetestandisnotafunction.

a. Explainhowtheverticallinetestshowsthatthisrelationis

NOTafunction.b. Nametwopointsonthegraphthatshowthatthisrelationis

NOTafunction.

2. Usethewordbanktocompletethesentencesbelow.

𝑥 −coordinate 𝑦 −coordinate 𝑥 −intercept 𝑦 −intercept solution

a. The____________________ofagraphisthelocationwherethegraphcrossesthe𝑥 −axis.b. The____________________ofagraphisthelocationwherethegraphcrossesthe𝑦 −axis.c. The____________________ofthe𝑦 −interceptisalwayszero.d. The____________________ofthe𝑥 −interceptisalwayszero.e. The𝑥 −interceptisthe______________toafunctionorgroup.


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3. Thegraphbelowrepresentsarockclimber’sheightassheascendsahill.

a. Theabovegraphis(circleone)

linear/nonlinear.Why?b. Whatisthe𝑦 −interceptandwhatdoes

the𝑦 −interceptrepresent?c. Whywouldtherenotbean𝑥 −intercept

forthissituation?

4. ThegraphbelowrepresentsthepathofaFrisbeethrownbytwofriends.

a. Theabovegraphis(circleone)linear/nonlinear.

Why?b. Whatisthe𝑦 −interceptandwhatdoesthe

𝑦 −interceptrepresent?c. Whatisthesolutiontothisgraphandwhatdoesit

representinthissituation?


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5. Considerthefollowinggraphofanabsolute

valuefunction.

a. Definethedomain.

b. Definetherange.

c. Whereisthegraphincreasing?

d. Whereisthegraphdecreasing?

e. Identifyanyrelativemaximums.

f. Identifyanyrelativeminimums.

6. Considerthefollowinggraphofaquadraticfunction.

a. Definethedomain.

b. Definetherange.

c. Whereisthegraphincreasing?

d. Whereisthegraphdecreasing?

e. Identifyanyrelativemaximums.

f. Identifyanyrelativeminimums.


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7. Considerthefollowinggraphofanexponentialfunction.

a. Definethedomainandrange.

b. Whereisthegraphincreasingordecreasing?

c. Identifyanyrelativemaximumsorminimums.

8. Considerthegraphbelowofapiecewisedefinedfunction.

Whichofthefollowingstatementsaretrue?Selectallthatapply.

£ Thegraphisincreasingwhen𝑥 < −2.£ Thegraphhasarelativeminimumat(0, 0).£ Thegraphhastworelativemaximums.


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£ Thegraphisdecreasingwhen −2 < 𝑥 < 0 ∪ [2 < 𝑥 < 7].£ Thedomainis−6 < 𝑥 ≤ 7.£ Therangeis𝑥 ≤ 4.

9. ThebanddirectorofEagleRockHighSchoolasksherstudentstohaveacarwashasafundraiser.Theyare

goingtocharge$7.50percar.Thegraphthatrepresentsthesituationisshownbelow.

Whichofthefollowingstatementsaboutthegrapharetrue?Selectallthatapply.

£ Thegraphrepresentsafunctionsinceevery

inputvaluecorrespondstoexactlyoneoutputvalue.

£ Thegraphhasan𝑥 −intercept.£ Thegraphhasa𝑦 −intercept.£ The𝑦 −interceptrepresentstheamountof

moneyraisedbythecarwash.£ Theamountofmoneyinthebandfundraiser

accountrepresentstheinputvalues.£ Thenumberofcarswashedrelatestothe

amountofmoneyinthebandfundraiseraccount.

10. Sketchthegraphofthefollowingfunctions.

a. 𝑓 𝑥 = 2𝑥 − 3

b. 𝑔 𝑥 = −4𝑥 + 5


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U3AverageRateofChange

Name:___________________________________ Date:________________Thesequestionswillgiveyoupracticetoimproveyourproficiencylevelforaportionofthefollowingstandard:MAFS.912.F-IF.2.6Calculateandinterprettheaveragerateofchangeofafunction(presentedsymbolicallyorasatable)overaspecifiedinterval.Estimatetherateofchangefromagraph.

===========FILLTHISOUTWITHYOURGROUP===========

Youwillworkwithateamof4students.Eachofyouwillbeplacedinaspecificyardline.Youwillpasstheballtotheteammatenexttoyouandwewillrecordthetimeittookfortheballtogettoyourteammate.Then,wewillrepeattheprocesswiththenext3teammates.Thepassshouldbesmoothand

Trial 𝒕(inseconds)

𝑫(𝒕)(inyards)

[a]

[b]

[c]

[d]

GraphyourdatainthequadrantIcoordinateplanebelow.Identifytheaveragerateofchangeovertheintervals[a,c],[b,d]and[a,d]


ALGEBRA1



1. Supposethatthecostofproducing𝒙tabletsisdefinedby𝑐 𝑥 = 200 + 10𝑥 + 0.2𝑥.,where𝑥representsthenumberoftabletsproduced.Thegraphbelowrepresentsthefunction.

PartA:Completetheboxestolabelthegraph.

PartB:Findtheaveragerateofchangeoverthe

interval[𝑎, 𝑏].PartC:Findtheaveragerateofchangeoverthe

interval[𝑏, 𝑐]

2. Refertoexercise#1andcomparetheaveragerateofchangeovertheinterval 𝑎, 𝑏 withhisaveragerate

ofchangeovertheinterval[𝑏, 𝑐].Whatdoesthisrepresentinreallife?

!"

#

(15,395)(20,480)

(25,575)


ALGEBRA1


3. Considerthetablefortheexponentialfunction,𝑝 𝑥 = 25,shownbelow.

PartA:Whichofthefollowingintervalswouldyouexpecttohavethegreatestrateofchange?Explainyourreasoning.PartB:Findtherateofchangeofthethreeintervalstodetermineifyourhypothesis

inPartAwascorrect.4. Considerthefollowinggraphofaquadraticfunction.PartA:Howcanyouquicklydeterminetheintervalswiththegreatestrateofchangebymakingobservations

fromthegraph?PartB:Whichintervalshavethegreatestrateofchange?PartC:Whichintervalshavetheleastrateofchange?PartD:Nametwointervalshaveequalratesofchange?

Point 𝒙 𝒑 𝒙

𝐴 0 1

𝐵 2 4

𝐶 4 16

𝐷 6 64

o [𝐴, 𝐵]o 𝐵, 𝐶 o [𝐶, 𝐷]

!

"

#

$

%


ALGEBRA1


===========WORKTHEFOLLOWINGPROBLEMSINDEPENDENTLY===========5. BradfordhoppedintohisFordMustangGTtorushtoDianaandtellherhowfantastichisstudentsdidon

theEndofCourseexam.HisMustangcangofrom0to60mphinabrief4.7seconds.Bradfordacceleratedtotopspeedquicklyandthenmaintainedaspeedof60mphuntilhearrivedatDiane’soffice.

PartA:Sketchthegraphofthesituationonthecoordinateplane.PartB:Overwhichintervalwouldyouhavethegreatestrateofchange?Justifyyouranswer.PartC:Whatpartofthesituationrepresentsthetimewhenthegraphhasnorateofchange?

6. Supposethatthecostofproducing𝒙tabletsisdefinedby𝑐 𝑥 = 200 + 10𝑥 + 0.2𝑥.,where𝒙represents

thenumberoftabletsproduced.Thetablebelowrepresentsthefunction.

𝒙 𝑪(𝒙)PartA:Whichintervalrepresentsthegreatestrateofchange?PartB:Whatdoesthistellyouaboutthesituation?

15 395

20 480

25 575

30 680

35 795


ALGEBRA1


Name:___________________________________ Date:________________Thesequestionswillgiveyoupracticetoimproveyourproficiencylevelforaportionofthefollowingstandard:MAFS.912.F-IF.2.6Calculateandinterprettheaveragerateofchangeofafunction(presentedsymbolicallyorasatable)overaspecifiedinterval.Estimatetherateofchangefromagraph.

===========FILLTHISOUTWITHYOURGROUP===========

Youwillworkwithateamof4students.Eachofyouwillbeplacedinaspecificyardline.Youwillpasstheballtotheteammatenexttoyouandwewillrecordthetimeittookfortheballtogettoyourteammate.Then,wewillrepeattheprocesswiththenext3teammates.Thepassshouldbesmoothand

Trial 𝒕(inseconds)

𝑫(𝒕)(inyards)

[a]

[b]

[c]

[d]

GraphyourdatainthequadrantIcoordinateplanebelow.Identifytheaveragerateofchangeovertheintervals[a,c],[b,d]and[a,d]


ALGEBRA1



1. Supposethatthecostofproducing𝒙tabletsisdefinedby𝑐 𝑥 = 200 + 10𝑥 + 0.2𝑥.,where𝑥representsthenumberoftabletsproduced.Thegraphbelowrepresentsthefunction.

PartA:Completetheboxestolabelthegraph.

PartB:Findtheaveragerateofchangeoverthe

interval[𝑎, 𝑏].PartC:Findtheaveragerateofchangeoverthe

interval[𝑏, 𝑐]

2. Refertoexercise#1andcomparetheaveragerateofchangeovertheinterval 𝑎, 𝑏 withhisaveragerate

ofchangeovertheinterval[𝑏, 𝑐].Whatdoesthisrepresentinreallife?

!"

#

(15,395)(20,480)

(25,575)


ALGEBRA1


3. Considerthetablefortheexponentialfunction,𝑝 𝑥 = 25,shownbelow.

PartA:Whichofthefollowingintervalswouldyouexpecttohavethegreatestrateofchange?Explainyourreasoning.PartB:Findtherateofchangeofthethreeintervalstodetermineifyourhypothesis

inPartAwascorrect.4. Considerthefollowinggraphofaquadraticfunction.PartA:Howcanyouquicklydeterminetheintervalswiththegreatestrateofchangebymakingobservations

fromthegraph?PartB:Whichintervalshavethegreatestrateofchange?PartC:Whichintervalshavetheleastrateofchange?PartD:Nametwointervalshaveequalratesofchange?

Point 𝒙 𝒑 𝒙

𝐴 0 1

𝐵 2 4

𝐶 4 16

𝐷 6 64

o [𝐴, 𝐵]o 𝐵, 𝐶 o [𝐶, 𝐷]

!

"

#

$

%


ALGEBRA1


===========WORKTHEFOLLOWINGPROBLEMSINDEPENDENTLY===========5. BradfordhoppedintohisFordMustangGTtorushtoDianaandtellherhowfantastichisstudentsdidon

theEndofCourseexam.HisMustangcangofrom0to60mphinabrief4.7seconds.Bradfordacceleratedtotopspeedquicklyandthenmaintainedaspeedof60mphuntilhearrivedatDiane’soffice.

PartA:Sketchthegraphofthesituationonthecoordinateplane.PartB:Overwhichintervalwouldyouhavethegreatestrateofchange?Justifyyouranswer.PartC:Whatpartofthesituationrepresentsthetimewhenthegraphhasnorateofchange?

6. Supposethatthecostofproducing𝒙tabletsisdefinedby𝑐 𝑥 = 200 + 10𝑥 + 0.2𝑥.,where𝒙represents

thenumberoftabletsproduced.Thetablebelowrepresentsthefunction.

𝒙 𝑪(𝒙)PartA:Whichintervalrepresentsthegreatestrateofchange?PartB:Whatdoesthistellyouaboutthesituation?

15 395

20 480

25 575

30 680

35 795


ALGEBRA1

U3TransformationsofFunctions

Name:___________________________________ Date:________________Thesequestionswillgiveyoupracticetoimproveyourproficiencylevelforaportionofthefollowingstandard:MAFS.912.F-BF.2.3Identifytheeffectonthegraphofreplacingby,,andforspecificvaluesof(bothpositiveandnegative);findthevalueofgiventhegraphs.Experimentwithcasesandillustrateanexplanationoftheeffectsonthegraphusingtechnology.Includerecognizingevenandoddfunctionsfromtheirgraphsandalgebraicexpressionsforthem.


1. Labelthefollowingastransformationsontheindependentvariableorthedependentvariableanddescribethe

transformation. Description

a. 𝑓 𝑥 + 3

b. 𝑓 𝑥 − 3

c. 𝑓(𝑥 + 3)

d. 𝑓(𝑥 − 3)

2. Thefollowingtablerepresentsthefunctionℎ 𝑥 .Completethetablefor𝑔(𝑥),given𝑔 𝑥 = ,

-ℎ 𝑥 .

𝒙 𝒉(𝒙) 𝒙 𝒈(𝒙)

−4 256

−2 16

0 0

3 81

6 1296

o Independent Variable o Dependent Variable





ALGEBRA1


3. Thefollowinggraphrepresentsthefunction𝑓 𝑥 .Sketchandlabelthefollowingfunctionsonthesamecoordinate

plane.a. 𝑓 𝑥 + 2 b. 𝑓(𝑥 − 5)c. 𝑓 𝑥 + 2d. 𝑓 𝑥 − 5

4. Thefollowingtablerepresentsthefunctionℎ 𝑥 .Completethetablefor𝑔(𝑥),given𝑔 𝑥 = ℎ ,

9𝑥 .Thefirsttwo

havebeendoneforyou!

𝒙 𝒉(𝒙) 𝒙 𝒉𝟏𝟒𝒙 𝒈(𝒙)

−2 3.25 −8 ℎ14∙ −8 = ℎ(−2) 3.25

−1 3.5 −4 ℎ14∙ −4 = ℎ(−1) 3.5

0 4

1 5

2 7

3 11

4 19


ALGEBRA1


5. Considerthefollowingtableofvalues.

𝒙 𝒇(𝒙) 𝒈(𝒙) 𝒉 𝒙 𝒎(𝒙)

−5 25 23 −25 32

−3 9 7 −9 16

0 0 −2 0 7

3 9 7 −9 16

5 25 23 −25 32

a. Write𝑔(𝑥)asatransformationof𝑓(𝑥).

b. Writeℎ(𝑥)asatransformationof𝑓(𝑥).

c. Write𝑚(𝑥)asatransformationof𝑓(𝑥).

6. Considerthefollowingtableofvaluesandwrite𝑚(𝑥)asatransformationof𝑓(𝑥).

𝒙 𝒇(𝒙) 𝒙 𝒎(𝒙)

−5 25 −7 25

−3 9 −5 9

0 0 −2 0

3 9 1 9

5 25 3 25


ALGEBRA1


==========WORKTHEFOLLOWINGPROBLEMSINDEPENDENTLY===========

7. Considerthefollowinggraph.

a. Writeℎ(𝑥)asatransformationof𝑔(𝑥).b. Write𝑔(𝑥)asatransformationof𝑓(𝑥).c. Writeℎ(𝑥)asatransformationof𝑓 𝑥 .

8. Matchthefollowingtransformationstotheircorrectdescription.

𝑓 𝑥 + 4 A. Thegraphisshiftedfourunitstotherightof𝑓(𝑥).

𝑓 𝑥 − 4 B. Thegraphisshiftedfourunitstotheleftof𝑓(𝑥).

𝑓(𝑥 − 4) C. Thegraphisshiftedfourunitsabove𝑓(𝑥).

𝑓(𝑥 + 4) D. Thegraphisshiftedfourunitsbelow𝑓(𝑥).

9. Thegraphof𝑔(𝑥)isshownbelow.Sketchthegraphofℎ 𝑥 = 𝑔 𝑥 + 3 − 6.

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