U 3 U U Answer Key - ROBERT BEACH-CHATTOOGA ......2. x = 20 feet 3. 25 feet 4. 1,500 cubic feet Practice 3.1.5 A: Solving Quadratic Equations by Factoring, p. U3-96 1. x = –6, x

U3-431© Walch Education GSE Algebra I Teacher Resource

Teacher Resource/Student Workbook Answer Key

UNIT 3 • MODELING AND ANALYZING QUADRATIC FUNCTIONS

Answer KeyPractice 3.1.2 A: Factoring Expressions by the Greatest Common Factor, p. U3-33

1. x2(3x+5)2. 2xy(x–4y)3. y2(y2+2)4. nocommonfactor5. xy2(x2–2xy+5)6. 7x(1–3xy)7. 15x2y8. 5x3+11y2

9. Samueldidnotfactoroutthegreatestcommonnumberfromeachterm.Hefactoredouta2,butheshouldhavefactoredouta4.

10. Arianadidnotfactoroutthecorrectcommonpowersofthevariables.Shefactoredoutxyz2,whensheshouldhavefactoredoutxyz.

Practice 3.1.2 B: Factoring Expressions by the Greatest Common Factor, p. U3-34

1. s(s2t2–3s+2t2)2. 2x3y(2+x2y2)3. nocommonfactor4. xy(x3+x2y +xy2+y3)5. nocommonfactor6. x4(1+x10)7. 13x3y–6z2

8. length: 3x2 feet;width:(x2–4x+5)feet9. 17x3+24y2cannotbefactoredbecausethereareno

commonnumbersorcommonvariablesbetweenthetwoterms.

10. (3a2b3–4c2+12)meters

Warm-Up 3.1.3, p. U3-351. (x2+8x+15)squarefeet2. (x2–16)squarefeet3. (x+7)feet

Practice 3.1.3 A: Factoring Expressions with a = 1, p. U3-55

1. (y+10)(y–10)2. (x–7)(x–2)3. (x+8)2

4. notfactorable5. 7(a–2)(a+2)6. 4(x+9)(x–1)7. 6(x+3y)(x–3y)8. (a–7)feetand(a–7)feet9. (2x+5)metersand(2x–5)meters

10. (y+5)inchesand(y–2)inches

Lesson 1: Creating and Solving Quadratic Equations in One Variable

Pre-Assessment, p. U3-11. d2. c3. a

4. d5. a

Warm-Up 3.1.1, p. U3-61. Thebaseoftheladderisabout7.75feetfromthehouse.2. Theladderreachesabout14.94feetupthesideofthehouse.

Practice 3.1.1 A: Taking the Square Root of Both Sides, p. U3-16

1. x=±92. norealsolutions3. x=±34. x=–4andx=–25. norealsolutions6. x=12.5andx=7.5

7. Aquadraticequationintheformax2+b=chastworeal,

rationalsolutionswhenc b

a

−isapositiveperfectsquare.

8. s= 2 10 ≈6.32centimeters

9. r=60ππ

≈4.37millimeters

10. a=10 3

3≈5.77inches

Practice 3.1.1 B: Taking the Square Root of Both Sides, p. U3-17

1. x=± 22. norealsolutions3. norealsolutions4. x = ±4 2 10 5. x=–66. x = ±5 77. Aquadraticequationintheformax2+b =chasonlyone

realsolutionwhenc=b.8. s=7inches9. r= 2 5 ≈4.47units

10. r= ±5ππ

≈1.26feet

Warm-Up 3.1.2, p. U3-181. Therearefourpossiblesetsofdimensions:

l =30in,w=1inl=15in,w=2inl=10in,w=3inl=6in,w=5in

2. w=5x inches3. l =10x2y inches

U3-432© Walch EducationGSE Algebra I Teacher Resource


5. y=5,y=3

6. a7

2= , a

1

3= −

7. x=10,x=–68. 0.375second9. 2seconds

10. 5seconds

Practice 3.1.5 B: Solving Quadratic Equations by Factoring, p. U3-97

1. x=4,x=9

2. y=–2, y7

3= −

3. b=0, b11

2=

4. x=2,x=–25. y=–1,y=8

6. n4

3= − , n

1

2= −

7. x=3,x=88. 1.25seconds9. 2seconds

10. 4seconds

Warm-Up 3.1.6, p. U3-981. Theareaofthepatiois(36x2)ft2.2. Thetotalareaofthepatiois(36x2+48x+16)ft2.

Practice 3.1.6 A: Completing the Square, p. U3-1091. c=812. c=144

3. c =225

4

4. c =1

4

5. x=–10andx=06. x=–13orx=1

7. x = − ±1

3

22

3

8. approximately6.89feet;youcannothaveanegativelengthofaporch,soonlyonevalueofxisreasonable.

9. afterapproximately2.12seconds;thenegativeanswerrepresentstimebeforetheballwaskicked,soonlyonevalueofxisreasonable.

10. approximately31.36mphandapproximately88.64mph

Practice 3.1.6 B: Completing the Square, p. U3-1101. c=1212. c=2500

3. c =81

4

4. c =4

25

5. x = ±4 14

Practice 3.1.3 B: Factoring Expressions with a = 1, p. U3-56

1. (n+6)(n–6)2. (x–5)(x–4)3. (y–10)2

4. notfactorable5. 8(b+1)(b–1)6. 5(x+4)(x–2)7. 10(n+3m)(n–3m)8. (n–12)feetand(n–12)feet9. (3y+10)metersand(3y–10)meters

10. (x+9)inchesand(x–2)inches

Warm-Up 3.1.4, p. U3-571. (x+1)inches2. (x+4)inchesand(x+6)inches3. Thewidthis8incheslargerthanthelength.

Practice 3.1.4 A: Factoring Expressions with a > 1, p. U3-78

1. (5x+7)(3x–1)2. (3n+2)(n+8)3. (2y–7)(2y–9)4. (3x–4)(x+5)5. 3(a+4)(a+9)6. (5n+2)2

7. 2(2y–7)(3y–1)8. length=(3x+7)inches;width=(3x+7)inches9. length=(5x–1)feet;width=(x +3)feet

10. length=(3x–2)meters;(2x+7)meters

Practice 3.1.4 B: Factoring Expressions with a > 1, p. U3-79

1. (2x+3)(4x–7)2. (8a–3)(a+2)3. (2y+5)(5y–1)4. (2x–7)(x–7)5. (4a–3)2

6. 5(x+3)(x+6)7. 2(5y–1)(3y+4)8. length=(3x+10)inches;width=(2x+1)inches9. length=(3x–2)feet;width=(x+8)feet

10. length=(5x–3)meters;width=(2x–3)meters

Warm-Up 3.1.5, p. U3-801. x–82. x =20feet

3. 25feet4. 1,500cubicfeet

Practice 3.1.5 A: Solving Quadratic Equations by Factoring, p. U3-96

1. x=–6,x =8

2. y5

2= ,y=–7

3. n=0, n9

5=

4. x=4,x=–4



Practice 3.1.8 A: Solving Quadratic Inequalities, p. U3-139

1. –3<x<2

– 6 – 4 – 2 20 4 6

2–3

2. x ≤−4

3 or x ≥

1

2

– 3 – 2 – 1 10 2 31/2–4/3

3. x≤–5orx≥5

– 5 50 10

5–5

4. –4<x<3

– 6 – 4 – 2 20 4

3–4

5. –20.15≤x≤0.15

– 20 – 10 100

0.15–20.15

6. x≤–4orx≥–3

– 5 – 4 – 3 – 2 – 1 10

–3–4

7. nosolution8. from0.29secondto1.71secondsintothejump9. from0.28secondto0.72secondintothedive

10. atspeedsbelow18.76milesperhour

Practice 3.1.8 B: Solving Quadratic Inequalities, p. U3-140

1. x ≤−1

2orx≥1

– 3 – 2 – 1 10 2 3

–1/2 1

2. − ≤ ≤7

52x

– 6 – 4 – 2 20 4 6

2–7/5

3. norealsolutions4. x≤–11orx≥–2

– 20 – 15 – 10 – 5 0 5

–2–11

5. x<–8.18orx>0.18

– 15 – 10 – 5 0 5

0.18–8.18

6. –3.79<x<0.79

– 6 – 4 – 2 20 4

0.79–3.79

6. x = − ±1

2

11

2

7. x=3andx=–78. 8.81feet;youcannothaveanegativelength,soonlyone

valueofxisreasonable.9. afterabout2.5seconds;thenegativeanswerrepresents

timebeforetheballwaskicked,soonlyonevalueofxisreasonable.

10. approximately13.49mphandapproximately61.51mph

Warm-Up 3.1.7, p. U3-1111. Theareaofthetriangularpatchis 4 3 squareinches,or

about6.93squareinches.2. Theamountoffabricneededforthe3additionalpatchesis

12 3 squareinches,orabout20.78squareinches.

Practice 3.1.7 A: Applying the Quadratic Formula, p. U3-123

1. 0;onereal,rationalsolution2. 1;tworeal,rationalsolutions

3. x =±3 41

4

4. x = − ±4 4 2

5. norealsolutions

6. x =3

2andx=–17

7. afterapproximately1.96seconds8. $109. afterapproximately1.30seconds

10. Thediscriminant,orthepartofthequadraticformulaundertheradical,tellsthenumberandtypeofrootsofaquadraticequation.

Practice 3.1.7 B: Applying the Quadratic Formula, p. U3-124

1. 13;tworeal,irrationalsolutions2. –80;norealsolutions3. x=–1

4. x=5

3− orx=–1

5. norealsolutions

6. x=1orx= −12

77. afterapproximately8.57seconds8. approximately$5.37andapproximately$18.639. afterapproximately1.58seconds

10. No,ifaquadratichastwosolutions,theymustbothberealorbothbenon-real.

Warm-Up 3.1.8, p. U3-1251. 0.6p+0.3r≤2402. Theworkerscanassemble600racinggamesatthemost.



Practice 3.2.1 A: Creating and Graphing Equations Using Standard Form, p. U3-172

1.

x

y

–7 –6 –5 –4 –3 –2 –1 0

–8

–6

–4

–2

2. xy

0 2 4 6 8

–40

–30

–20

–10

7. x≤–0.91orx≥1.91

– 4 – 3 – 2 – 1 10 2 3

1.91–0.91

8. from0.12secondto8.45secondsaftertakeoff9. Theballoonisnevermorethan6feetabovetheground.

10. Thetimeofacompleteswingwillbegreaterthan4.49seconds.

Progress Assessment, p. U3-1411. a2. b3. b4. a5. c

6. c7. a8. d9. c

10. b11. Methodsmayvary.Recommendedmethodsarelisted.

a.Solvebyfactoring;x=–1orx=–19b.Solvebycompletingthesquare; x = − ±10 85

c.Solvebyusingthequadraticformula; x =− ±10 79

3

d.Solvebytakingsquareroots; x = ±5

3

Lesson 2: Creating Quadratic Equations in Two or More Variables

Pre-Assessment, p. U3-1431. b2. b3. a

4. d5. c

Warm-Up 3.2.1, p. U3-1461.

y

0 2 4 6 8 10 12

6

4

2

8

10

12

14

Distance from basket in feet

Ball

heig

ht in

feet

x

2. Theballwillbeabout13.5feethigh.



3.

x

y

0 1 2 3 4 5 6

2

4

6

8

10

4. y-intercept:3;vertex: (2,–1);thevertexisaminimumbecausea>0.

5. y-intercept:0;vertex: (3,–9);ithasaminimumbecausea>0.

6. y-intercept:–6;vertex: (–1,1);ithasamaximumbecausea<0.

7. Yes,they-interceptis–28andthegraphopensupbecausea>0.

8. y=–x2–6x–29. 328feet

10. $36.67

Practice 3.2.1 B: Creating and Graphing Equations Using Standard Form, p. U3-174

1.

x

y

–4 –2 0 2 4

2

4

6

2.

x

y

–4 –2 0 2 4

–1

1

2

3

4

5

3.

x

y

–4 –2 0 2 4

2

4

6

4. y-intercept:0;vertex: (–1.75,6.125);thevertexisamaximumbecausea<0.

5. y-intercept:7;vertex: (–5,–93);thevertexisaminimumbecausea>0.

6. y-intercept:3;vertex: (1,5);thevertexisamaximumbecausea<0.

7. No.They-interceptis–5becausetheconstanttermis–5.However,thegraphshowsay-interceptof–10.

8. y=–3x2+30x–699. approximately356feet

10. 5,000pairsofsocks



Warm-Up 3.2.2, p. U3-1761. y=0.002x+25002.

0 50,000 100,000 150,000 200,000 250,000 300,000 350,000

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

4,500

5,000

Sales in dollars

Peyt

on’s

take

-hom

e pa

y in

dol

lars

3. x-intercept:–1,250,000;y-intercept:2,5004. Thex-interceptisthetotalamountofmonthlycarsales

whenPeytonhasmade$0inamonth.Thisisimpossiblesinceshehasafixedmonthlysalaryof$2,500.They-interceptisthetotalmonthlytake-homepaywhenPeytonsells0cars.

Practice 3.2.2 A: Creating and Graphing Equations Using the x-intercepts, p. U3-191

1. x-intercepts:3,–6;axisofsymmetry:x=–1.52. x-intercepts:2/3,–2/3;axisofsymmetry:x=03. y=0.4x2+1.6x+1.64. y=x2+7x–605.

x

y

–8 –6 –4 –2 0 2 4 6 8

–10

–8

–6

–4

–2

0

2

4

6

8

10

6.

x

y

–8 –6 –4 –2 0 2 4 6 8

–10

–8

–6

–4

–2

0

2

4

6

8

10

7. y=–2x2+88. y=x2–369. 5inchesby7inches

10. 25seconds

Practice 3.2.2 B: Creating and Graphing Equations Using the x-intercepts, p. U3-193

1. x-intercepts:1/16,7;axisofsymmetry:x=3.532. x-intercepts:3/4,–7/2;axisofsymmetry:x=–1.3753. y=x2+6x+84. y=x2–20x+755.

x

y

–8 –6 –4 –2 0 2 4 6 8

–8

–6

–4

–2

2

4

6

8



8.

x

y

–2 0 2 4 6

–4

–2

0

2

4

6

(2, 5)

Axis of symmetry: x = 2

(3, 3)(1, 3)

(4, –3)(0, –3)

9. y x= − −( ) +25

14424 1002

10. y=–2(x–3)2+18

Practice 3.2.3 B: Creating and Graphing Equations Using Vertex Form, p. U3-210

1. Vertex:(–4,1);itisamaximumbecausea<0.2. Vertex:(2,0);itisaminimumbecausea>0.3. y=0.96(x–5.5)2–64. y=5(x–2)2

5. f(x)=(x–1)2–36. g(x)=0.3(x+2)2

7.

x

y

–8 –6 –4 –2 00

5

10

15

20Axis of symmetry: x = –4

V (–4, 2)

(–8, 18) (0, 18)

(–1, 11)(–7, 11)

6.

x

y

–8 –6 –4 –2 0 2 4 6 8

–10

–8

–6

–4

–2

0

2

4

6

8

10

7. y=3x2+15x–188. y=5x2–30x+409. 8feet

10. 1.25seconds

Warm-Up 3.2.3, p. U3-1961. $3.25 2. $5,070

Practice 3.2.3 A: Creating and Graphing Equations Using Vertex Form, p. U3-209

1. Vertex:(–3,–3);itisamaximumbecausea<0.2. Vertex:(6,6);itisaminimumbecausea>0.

3. y=0.89(x+2.50)2–3or y x= +

−8

9

5

23

2

4. y=–2(x–2)2+105. f(x)=(x+3)2

6. g(x)=3(x–2)2–37.

x

y

–5 0 5

–5

0

5

10

(0, 8)

Axis of symmetry: x = 0

(1, 6)(–1, 6)

(–2, 0) (2, 0)



Progress Assessment, p. U3-2261. a2. c3. b4. c5. a

6. b7. c8. c9. a

10. b11. a.vertex:(1.5,–5);axisofsymmetry:x=1.5

b.y=4x2–12x+4c.x-intercepts:0.38and2.62.Theserepresenthowmuch

timehaselapsedfromthestartofthepelican’sdescent,indicatingwhenthepelicanentersandexitsthewater,respectively.

8.xy

0 2 4 6 8 10

–25

–20

–15

–10

–5

Axis of symmetry:x = 5

V (5, –2.5)

(9, –18.5)(1, –18.5)

(10, –27.5)(0, –27.5)

9. y=–2(x–6)2+7210. y=–3.49(x–10.5)2+385

Warm-Up 3.2.4, p. U3-2111. Rearrangetheformulasothath isisolated.

2. hA

b b=

+2

1 2

3. Thewallis30feettall.

Practice 3.2.4 A: Rearranging Formulas Revisited, p. U3-223

1. x y= ± −25 2

2. x y= − ±3

3. xy

= − ±77

4

4. xy

= − ±+

190

2

5. rV

h=

3

π

6. rA

=π

7. b c a= −2 2

8. vE

mk=

2

9. x hy k

a= ±

−

10. ba y k

a x h=

−( )− −( )

2 2

2 2

Practice 3.2.4 B: Rearranging Formulas Revisited, p. U3-224

1. x y= ± −360 4 2

2. xy

= ±+9

16

3. xy

= − ±74

4. xy

= ±+

6 520

2 5.

.

5. V PR= ±

6. rA

=2

θ

7. c a b= +2 2

8. v ar=

9. y k r x h= ± − −( )2 2

10. rSA

4π=

Lesson 3: Interpreting and Analyzing Quadratic FunctionsPre-Assessment, p. U3-229

1. a2. b3. c

4. a5. c

Warm-Up 3.3.1, p. U3-2341. Thex-interceptsare0and8.2. Theequationfortheaxisofsymmetryisx=4.Itismidway

betweenthex-intercepts.3. Thevertexis(4,6).4. Yes,itwill.Sketchtheboulder’spathusing(0,0),(4,6),

and(8,0);thepathisabovetheobstacle.5. No,itwillnot.Ifyousubstitutex=7intotheequationof

theparabola,youfindthatthey-valueoftheboulderisslightlyabovetheboxes.

Practice 3.3.1 A: Interpreting Key Features of Quadratic Functions, p. U3-254

1. x>0.3,x<0.3;minimum=–5.3;x-intercepts:–1and1.67;y-intercept:–5;neither

2. x<1.667,x>1.667;maximum=9.333;x-intercepts:–0.097and3.431;y-intercept:1;neither

3. x>–1,x<–1;minimum=6;nox-intercepts;y-intercept:11;neither

4. x>1,x<1;minimum=–13;x-intercepts:–1.55and3.55;y-intercept:–11;neither

5. x=–0.67;x>–0.67,x<–0.676. x=1.125;x<1.125,x>1.1257. x=0.375;x>0.375,x<0.3758. x=0.15;x>0.15,x<0.159. Theheightofthesoftballincreasesuntilitreachesa

distanceof25feet,andthenitdecreases.10. Themaximumtemperatureisabout62.5˚.

Practice 3.3.1 B: Interpreting Key Features of Quadratic Functions, p. U3-256

1. x>1.5,x<1.5;minimum=–8.25;x-intercepts:–1.37and4.37;y-intercept:–6;neither

2. x<–2,x>–2;maximum=11;x-intercepts:–5.32and1.32;y-intercept:7;neither



9. about3seconds10. 25increases;5.5increases

Warm-Up 3.3.3, p. U3-2751. –1/22. G(x)=(–1/2)x+653. Thecar’smileagewillalwaysbereducedby1milepergallon

forevery2milesperhourthespeedofthecarincreases.

Practice 3.3.3 A: Identifying the Average Rate of Change, p. U3-288

1. 42. 63. –44. 05. –12

6. 77. betweenx=0andx=28. betweenx=–2andx=09. betweenx=0andx=2

10. –80ft/s

Practice 3.3.3 B: Identifying the Average Rate of Change, p. U3-290

1. –162. 83. –3/24. –185. –26

6. 27. betweenx=–1andx=08. betweenx=–2andx=–19. betweenx=–1andx=0

10. –56ft/s

Warm-Up 3.3.4, p. U3-2921. Thevertexis(4,5).2. Thesecondx-interceptis9.3. Themaximumis5.4.

16

14

12

10

8

6

4

2

0

2

4

6

8

10

12

14

10 5 5 10

(4, 5) (6, 5)

y

x

5. No,thewaterwillnotcleartheboxes.Iftheboxeswerenotintheway,thewaterwouldstillnotfillthebirdbathbecausethebirdbathistootall.

Practice 3.3.4 A: Writing Equivalent Forms of Quadratic Functions, p. U3-311

1. a.8b.(3,–1)c.minimum

2. a.–2and4b.4c.x=1d.(1,4.5)

3. x<1,x>1;maximum=16;x-intercepts:–1and3;y-intercept:12;neither

4. x>0,x<0;minimum=0;x-intercept:0;y-intercept:0;even

5. x=–16.3;x>–16.3,x<–16.36. x=2.25;x>2.25,x<2.257. x=–0.5;x<–0.5,x>–0.58. x=–0.02;x>–0.02,x<–0.029. Theheightofthefootballisincreasingforthefirstsecond

andthenitisdecreasing.10. Theflarewillexplodeat256feet.

Warm-Up 3.3.2, p. U3-2581. y=3x+642. allwholenumbers3. Thedomainrepresentsthepossiblenumberofgamesthat

Ginacouldsellinone8-hourworkday.

Practice 3.3.2 A: Identifying the Domain and Range of a Quadratic Function, p. U3-269

1. Domain:allrealnumbersor −∞< <∞x ;range:allrealnumbersgreaterthanorequalto13.25,or–∞<y≤13.25

2. Domain:allrealnumbersor −∞< <∞x ;range:allrealnumbersupto180.75,or–∞<y≤180.75

3. Domain:allrealnumbersor −∞< <∞x ;range:allrealnumbersuptoabout13.6,or–∞<f(x)≤13.6

4. Domain:allrealnumbersor −∞< <∞x ;range:allrealnumbersgreaterthanorequalto–44,or–44≤g(x)<∞

5. Domain:allrealnumbers; −∞< <∞x ;range:allrealnumbersuptoabout5.3,or–∞<y≤5.3

6. Domain:allrealnumbers; −∞< <∞x ;range:allrealnumbersgreaterthanorequalto–9.6,or–9.6≤y<∞


8. lessthan3.9seconds9. 9.4seconds

10. Domain:8.67≤x≤28.83;range:0≤P(x)≤1625;x=$18.75

Practice 3.3.2 B: Identifying the Domain and Range of a Quadratic Function, p. U3-272

1. Domain:allrealnumbersor −∞< <∞x ;range:allrealnumbersgreaterthanorequaltoabout0.7,or0.7≤y<∞

2. Domain:allrealnumbersor −∞< <∞x ;range:allrealnumbersuptoabout141.3,or–∞<y≤141.3

3. Domain:allrealnumbersor −∞< <∞x ;range:allrealnumbersgreaterthanorequaltoabout–4.4,or–4.4≤f(x)<∞

4. Domain:allrealnumbersor −∞< <∞x ;range:allrealnumbersgreaterthanorequalto–27,or–27≤g(x)<∞


6. Domain:allrealnumbers; −∞< <∞x ;range:allrealnumbersgreaterthanorequalto–15,or–15≤y<∞

7. Domain:allrealnumbers; −∞< <∞x ;range:allrealnumbersgreaterthanorequalto–10,or–10≤y<∞

8. lessthan10seconds



6. –1and2;thex-interceptsrepresentthetimeswhenthesnowboarderisontheground.Thex-intercept–1doesnotmakesensebecausetimecannotbenegative;2seconds.

0 1 2 3 4

5

10

15

20

25

30

35

7. (3,25);thepaperairplanewillreachamaximumheightof25feetaftertraveling3feetinthehorizontaldirection.

0 1 2 3 4 5 6 7 8 9

5

10

15

20

25

30

35

40

3. a.(1,10)b.maximum

4. (7,–9);afterthebirdtravelsfor7seconds,itreachesitsminimumheight,whichis9feetbelowthesurfaceofthewater.

-5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

5. –2and40;thex-interceptsrepresentthetimeswhentheheightis0miles,whichiswhenthemissileisontheground.Thex-interceptof–2doesnotmakesenseintheproblembecausetimecannotbenegative;38seconds.

0 5 10 15 20 25 30 35 40 45

50

100

150

200

250

300

350

400

450

500



Practice 3.3.4 B: Writing Equivalent Forms of Quadratic Functions, p. U3-313

1. a.12b.(4,–4)c.minimum

2. a.3and–5b.30c.x=–1d.(–1,32)

3. a.(3,0)b.maximum

4. (5,1);thebutterflyreachesitsminimumheightof1footabovetheground5secondsafteryouseeit.

-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10-1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

5. (1,52);thecliffdiverreachesamaximumheightof52feet1secondafterstartingthedive;2seconds

0 1 2 3 4 5 6 7 8 9 10

-5

5

10

15

20

25

30

35

40

45

50

55

8. (1,16);thegolfballwillreachamaximumheightof16feet1secondafterbeinghit.

0 1 2 3 4 5 6

2

4

6

8

10

12

14

16

18

20

9. –5and15;thex-interceptsrepresentthenumberofdollarsinpriceincrease/decreasethatwouldresultinnorevenue;5.

– 5 50 10 15

500

450

550

400

350

300

250

200

150

100

50

– 50

10. (7,289):themaximumrevenueof$289willoccurwhenthepriceisreducedby$7.

0 5 10 15 20 25

50

100

150

200

250

300

350



8. (2,4);thefrogreachesitsmaximumheightof4feetafter2seconds;x=2;4seconds

-1 0 1 2 3 4 5 6

-1

1

2

3

4

5

9. (5,200);therevenuereachesitsmaximumvalueof$200whenthepriceisincreasedby$5;x=5;$10

0 2 4 6 8 10 12 14 16 18 20

-20

20

40

60

80

100

120

140

160

180

200

220

6. 2and50;thex-interceptsrepresentthenumberofwidgetssoldthatresultinnorevenue;26widgets

-5 0 5 10 15 20 25 30 35 40 45 50 55

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

4,500

5,000

5,500

7. (20,12);thefootballreachesamaximumheightof12feetafterithastraveled20feetinthehorizontaldirection;40feet

-5 0 5 10 15 20 25 30 35 40

-2

2

4

6

8

10

12



3. Therevenueswouldincreaseby$2.Thiswouldaffectthey-intercept,andthegraphwouldmoveup2units.

Practice 3.4.1 A: Replacing f(x) with f(x) + k and f(x + k), p. U3-339

1. y=(x+2)22. y=x2+33. y=(x–5)2–24. (2,0)

0 1 2 3 4 5

1

2

3

4

5

5. (0,–4)

–5 –4 –3 –2 –1 0 1 2 3 4 5

–10–9–8–7–6–5–4–3–2–1

123456789

10

6. (5,–2)

0 1 2 3 4 5 6 7 8 9

–3

–2

–1

1

2

3

4

5

6

7

8

9

10

Lesson 4: Transforming FunctionsPre-Assessment, p. U3-318

1. a2. d3. b

4. a5. c

Warm-Up 3.4.1, p. U3-3221. f(x)=5x–102.

0 20 40 60 80 100 120 140

50

100

150

200

250

300

350

400

450

500

550

600

650

700

750

Number of tags sold

y

x

Pro�

t in

dolla

rs ($

)

10. –10and26;thex-interceptsrepresentpricedecreasesthatresultinnorevenue.Thex-intercept–10doesnotmakesenseinthecontextoftheproblembecauseapricedecreaseshouldnotbenegative;x=8;$16

0 5 10 15 20 25 30

-50

50

100

150

200

250

300

350

Progress Assessment, p. U3-3151. b2. c3. c4. d5. d

6. b7. b8. c9. c

10. a11. a.–1.2mpg

b.valuesofsbetween48and70mphc.G(s)=33mpg,ats=48mph



6.

–5 –4 –3 –2 –1 0 1 2 3 4 5

–10–9–8–7–6–5–4–3–2–1

123456789

10

7. y=–(x+2)2+5.Forthehorizontaltranslation,k=2.Fortheverticaltranslation,k=5.

8. f(x)=–0.0008(x–12)2+0.24(x–12)9. f(x)=–0.05x2+x+19;thepaperwadwilllandabout

32feetaway.10. f(x)=–0.4(x–9)2+14;yes,itwillhitthetarget.

Warm-Up 3.4.2, p. U3-3431. 576ft22. 1,728ft2

3. 1,936ft2

Practice 3.4.2 A: Replacing f(x) with k • f(x) and f(k • x), p. U3-363

1. 2•f(x)=2x2+2x–12.Sincetheentirefunctionisbeingmultipliedby2,thex-interceptsofthetwofunctionsarethesame.However,thesecondparabola,2•f(x),isstretchedverticallybecauseeachy-coordinatehasbeenmultipliedby2.Asaresult,thenewparabolaisnarrowerthantheoriginalparabola.

–5 –4 –3 –2 –1 0 1 2 3 4 5

–14–13–12–11–10

–9–8–7–6–5–4–3–2–1

123456789

10

2 • f(x)

f(x)

7. y=–(x+4)2;k=48. f(x)=–0.0009(x–18)2+0.2088(x–18)9. f(x)=–0.03x2+1.3x+16;theballwilllandabout53feetaway.

10. f x x( ) ( )= − − +1

813 102 ;yes,itwilllandinthebasket.

Practice 3.4.1 B: Replacing f(x) with f(x) + k and f(x + k), p. U3-341

1. y=(x–3)22. y=x2–43. y=(x+6)2–14.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1

–3

–2

–1

1

2

3

4

5

6

7

8

9

10

5.

–5 –4 –3 –2 –1 0 1 2 3 4 5

–3

–2

–1

1

2

3

4

5

6

7

8

9

10



6. Graphoff(x)andg(x):

–5 –4 –3 –2 –1 0 1 2 3 4 5

–5

–4

–3

–2

–1

1

2

3

4

5

6

7

8

9

10

f(x) g(x)

Thefunctiong(x)isobtainedbymultiplyingthex-valuesinf(x)by–2.Thismeansthatthey-interceptwillbethesame,theminimumy-valuewillbethesame,butthex-interceptwillchangeandtheintervalbecomesnarrower.Thegraphofg(x)willbecompressedhorizontallybyascalefactorof1/2,andthegraphwillbereflectedoverthey-axis.


–5 –4 –3 –2 –1 0 1 2 3 4 5

–5

–4

–3

–2

–1

1

2

3

4

5

6

7

8

9

10

f(x)

g(x)

Sinceg(x)isamultipleoff(x),itwillhavethesame

x-interceptsasf(x):–1and1.Sincethevalueofkis−1

2,the

graphwillbecompressedverticallybyafactorof1

2,andit

willreflectoverthex-axisbecausek<0.

8. Ifthesidesoftheoriginalpenweredoubled,thenthesidelengthswouldbe2xand4x.Theareawouldbef(2x)=2(2x)2=8x2.Ifthefarmerweretobuildasecondpenofthesamesize,theareaofthetwopenswouldbe2•f(x)=2(2x2)=4x2.Doublingthesidelengthgivesalargerarea.

2. f(3x)=9x2+3x.Sincethevariablexismultipliedby3,the

graphiscompressedhorizontallybyafactorof1

3.Thatis,

eachx-valueismultipliedby1

3,sothewidthoftheparabola

becomesnarrower.They-interceptremainsthesame,asdo

theminimumvaluesofthefunctions.Thex-interceptschange

andtheintervalbetweenthembecomesnarrower.

–5 –4 –3 –2 –1 0 1 2 3 4 5

1

2

3

4

5

6

7

8

9

10

f(3x)f(x)

3. Answersmayvary.Applythetransformationk•f(x),where0<k<1.Sampleanswer:1/2•f(x)=g(x)=0.5x2–0.5x–1

4. Answersmayvary.Applythetransformationf(k•x),wherek>1.Sampleanswer:f(2x)=g(x)=4x2+6x–4


–5 –4 –3 –2 –1 0 1 2 3 4 5

–10–9–8–7–6–5–4–3–2–1

123456789

10

g(x)

f(x)

Sinceg(x)isamultipleoff(x),itwillhavethesamex-interceptsasf(x): –1and2.Sincethevalueofkis–2,thegraphwillbestretchedverticallybyafactorof2,anditwillreflectoverthex-axisbecausek<0.



2. f(2x)=4x2–6x +2.Sincethevariablexismultipliedby

2,thegraphshrinkshorizontallybyafactorof1

2.That

is,eachx-valueismultipliedby1

2,sothewidthofthe

parabolabecomesnarrower.They-interceptsremainthe

same,asdotheminimumvaluesofthefunctions.The

x-interceptschangeandtheintervalbecomesnarrower.

–5 –4 –3 –2 –1 0 1 2 3 4 5

1

2

3

4

5

6

7

8

9

10

f(x)f(2x)

3. Answersmayvary.Applythetransformationk•f(x),wherek>1.Sampleanswer:2•f(x)=g(x)=2x2–8x+6

4. Answersmayvary.Applythetransformationf(k•x),wherek>1.Sampleanswer:f(2x)=g(x)=4x2–2x–1


–5 –4 –3 –2 –1 0 1 2 3 4 5

–10–9–8–7–6–5–4–3–2–1

123456789

10

f(x)

g(x)

Sinceg(x)isamultipleoff(x),itwillhavethesamex-interceptsasf(x): –3and1.Sincethevalueofkis–2,thegraphwillbestretchedverticallybyafactorof2,anditwillreflectoverthex-axisbecausek<0.

9. f(x)representstheprofit,soadoublingoftheprofitismodeledby2•f(x).

10. Neitherplayeriscorrectbecauseneitheroftheequationsleadstoaballgoingfartherthantheoriginalball.Jada’sequation,f(2x)=–0.01(2x)2+0.98(2x)+2,leadstoaballgoingthesameheightastheoriginalballbutonlyhalfasfar;Jayla’sequation,2•f(x)=2(–0.01x2+0.98x +2),leadstoaballgoingthesamedistanceastheoriginalballbuttwiceashigh.

0 10 20 30 40 50 60 70 80 90 100

5

10

15

20

25

30

35

40

45

50

f(x)

2 • f(x)

f(2x)

Practice 3.4.2 B: Replacing f(x) with k • f(x) and f(k • x), p. U3-366

1. 3•f(x)=3x2–9x–12.Sincetheentirefunctionisbeingmultipliedby3,thex-interceptsofthetwofunctionsarethesame(–1and4).However,thesecondparabola,3•f(x),isstretchedverticallybecauseeachy-coordinateismultipliedby3.Asaresult,thenewparabolaisnarrowerthantheoriginalparabola.

–5 –4 –3 –2 –1 0 1 2 3 4 5

–20

–18

–16

–14

–12

–10

–8

–6

–4

–2

2

4

6

8

10

3 • f(x)

f(x)



8. Theoriginalclosethassidelengthsofxand3x.Lettheareaoftheoriginalclosetbef(x)=3x2.Ifthesidesoftheoriginalclosetweretripled,thenthesidelengthswouldbe3xand9x.Theareawouldbef(3x)=3(3x)2=27x2.Ifthecelebritybuildstwoadditionalclosetsofthesamesize,theareaofthethreeclosetswouldbe3•f(x)=3(3x2)=9x2.Triplingthesidelengthgivesalargerarea.

9. f(x)representstheprofit,soearninghalfoftheprofitis

modeledby1

2•f(x).

10. Zioniscorrectbecausehisequation, f x1

2

,leadstoa

beanbaggoingtwiceasfarastheoriginalbeanbag.This

beanbagreachesthesameheightastheoriginalbeanbag.

Zavier’sequation,2•f(x),leadstoabeanbaggoingthe

samedistanceastheoriginalbeanbagbuttwiceashigh.

0 40 80 120 160 200 240 280 320 360 400–10

10

20

30

40

50

60

70

80

90

100

110

2 • f(x)

f(x)f( x)1

2

Progress Assessment, p. U3-3691. b2. b3. a4. a5. d

6. d7. b8. b9. c

10. b11. a.f(k • x),where0<k<1tostretchtheparabola

horizontallysothatthepathoftheballextendstoreachthehoop.

b.k =19

20


–5 –4 –3 –2 –1 0 1 2 3 4 5

–10–9–8–7–6–5–4–3–2–1

123456789

10

f(x) g(x)

Thefunctiong(x)isobtainedbymultiplyingthex-valuesinf(x)by–2.Thismeansthatthey-interceptwillbethesame,theminimumy-valuewillbethesame,thex-interceptswillchangeandtheintervalbetweenthemwillbecomenarrower.Thegraphofg(x)willbecompressedhorizontallybyafactorof1/2,andthegraphwillbereflectedoverthey-axis.


–5 –4 –3 –2 –1 0 1 2 3 4 5

–10–9–8–7–6–5–4–3–2–1

123456789

10

f(x)

g(x)

Sinceg(x)isamultipleoff(x),itwillhavethesame

x-interceptsasf(x): –2and2.Sincethevalueofkis−1

2,the

graphwillbecondensedverticallybyafactorof1

2,andit

willreflectoverthex-axisbecausek<0.



Practice 3.5.2 A: Comparing Properties of Quadratic Functions Given in Different Forms, p. U3-415

1. thefunctionwitha>02. thefunctionwithnox-intercepts3. thefunctionwithx-interceptspand–p4. No;thefunctionsdonotcrossanywherebuttheorigin.5. ModelAhastheshorteststoppingdistanceatx=25.It

willalwayshavetheshorteststoppingdistancebecausethefunctionsforthestoppingdistancesdonotcrossanywherebuttheorigin.

6. ModelBhasthelongeststoppingdistanceatx=55.Itwillalwayshavethelongeststoppingdistancebecausethefunctionsforthestoppingdistancesdonotcrossanywherebuttheorigin.

7. TypeC8. TypeB9. TypeA:(2,0)

TypeB:(0,0)TypeC:(4,0)

10. TypeA:(14,0)TypeB:(16,0)TypeC:(12,0)

Practice 3.5.2 B: Comparing Properties of Quadratic Functions Given in Different Forms, p. U3-418

1. thefunctionwithnox-intercepts2. thefunctionwitha>03. thefunctionwithx-interceptspand–3p4. No;thefunctionsdonotcrossanywherebuttheorigin.5. ModelBhastheshorteststoppingdistanceatx=25.It

willalwayshavetheshorteststoppingdistancebecausethefunctionsforthestoppingdistancesdonotcrossanywherebuttheorigin.

6. ModelChasthelongeststoppingdistanceatx=55.Itwillalwayshavethelongeststoppingdistancebecausethefunctionsforthestoppingdistancesdonotcrossanywherebuttheorigin.

7. TypeB8. TypeC9. TypeA:(3,0)

TypeB:(5,0)TypeC:(0,0)

10. TypeA:(13,0)TypeB:(11,0)TypeC:(16,0)

Progress Assessment, p. U3-4211. b2. a3. c4. b5. d

6. c7. d8. a9. a

10. b11. a.Q(x)hasaloweridealsaleprice.

b.P(x)predictsahighermaximumprofit.

Lesson 5: Building and Comparing Quadratic Functions

Pre-Assessment, p. U3-3741. b2. c3. d

4. a5. c

Warm-Up 3.5.1, p. U3-3781. 12seconds2. 13seconds

Practice 3.5.1 A: Building Quadratic Functions From Context, p. U3-396

1. f(x)=2x2–13x–242. g(3)=63. f(x)=x(x+2);theintegersare60and62.4. vertex:(2100,20); y-intercept:(0,500)

5. f x x( )2

18,375( 2100) 202= − +

6. vertex:(17,16);y-intercept:(0,6)

7. f x x( )10

289( 17) 162= − − +

8. f x x x( )1

2base • height

1

2(16 )= = −

9. 32ft2

10. 8feetforboth

Practice 3.5.1 B: Building Quadratic Functions From Context, p. U3-398

1. f(x)=10x2+26x+122. g(10)=1583. f(x)=x(x+1);theintegersare91and92.4. vertex:(800,40); y-intercept:(0,140)

5. f x x( )1

6400( 800) 402= − +

6. vertex:(23,20);y-intercept:(0,5)

7. f x x( )15

529( 23) 202= − − +

8. f x x x x x( ) length • width •1

2(32 )

1

2(32 )= = − = −

9. 128ft2

10. Thesideparalleltothehouseshouldbe16feetlong,andthesidesperpendiculartothehouseshouldeachbe8feetlong.

Warm-Up 3.5.2, p. U3-4001. thefirstbird2. thesecondbird3. thesecondbird



Unit Assessmentp. U3-426

1. c2. a3. d4. d5. c6. b

7. b8. c9. a

10. d11. d12. b

13. a.Thefunctionisincreasingwhenx<0.b.Thefunctionisdecreasingwhenx>0.c. Thevertexis(0,144)andrepresentsthemaximum

heightofthecrab,144feet,beforeitisdropped.d.Areasonabledomainforthisscenariois0≤x≤3since

thecrabhitstherockat3seconds.e.Theaveragerateofchangeis–56feet/second.

14. a.Thevertexis(1,18)andrepresentsthemaximumheightofthebaseball,18feet,1secondafteritwashit.

b.Theequationthatrepresentstheaxisofsymmetryisx=1.c.

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25

2

4

6

8

10

12

14

16

18

Time in seconds

Hei

ght i

n fe

et

15. a.Q(x)hasahigheridealsaleprice.b.P(x)predictsahighermaximumprofit.c.Thedomainrepresentsthesetofpossiblesaleprices,

x,thatyieldaprofit.Whilenegativeprofitispossible,itisnotdesirable.Therefore,thedomainwillbevaluesofxthatgivenon-negativeprofit.ThedomainofP(x)is16≤x≤40,andthedomainofQ(x)is20≤x≤50.

d.Therangerepresentstheprofitatvarioussaleprices.Whileitispossibletohavenegativeprofit,itisnotdesirable.Therefore,therangeofeachwillbethesetofpositiveoutputs.TherangeofP(x)is0≤P(x)≤5760,andtherangeofQ(x)is0≤Q(x)≤4500.

Documents

U 3 U U Answer Key - ROBERT BEACH-CHATTOOGA ......2. x = 20 feet 3. 25 feet 4. 1,500 cubic feet Practice 3.1.5 A: Solving Quadratic Equations by Factoring, p. U3-96 1. x = –6, x