lesson 11.1 assignment - vvsd · lesson 11.1 assignment page 2 ... the quickgrow fertilizer company...

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Chapter 11 Assignments 153

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Lesson 11.1 Assignment

Name Date

Up and Down or Down and UpExploring Quadratic Functions

1. The citizens of Herrington County are wild about their dogs. They have an existing dog park for dogs to play, but have decided to build another one so that one park will be for small dogs and the other will be for large dogs. The plan is to build a rectangular fenced in area that will be adjacent to the existing dog park. The sketch is shown below. The county has enough money in the budget to buy 1000 feet of fencing.

w

l

w

Existing Dog Park New Dog Park

a. Determine the length of the new dog park, l, in terms of the width, w.

2w1l51000

l510002 2w

b. Write a function for the area of the new dog park, A(w), in terms of the width, w. Write the function in standard quadratic form. Does this function have an absolute minimum or an absolute maximum? Explain your answer.

A(w)5l?w

A(w)5(100022w)w

A(w)51000w22w2

A(w)522w21 1000w

Thisfunctionhasanabsolutemaximum.Becausethecoefficientofthesquared-termisnegative,theparabolaopensdownwardwhichresultsinanabsolutemaximum.

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154 Chapter 11 Assignments

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Lesson 11.1 Assignment page 2

c. Determine the x-intercepts of the function. Explain what each means in terms of the problem situation.

Thex-interceptsare(0,0)and(500,0).

Eachrepresentsthewidthinfeetforwhichtheareaoftherectangularparkis0squarefeet.

Thex-intercept(0,0)meansthatifthewidthoftheparkis0feetthentheareawillbe0squarefeet,whichmakessensebecausetherewillbenoenclosedareaifthereisnowidth.

Thex-intercept(500,0)meansthatifthewidthoftheparkis500feetthentheareawillbe0squarefeet.Thismakessensebecauseifbothsidesoftheparkare500feetwide,thentherewillbenofencingleftforthelengthsotherewillbenoareatoenclose.

d. What should the dimensions of the dog park be to maximize the area? What is the maximum area of the park?

Thewidthofthedogparkshouldbe250feetandthelengthofthedogparkshouldbe10002 2(250)5 100025005500feet.Themaximumareaoftheparkis125,000squarefeet.

e. Sketch the graph of the function. Label the axes, the absolute maximum or minimum, the x-intercepts, and the y-intercept.

0 100Width (feet)

(250, 125,000)

(0, 500)(0, 0)

Are

a (s

qua

re f

eet)

200 300 400

20,000

40,000

60,000

80,000

100,000

120,000

140,000

160,000

180,000

x

y

f. Use the graph to determine the dimensions of the park if the area was restricted to 105,000 square feet.

Iftheareais105,000squarefeetthewidthcouldeitherbe150feetor350feet.

Sotheparkcouldeitherbe150feetwideand10002 2(150)5 100023005700feetlongortheparkcouldbe350feetwideand10002 2(350)5 100027005300feetlong.

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Name Date

Just U and IComparing Linear and Quadratic Functions

1. The Quickgrow Fertilizer Company is working on different formulas for flower fertilizers. The table shows the growth of unfertilized plant A and the growth of fertilized plant B.

Time(days)

HeightofplantA(centimeters)

HeightofplantB(centimeters)

0 4 3

1 6 4

2 8 6

3 10 9

4 12 13

5 14 18

6 16 24

a. Which plant height would be represented by a linear function? Which would be represented by a quadratic function? Explain your reasoning.

TheheightofplantAwouldberepresentedbyalinearfunctionbecausethefirstdifferencesareallthesameandtheseconddifferencesareall0.

TheheightofplantBwouldberepresentedbyaquadraticfunctionbecausethefirstdifferencesarechangingbuttheseconddifferencesareallthesame.

b. Would the function A(x) 5 22x 1 4 or A(x) 5 2x 1 4 represent the growth of plant A? Explain using leading coefficients.

ThelinearfunctionA(x)52x14wouldrepresentthegrowthofplantAbecausetheheightoftheplantisincreasingasthenumberofdaysincreasessotheleadingcoefficientmustbepositive.

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c. Would Graph A or Graph B represent the growth of plant B? Explain using second differences.

0 2 4 6Number of Days

Hei

ght

(cen

timet

ers)

8 10 12 14 16 18

4

8

12

16

20

24

Graph A

Graph B

28

32

36

x

y

GraphBwouldrepresentthegrowthofPlantB.Becausetheseconddifferencesintheplantgrowtharepositive,thegraphwouldopenupward.

2. The Quickgrow Fertilizer Company has run into problems while experimenting with a type of fertilizer that is supposed to increase yield of pepper plants. The yield for plant C can be represented by the function C(x) 5 212.5x 1 100. The yield for plant D can be represented by the function D(x) 5 23x2 1 21x 1 50. The graphs of the yields for both plants are shown.

0 1 2 3Fertilizer (tsp)

Yie

ld (n

umb

er o

f pep

per

s)

4 5 6 7 8 9

10

20

30

40

50

60

70

80

90

x

y

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Name Date

a. Determine the y-intercept(s) of each function and describe the meaning of each in terms of the problem situation.

They-interceptforthelinearfunctionisat(0,100).Thismeansthatwhennofertilizerwasputontheplantityielded100peppers.

They-interceptforthequadraticfunctionisat(0,50).Thismeansthatwhennofertilizerwasputontheplantityielded50peppers.

b. Determine the x-intercept of the linear function algebraically. Describe the meaning in terms of the yield for plant C.

C(x)5212.5x1100

05212.5x1100

21005212.5x

85 x

Thex-interceptofthelinearfunctionisat(8,0).Thismeansthatwhen8teaspoonsofthefertilizerwereputonplantCitdidnotyieldanypeppers.

c. Determine the x-intercept(s) of the quadratic function. Then, describe the meaning of each in terms of the problem situation.

Thex-interceptsofthequadraticfunctionareatabout(21.9,0)and(8.9,0).Thefirstinterceptdoesnotmakesensebecauseyoucannotputanegativeamountoffertilizerontheplant.Thesecondinterceptmeansthatwhen8.9teaspoonswereputontheplantitdidnotyieldanypeppers.

d. Determine the absolute maximum of the quadratic function. Explain what it means in terms of the yield for plant D.

Theabsolutemaximumofthequadraticfunctionisat(3.5,86.75).ThismeansthatplantDwouldyieldamaximumofabout87peppersif3.5teaspoonsoffertilizerwereputonit.

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Name Date

Walking the . . . Curve?Domain, Range, Zeros, and Intercepts

1. A masking tape company has to decide how many hundreds of rolls of tape to produce each day. The company knows that the costs to produce the tape go down the more rolls they make. However, the overall cost to the company increases if they make too many rolls due to the cost of storing overstock. The company determined that the cost to produce x hundreds of units a day could be represented by the function f(x) 5 0.04 x 2 2 16x 1 15,000.

a. Graph the function. Sketch the graph and label the axes.

0 200Rolls of Tape

Co

st (d

olla

rs)

400 600 800

5000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

45,000

x

y

b. What are the domain and range of the function in terms of the graph?

Thedomainofthefunctionisallrealnumbersfromnegativeinfinitytopositiveinfinity.Therangeofthefunctionisallrealnumbersgreaterthanorequalto13,400.

c. What are the domain and range of the function in terms of the problem situation?

Thedomainintermsofthenumberofrollsoftapeproducedisanyintegergreaterthanorequalto0.Thecompanycannotmakeanegativenumberofrollsoftapeorpartsofrollsoftape.Therangeintermsoftheprofitisallrealnumbersgreaterthanorequalto13,400.Thismeansthatthecostsassociatedwithmakingrollsoftapewillneverbelessthan$13,400.

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d. Over what interval does the cost of making the rolls of tape decrease? Increase?

Thecostofmakingtherollsoftapedecreasesovertheinterval[0,200)andincreasesovertheinterval(200,`).

e. How many rolls of tape should the company make to minimize the cost?

Thecompanyshouldmake20,000rollsoftapedailytominimizecost.

f. What is the minimum cost to the company? What does this number represent for the function?

Theminimumcostis$13,400.Thisistheabsoluteminimumofthefunction.

g. Determine the x-intercept(s) of this function and describe what they mean in terms of the cost to the company.

Therearenox-interceptsforthisfunction.Thismeansthatthecostofproducingtherollsoftapewillneverequal0.Thisisbecauseoffixedcostsassociatedwithoverhead,wages,etc.

2. The profit a masking tape company makes from producing and selling x hundred rolls of tape can be represented by the function g(x) 5 20.1 x 2 1 100x 2 15,000. The graph of the profit function is shown.

0 1200

10,000

210,000

220,000

230,000

240,000

20,000

30,000

40,000

400 800x

y

a. What is the domain of this function? What is the domain for the problem situation?

Thedomainofthefunctionisallrealnumbers.Thedomainfortheproblemsituationisallintegersgreaterthanorequaltozero.Theycannotmakeanegativenumberofrollsoftapeorpartsofrolls.

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Name Date

b. What is the range of this function? What is the range for the problem situation?

Therangeofthefunctionisallrealnumberslessthanorequalto10,000.Therangefortheproblemsituationisallrealnumberslessthanorequalto10,000.

c. Over what interval does the profit increase? Decrease?

Theprofitincreasesovertheinterval[0,500).Theprofitdecreasesovertheinterval(500,`).

d. How many rolls of tape must they produce and sell to make a profit of $1590?

Theymustmakeeither21,000or79,000rollsoftapetomakeaprofitof$1590.

e. Determine the x-intercepts of the function. Describe what the x-intercepts mean in terms of this problem situation.

Thex-interceptsareabout(183.77,0)and(816.23,0).Thex-interceptsindicatethatifthecompanymakesaround18,377or81,623rollsoftapetheywillbreak-even,orhavezeroprofit.

f. Over what interval(s) is there a negative profit? Over what interval(s) is there a positive profit?

Thecompanyhasanegativeprofitovertheintervals[0,183.77)and(816.23,`)andapositiveprofitovertheinterval(183.77,816.23).

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Are You Afraid of Ghosts?Factored Form of a Quadratic Function

1. The Squeaky Clean Car Wash charges $20 for its deluxe wash. At this price, the car wash averages 200 customers per day. The car wash has determined that for every $0.25 that they decrease the price of the wash, they will see an increase of 5 customers per day.

a. Complete the table to determine the revenue the car wash will see based on the number of price decreases.

NumberofPriceDecreases

PriceofCarWash(dollars)

NumberofCustomers

Revenue(dollars)

0 20 200 4000

1 19.75 205 4048.75

4 19 220 4180

15 16.25 275 4468.75

30 12.5 350 4375

60 5 500 2500

75 1.25 575 718.75

n 2020.25n 20015n (2020.25n)(20015n)

b. The manager of the car wash thinks that he should just keep decreasing the price to bring in the most customers. Based on the table, is this a good decision? Explain your reasoning.

No.Themanagershouldnotkeepdecreasingtheprice.Atsomepointthepricewillbetoolowandeventhoughthenumbersofcustomersincreasetherevenuestartstodecrease.

c. Write the revenue for the car wash as a function, R(n), in standard form. Then, identify a, b, and c for the function.

R(n)5(2020.25n)(20015n)

R(n)520(200)120(5n)1 (20.25n)(200)1 (20.25n)(5n)

R(n)540001100n 2 50n2 1.25n2

R(n)521.25n21 50n1 4000

a521.25,b5 50,andc5 4000


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d. Based on the equation, will the graph of the revenue function open up or down? Explain your reasoning.

Thegraphoftherevenuewillopendownbecausethecoefficientofthex2termisnegative.

e. Graph the revenue function. Then sketch the graph and label the axes.

0 10 20 30Number of Decreases

Rev

enue

(do

llars

)

40 50 60 70 80 90

500

1000

1500

2000

2500

3000

3500

4000

4500

x

y

f. At what number of price decreases will the revenue be maximized? What is the maximum revenue the car wash will see at that price?

Therevenuewillbemaximizedwhenthereare20pricedecreases.Themaximumrevenuewillbe$4500.

g. What should the price be to maximize revenue? How many customers will the car wash see at that price? Show your work.

2020.25(20)52025515

Thepriceshouldbe$15tomaximizerevenue.

20015(20)520011005300

Thecarwashwillsee300customerswhenthepriceis$15.

h. Determine the x-intercepts of the revenue function. Explain what they mean in terms of the problem situation.

Thex-interceptsare240and80.Theintercept240doesnotmakesensetothisproblembecausetherecannotbeanegativenumberofdecreasesinprice.Theintercept200meansthatafter200pricedecreases,therevenuewillequal$0.Thisisbecausethepriceofthecarwashwillbe$0after200pricedecreases.

i. Determine the y-intercept of the revenue function. Explain what it means in terms of the problem situation.

They-interceptis4000.Thismeansthatifthereare0pricedecreases,thecarwashcanexpectrevenueof$4000.

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Just Watch that Pumpkin Fly!Investigating the Vertex of a Quadratic Function

Investing in the stock market is always a risk. Sometimes there can be big payouts but other times you can end up losing it all.

1. Maya has saved up some money and decides to take a risk and invest in some stocks. She invests her money in Doogle, a popular computer company. Unfortunately she lost it all over a matter of months. The change in her money during this investment can be represented by the function v(x) 5 75 1 72x 2 3 x 2 , where v is the value of her investment and x is the time in months.

a. How much money did Maya first invest in the company? What does this value represent in the function?

Mayastartswith$75.Thevalue(0,75)representsthey-interceptofthefunction.

b. Determine the x-intercepts of the function. Explain what each intercept means in terms of the problem situation.


Thex-intercept(21,0)meansthatshehad0dollarsonemonthbeforeinvestinginthecompany.Thereisnowaytoknowthat,sothisdoesnotmakesense.

Thex-intercept(25,0)meansthatafter25monthsthevalueofherinvestmentwas0dollars.ThismakessensetotheproblemsituationbecauseMayalostallhermoney.

c. Determine the vertex. Explain what it means in terms of the problem situation.

Thevertexis(12,507).Thismeansthatherinvestmentreacheditsmaximumvalueof$507after12months.

d. Determine when her portfolio reached a value of $360.

Theportfolioreachedavalueof$360whenithadbeeninvestedfor5monthsandwhenithadbeeninvestedfor19months.


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2. Jack invested some of his money in Home-mart, a large home improvement store in his town. A few years after investing, the company went out of business and Jack lost all his money. The growth and decline of his money over this time can be represented by the function v(x) 5 22 x 2 1 98x 1 100, where v is the value of his investment and x is the time in months.

a. Describe this function in terms of the problem situation. Include information regarding the y-intercept, the x-intercepts, and whether the function has an absolute maximum or absolute minimum.

Jackstartswith$100.Thevalue(0,100)representsthey-interceptofthefunction.


Thex-intercept(21,0)meansthathehad0dollarsonemonthbeforeinvestinginthecompany.Thereisnowaytoknowthat,sothisdoesnotmakesense.

Thex-intercept(50,0)meansthatafter50monthsthevalueofhisinvestmentwas0dollars.ThismakessenseintheproblembecauseJackendeduplosingallhismoney.

Thefunctionhasanabsolutemaximum.Thisrepresentswhentheinvestmentwasatitsgreatest.

b. Determine the axis of symmetry for this parabola. Then determine the vertex and explain what it means in terms of the problem situation.

Theaxisofsymmetryisx524.5because21150________2 549___

2524.5.

They-coordinatewhenx524.5is:

v(24.5)522(24.5)2 1 98(24.5) 1 100

521200.5 1 2401 1 100

51300.5

Thevertexis(24.5,1300.5).Thismeansthatwhenthemoneyhasbeenintheportfoliofor24.5monthsthevaluewillbeamaximumat$1300.50.

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Name Date

c. Jack’s account has $1288 in it after 22 months. Use the axis of symmetry to determine another time when the account will have $1288 in it. Show and explain your work.

Anotherpointontheparabolawithay-valueof$1288isasymmetricpointto(22,1288).

Thex-coordinateis:

221a_______2 524.5

221a549

a527

Theaccountwillalsohave$1288initwhenthemoneyhasbeenintheaccountfor27months.

d. Draw a graph of this function on the grid provided. Label the axes, vertex, axis of symmetry, x-intercepts, and the set of symmetric points that you determined.

0 20

Time (months)

(22, 1288) (27, 1288)(24.5, 1300.5)

(50, 0)(21, 0)Val

ue o

f P

ort

folio

(do

llars

)

40 60 8070503010

200

400

600

800

1000

1200

1400

1600

x

yx 5 24.5

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The Form Is “Key”Vertex Form of a Quadratic Function

1. A concert venue can hold up to 20,000 people. The concert will sell out if tickets are sold for $40 a piece. In order to make more money, the venue would like to increase the ticket price. They determine that for every one dollar increase in price, 200 fewer people will attend. If x represents the number of one dollar increases in the price, then the revenue that the concert will bring in is represented by the function R(x) 5 (20,000 2 200x)(40 1 x).

a. Rewrite this function in the correct factored form. Then state the key characteristics you can determine from the equation and what they mean in terms of the problem situation.

R(x)52200(x2100)(x140)

Theparabolaopensdownward,whichmeansthattherevenuewillincrease,reachamaximum,andthendecreaseagainasthenumberofonedollarincreasesinpricegoesup.

Thex-interceptsare(100,0)and(240,0).Thefirstintercept(100,0)meansthatifthereare40onedollarpriceincreases,thentherevenuewillbe0dollarsbecausenoonewillcometotheconcert.Thesecondintercept(240,0)doesnotmakesensebecausetherecannotbeanegativenumberofonedollarpriceincreases.

b. Determine the vertex for the function. Explain what it means in terms of the problem situation. Then use it to rewrite the function in vertex form.

Thevertexisat(30,980,000).Thismeanswhenthereare30onedollarincreasesinprice,therevenuewillbeatamaximumof$980,000.

ThevertexformisR(x)52200(x230)21980,000.

c. Determine the y-intercept for the function. Explain what it means in terms of the problem situation.

They-interceptis(0,800,000).Thismeansthatwhentherearenotanyonedollarincreasesinpricetherevenuewillbe$800,000.

d. Which of the following functions must be the revenue function written in standard form? Explain your reasoning.

R(x) 5 2200 x 2 1 12,000x 2 800,000 or R(x) 5 2200 x 2 1 12,000x 1 800,000

Therevenuefunctionwritteninstandardformmustbethesecondfunction,R(x)52200x2112,000x1800,000,becausethisfunctionhasapositivey-intercept,meaningwhenthereare0onedollarincreasestherevenueisapositiveamount.


Name Date

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2. Perez throws a softball up in the air. The height of the ball in meters can be determined by the function h(t) 5 24.9(t 2 3 ) 2 1 60, where t is the time it is in the air in seconds.

a. Identify the form of this quadratic function. Then state all you know about the key characteristics, based only on the given equation of the function. Explain what they mean in terms of the problem situation.

Thisquadraticfunctionisinvertexform.

Theparabolaopensdownward,whichmeansthattheheightoftheballwillincrease,reachamaximum,andthendecreaseagainastimeincreases.

Thevertexoftheparabolais(3,60).Thismeansthatwhentheballhasbeenintheair3seconds,itwillreachamaximumheightof60meters.

b. Determine the x-intercept(s) of the function. Explain what they mean in terms of the problem situation. Then, write the function in factored form.

Thex-interceptsareabout(20.5,0)and(6.5,0).Thefirstinterceptdoesnotmakesensetotheproblembecausetimecannotbenegative.Thesecondinterceptmeansthattheballwillhitthegroundin6.5seconds.

Thefunctioninfactoredformish(t)524.9(t10.5)(t26.5).

c. Use your graphing calculator to determine the y-intercept for the function. Explain what it means in terms of the problem situation.

They-interceptofthefunctionisabout(0,15.9).Thismeansthattheinitialheightoftheballwas15.9meters.

d. Which of the following functions must be the revenue function written in standard form? Explain your reasoning.

h(t) 5 24.9 t 2 1 29.4t 1 15.9 or h(t) 5 24.9 t 2 1 29.4t 2 15.9

Theheightfunctionwritteninstandardformmustbethefirstfunction,h(t)524.9t2129.4t115.9,becausethisfunctionhasapositivey-intercept,meaningwhentimeiszerotheheightoftheballis15.9meters.

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Name Date

3. A company knows that the more it advertises the more product it will sell. However, advertising more will also cost more money, which then takes away some of the profit. The profit will therefore follow the path of a parabola, because it will increase from more advertising but eventually decrease if too much money is spent on advertising. The profit (in thousands of dollars) can be represented by the function P(x) 5 22 x 2 1 14x 1 60, where x represents the amount of money spent (in thousands of dollars).

a. Identify the form of this quadratic function. Then state all you know about the key characteristics, based only on the given equation of the function. Explain what they mean in terms of the problem situation.

Thisquadraticfunctionisinstandardform.

Theparabolaopensdownward,whichmeansthattheprofitwillincrease,reachamaximum,andthendecreaseagainastheamountofmoneyspentonadvertisingincreases.

They-interceptoftheparabolais(0,60).Thismeansthatwhenthecompanyspends0dollarsonadvertisingtheprofitwillbe$60,000.

b. Determine the x-intercept(s) of the function. Explain what they mean in terms of the problem situation. Then, write the function in factored form.

Thex-interceptsare(23,0)and(10,0).Thefirstinterceptdoesnotmakesensetotheproblembecausemoneycannotbenegative.Thesecondinterceptmeansthatwhen$10,000isspentonadvertisingtheprofitwillbe0dollars.

ThefunctioninfactoredformisP(x)522(x13)(x210).

c. Determine the vertex of the function. Explain what it means in terms of the problem situation. Then write the function in vertex form.

Thevertexofthefunctionis(3.5,84.5).Thismeansthatwhen$3500isspentonadvertisingtheprofitwillbeatamaximumof$84,500.

ThefunctioninvertexformisP(x)522(x23.5)2184.5.

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More Than Meets the EyeTransformations of Quadratic Functions

1. The owners of a botanical park would like to create a walkway around one of their premier gardens. The garden is 100 feet long and 100 feet wide. The drawing below shows the layout of the garden and walkway.

x

100 ft

100 ft

a. Determine the function, A(x), that represents the total area of the garden and walkway, Let x represent the width of the walkway. Then, write the quadratic function in vertex form.

A(x)5(2x1100)(2x1100)

A(x)52(x150)(2)(x150)

A(x)54(x150)2

b. Graph the function with the bounds [280, 80] X [210, 1000], with an X-scale of 10 and a Y-scale of 100. Sketch the graph on the coordinate plane provided. Also sketch the graph of the basic function f(x) 5 x 2 on the same coordinate plane. Label the graphs.

x

y

A(x) 5 4(x 1 50)2

f(x) 5 x2


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c. Describe how the graph of A(x) compares to the graph of f(x) and define the types of transformations the changes represent.

ThegraphofA(x)istranslated50unitstotheleftoff(x).Thisisahorizontaltranslation.

Eachy-coordinateofthegraphofA(x)is4timesthey-coordinateofthegraphoff(x).Thisisaverticaldilationwithadilationfactorof4.

2. A physics class has been assigned the task of creating a container that will protect an egg that their teacher will drop from the roof of their school. The graph below shows the basic function f(x) 5 x2 , and also shows the function h(x) which represents the height of the egg with respect to x, the time it is in the air.

28 26 24 22 0 2

50

250

2100

2150

2200

100

150

200

4 6 8x

y

f(x)

h(x)

a. Describe the types of transformations performed on f(x) to result in h(x).

Thegraphoff(x)hasbeenreflectedabouttheliney50,hasbeentranslated200unitsup,andhasbeenstretchedverticallyinordertogetthegraphofh(x).

b. If the dilation factor is 16, write the function h(x) that represents the height of the egg.

h(x)5216x21200

8070_Assign_CH11_153-176.indd 174 4/20/12 9:46 AM

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012

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11

3. A company’s revenues are dependent on the amount of product they sell, x. Use the given characteristics to write a function R(x) in vertex form, which represents the company’s revenue with respect to x. Then, sketch the graph of R(x) and the basic function f(x) 5 x2 on the grid.

• The function is quadratic.

• The function is continuous.

• The function has an absolute maximum.

• The function is translated 70 units up and 100 units to the right from f(x)521 __5 x2.

• The function is vertically stretched with a dilation factor of1 __ 5 .

Equation: R(x) 5 21__5

(x2100)2170

x

y

f(x) 5 x2

R(x) 5 2 (x 2 100)2 1 7015


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8070_Assign_CH11_153-176.indd 175 4/20/12 9:46 AM

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