Chapter 3 Find the axis of symmetry of the parabola defined by the given

Ch.3 PolynomialandRationalFunctions

3.1 QuadraticFunctions

1 RecognizeCharacteristicsofParabolas

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Thegraphofaquadraticfunctionisgiven.Determinethefunctionʹsequation.

1)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A) f(x)=(x+2)2+2 B) g(x)=(x+2)2–2 C) h(x)=(x–2)2+2 D) j(x)=(x–2)2–2

2)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A) g(x)=(x+1)2–1 B) f(x)=(x+1)2+1 C) h(x)=(x–1)2+1 D) j(x)=(x–1)2–1

3)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A) h(x)=(x–2)2+2 B) g(x)=(x+2)2–2 C) f(x)=(x+2)2+2 D) j(x)=(x–2)2–2

Page1

4)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A) j(x)=(x–2)2–2 B) g(x)=(x+2)2–2 C) h(x)=(x–2)2+2 D) f(x)=(x+2)2+2

5)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A) f(x)=x2–2x+1 B) g(x)=x2+2x+1 C) h(x)=x2–1 D) j(x)=x2+1

6)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A) g(x)=x2+4x+4 B) f(x)=x2–4x+4 C) h(x)=x2–2 D) j(x)=x2+2

Page2

7)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A) h(x)=x2–1 B) g(x)=x2+2x+1 C) f(x)=x2–2x+1 D) j(x)=x2+1

8)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A) j(x)=x2+1 B) g(x)=x2+2x+1 C) h(x)=x2–1 D) f(x)=x2–2x+1

9)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A) j(x)=–x2+3 B) g(x)=–x2+6x+9 C) h(x)=–x2–3 D) f(x)=–x2–6x–9

Page3

10)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A) h(x)=–x2–2 B) g(x)=–x2+4x+4 C) j(x)=–x2+2 D) f(x)=–x2–4x–4

Findthecoordinatesofthevertexfortheparaboladefinedbythegivenquadraticfunction.

11) f(x)=(x+2)2+2

A) (–2

,

2) B) (–2

,

–2) C) (0,2) D) (2

,

0)

12) f(x)=x2+2

A) (0,2) B) (–2

,

0) C) (0,–2) D) (2

,

0)

13) f(x)=(x+5)2+4

A) (–5

,

4) B) (–4

,

5) C) (4

,

–25) D) (4

,

–5)

14) f(x)=8–(x+5)2

A) (–5

,

8) B) (5

,

8) C) (8

,

5) D) (8

,

–5)

15) f(x)=(x+3)2–5

A) (–3

,

–5) B) (3

,

5) C) (3

,

–5) D) (–3

,

5)

16) y+4=(x–2)2

A) (2

,

–4) B) (–2

,

–4) C) (4

,

2) D) (4

,

–2)

17) f(x)=11(x–5)2+4

A) (5

,

4) B) (11,5) C) (–5

,

4) D) (4

,

–5)

18) f(x)=–7(x–3)2–4

A) (3

,

–4) B) (–4

,

3) C) (–3

,

–4) D) (–7,–3)

19) f(x)=x2–2

A) (0,–2) B) (1,0) C) (0,2) D) (2

,

0)

20) f(x)=x2–2x–4

A) (1

,

–5) B) (–1

,

–1) C) (–2

,

4) D) (1

,

–7)

21) f(x)=–x2–10x–2

A) (–5

,

23) B) (5

,

–77) C) (–10

,

–2) D) (5

,

–27)

22) f(x)=3–x2–2x

A) (–1

,

4) B) (1

,

4) C) (–1

,

–4) D) (1

,

–4)

Page4

23) f(x)=–2x2+4x+7

A) (1

,

9) B) (–1

,

1) C) (2

,

3) D) (–2

,

–9)

Findtheaxisofsymmetryoftheparaboladefinedbythegivenquadraticfunction.

24) f(x)=x2+5

A) x=0B)x=5C)x= –5D)y=5

25) f(x)=(x+2)2+7

A) x=–2B)x=2C)y=7D)y= –7

26) f(x)=7–(x+4)2

A) x=–4B)x=4C)x=7D)x= –7

27) f(x)=(x+1)2–6

A) x=–1B)x=1C)x= –6D)x=6

28) y+4=(x+2)2

A) x=–2B)x=2C)y=4D)y= –4

29) f(x)=11(x–3)2+9

A) x=3B)x=11 C) x= –3D)x=9

30) f(x)=–7(x–3)2–8

A) x=3B)x=–8C)x= –3D)x= –7

31) f(x)=x2–14x+3

A) x=7B)x=–7C)x= –14 D) x= –46

32) f(x)=–x2–6x+1

A) x=–3B)x=3C)x=–6D)x=10

33) f(x)=7x2–14x+5

A) x=1B)x=–1C)x=2D)x=–2

Findtherangeofthequadraticfunction.

34) f(x)=x2+1

A) [1

,

∞)B)(

–∞

,

1] C) [–1

,

∞) D) [0,∞)

35) f(x)=(x+2)2+8

A) [8

,

∞)B)[

–8

,

∞)C)[2

,

∞)D)[

–2

,

∞)

36) f(x)=4–(x+3)2

A) (–∞

,

4] B) [4

,

∞)C)(

–∞

,

3] D) [–3

,

∞)

37) f(x)=(x+9)2–4

A) [–4

,

∞)B)(

–∞

,

–9] C) (–∞

,

–4] D) [–9

,

∞)

38) y+9=(x+3)2

A) [–9

,

∞)B)(

–∞

,

3] C) [9

,

∞)D)(

–∞

,

9]

Page5

39) f(x)=11(x–2)2+9

A) [9

,

∞)B)[2

,

∞)C)(

–∞

,

9] D) [–9

,

∞)

40) f(x)=–7(x–3)2–7

A) (–∞

,

–7] B) (–∞

,

3] C) [–7

,

∞)D)[

–3

,

∞)

41) f(x)=x2+12x+9

A) [–27

,

∞)B)[6

,

∞)C)(

–∞

,

–27] D) (–∞

,

–99]

42) f(x)=–x2–6x+6

A) (–∞

,

15] B) [15

,

∞)C)[

–3

,

∞)D)(

–∞

,

–3]

43) f(x)=4x2+2x–7

A) [–29

4,∞)B)(

–∞,–29

4]C)[

–1

4,∞)D)(

–∞,–1

4]

44) f(x)=–2x2–4x

A) (–∞

,

2] B) (–∞

,

–2] C) (–∞

,

–1] D) (–∞

,

1]

Findthex–intercepts(ifany)forthegraphofthequadraticfunction.

45) f(x)=x2–1

A) (–1

,

0)and(1

,

0) B) (–1

,

0) C) (1

,

0) D) Nox–intercepts

46) f(x)=(x+1)2–1

A) (0,0)and(–2

,

0) B) (0,0)and(2

,

0) C) (0,0)and(–1

,

0) D) (2

,

0)and(–2

,

0)

47) y+4=(x–2)2

A) (0,0)and(4

,

0) B) (0,0)and(–4

,

0) C) (–4

,

0)and(4

,

0) D) (0,0)

48) f(x)=6+5x+x2

A) (–3

,

0)and(–2

,

0) B) (3

,

0)and(2

,

0) C) (3

,

0)and(–2

,

0) D) (–3

,

0)and(2

,

0)

49) f(x)=x2+12x+15Giveyouranswersinexactform.

A) (–6±21,0) B) (6+21,0) C) (6±15,0) D) (–12±15,0)

50) f(x)=–x2+9x–20

A) (4

,

0)and(5

,

0) B) (–4

,

0)and(–5

,

0) C) (4

,

0)and(–5

,

0) D) Nox–intercepts

51) f(x)=2x2+15x+28

A) (–4

,

0)and(–3.5

,

0) B) (–4

,

0)and(3.5

,

0) C) (–7

,

0)and(–2

,

0) D) (–7

,

0)and(2

,

0)

52) f(x)=2x2+14x+24

A) (–3

,

0)and(–4

,

0) B) (3

,

0)and(4

,

0) C) (–3

,

0)and(4

,

0) D) (3

,

0)and(–4

,

0)

53) 5x2+8x+2=0

Giveyouranswersinexactform.

A) –4±6

5,0 B) –4±6

10 ,0 C) –8±6

5,0 D) –4±26

5,0

Page6

Findthey–interceptforthegraphofthequadraticfunction.

54) f(x)=–x2–2x+8

A) (0,8) B) (8

,

0) C) (0,–4) D) (0,–8)

55) y+4=(x–2)2

A) (0,0) B) (0,–4) C) (0,4) D) (4

,

0)

56) f(x)=6+5x+x2

A) (0,6) B) (0,3) C) (0,–6) D) (0,5)

57) f(x)=x2+7x–10

A) (0,–10) B) (0,2) C) (0,10) D) (0,7)

58) f(x)=(x+1)2–1

A) (0,0) B) (0,2) C) (0,1) D) (0,–1)

59) f(x)=3x2–4x–7

A) (0,–7) B) (0,7) C) 0,7

3D) 0,–7

3

Findthedomainandrangeofthequadraticfunctionwhosegraphisdescribed.

60) Thevertexis(1

,

–13)andthegraphopensup.

A) Domain:(–∞

,

∞)

Range:[–13,∞)

B) Domain:[1

,

∞)

Range:[–13,∞)

C) Domain:(–∞

,

∞)

Range:(–∞,–13]

D) Domain:(–∞

,

∞)

Range:[1,∞)

61) Thevertexis(–1

,

0)andthegraphopensdown.

A) Domain:(–∞

,

∞)

Range:(–∞,0]

B) Domain:(–∞

,

–1]

Range:(–∞,0]

C) Domain:(–∞

,

∞)

Range:[0,∞)

D) Domain:(–∞

,

∞)

Range:(–∞,–1]

62) Theminimumis4atx=1.

A) Domain:(–∞

,

∞)

Range:[4,∞)

B) Domain:[1

,

∞)

Range:[4,∞)

C) Domain:(–∞

,

∞)

Range:(–∞,4]

D) Domain:(–∞

,

∞)

Range:[1,∞)

63) Themaximumis4atx=1

A) Domain:(–∞

,

∞)

Range:(–∞,4]

B) Domain:(–∞

,

1]

Range:(–∞,4]

C) Domain:(–∞

,

∞)

Range:[4,∞)

D) Domain:(–∞

,

∞)

Range:(–∞,1]

Solvetheproblem.

64) Writeanequationinstandardformoftheparabolathathasthesameshapeasthegraphoff(x)=11x2,but

whichhasitsvertexat(5,6).

A) f(x)=11(x–5)2+6 B) f(x)=11(x+5)2+6

C) f(x)=(11x+5)2+6 D) f(x)=11(x+6)2+5

65) Writeanequationinstandardformoftheparabolathathasthesameshapeasthegraphoff(x)=5x2,but

whichhasaminimumof4atx=2.

A) f(x)=5(x–2)2+4 B) f(x)=5(x+2)2+4

C) f(x)=–5(x–2)2+4 D) f(x)=5(x+4)2–2

Page7

66) Writeanequationinstandardformoftheparabolathathasthesameshapeasthegraphoff(x)=–7x2,but

whichhasamaximumof9atx=5.

A) f(x)=–7(x–5)2+9 B) f(x)=–7(x+5)2+9

C) f(x)=7(x–5)2+9 D) f(x)=–7(x–5)2–9

2 GraphParabolas

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Usethevertexandinterceptstosketchthegraphofthequadraticfunction.

1) y+1=(x+5)2

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page8

2) f(x)=–2(x+1)2+3

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page9

3) f(x)=(x–1)2–4

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

4) f(x)=4–(x–2)2

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page11

5) f(x)=x2+6x+8

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page12

6) f(x)=–x2–4x–3

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page13

7) f(x)=x2–2x–3

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page14

8) f(x)=–4x+3+x2

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page15

9) f(x)=–x2+4x–3

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

10) f(x)=8–x2–2x

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page16

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

11) f(x)=2+3x+x2

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page17

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

12) f(x)=2x2–8x+2

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page18

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10



B)

x

-10 10

y

10

-10

x

-10 10

y

10

-10



C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10



D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

3 DetermineaQuadraticFunctionʹsMinimumorMaximumValue

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Determinewhetherthegivenquadraticfunctionhasaminimumvalueormaximumvalue.Thenfindthecoordinatesof

theminimumormaximumpoint.

1) f(x)=x2+2x–1

A) minimum;–1

,

–2 B) maximum;–1

,

–2

C) minimum;–2

,

–1 D) maximum;–2

,

–1

2) f(x)=–x2+3x–9

A) maximum;3

2,–27

4B) minimum;3

2,–27

4

C) minimum;–27

4,3

2D) maximum;–27

4,3

2

3) f(x)=2x2–2x+2

A) minimum;1

2,3

2B) maximum;1

2,3

2C) minimum;3

2,1

2D) maximum;3

2,1

2

4) f(x)=4x2–8x

A) minimum;1

,

–4 B) maximum;1

,

–4

C) minimum; –1

,

–4 D) maximum; –1

,

–4

Page19

5) f(x)=–5x2+10x

A) maximum;1

,

5 B) minimum;1

,

5

C) minimum;–1

,

–5 D) maximum;–1

,

–5

4 SolveProblemsInvolvingaQuadraticFunctionʹsMinimumorMaximumValue

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

1) Youhave340feetoffencingtoenclosearectangularregion.Findthedimensionsoftherectanglethat

maximizetheenclosedarea.

A) 85ftby85ft B) 170ftby170 ft C) 170 ftby42.5 ft D) 87ftby83 ft

2) Adeveloperwantstoenclosearectangulargrassylotthatbordersacitystreetforparking.Ifthedeveloperhas

232feetoffencinganddoesnotfencethesidealongthestreet,whatisthelargestareathatcanbeenclosed?

A) 6728ft2B) 13,456ft2C) 3364ft2D) 10,092ft2

3) Youhave144feetoffencingtoenclosearectangularregion.Whatisthemaximumarea?

A) 1296squarefeet B) 5184 squarefeet C) 20,736 squarefeet D) 1292 squarefeet

4) Youhave72feetoffencingtoenclosearectangularplotthatbordersonariver.Ifyoudonotfencetheside

alongtheriver,findthelengthandwidthoftheplotthatwillmaximizethearea.

A) length:36feet,width:18feet B) length:54 feet,width:18feet

C) length:36feet,width:36feet D) length:18 feet,width:18feet

5) Araingutterismadefromsheetsofaluminumthatare18incheswidebyturninguptheedgestoformright

angles.Determinethedepthofthegutterthatwillmaximizeitscross–sectionalareaandallowthegreatest

amountofwatertoflow.

A) 4.5inches B) 4inches C) 5inches D) 5.5inches

6) Arectangularplaygroundistobefencedoffanddividedintwobyanotherfenceparalleltoonesideofthe

playground.648feetoffencingisused.Findthedimensionsoftheplaygroundthatmaximizethetotal

enclosedarea.

A) 108ftby162ft B) 162ftby162 ft C) 54 ftby243 ft D) 81ftby162 ft

7) Arectangularplaygroundistobefencedoffanddividedintwobyanotherfenceparalleltoonesideofthe

playground.576feetoffencingisused.Findthemaximumareaoftheplayground.

A) 13,824ft2B) 20,736ft2C) 10,368ft2D) 15,552ft2

8) Thecostinmillionsofdollarsforacompanytomanufacturexthousandautomobilesisgivenbythefunction

C(x)=3x2–18x+81.Findthenumberofautomobilesthatmustbeproducedtominimizethecost.

A) 3thousandautomobiles B) 6 thousandautomobiles

C) 54thousandautomobiles D) 9 thousandautomobiles

9) InoneU.S.city,thequadraticfunctionf(x)=0.0038x2–0.41x+36.47modelsthemedian,oraverage,age,y,at

whichmenwerefirstmarriedxyearsafter1900.Inwhichyearwasthisaverageageataminimum?(Roundto

thenearestyear.)Whatwastheaverageageatfirstmarriageforthatyear?(Roundtothenearesttenth.)

A) 1954

,

25.4yearsold B) 1954

,

47.5 yearsold

C) 1936,47.5yearsold D) 1951

,

36yearsold

10) Theprofitthatthevendormakesperdaybysellingxpretzelsisgivenbythefunction

P(x)=–0.004x2+3.2x–400.Findthenumberofpretzelsthatmustbesoldtomaximizeprofit.

A) 400pretzels B) 800pretzels C) 1.6 pretzels D) 240 pretzels

Page20

11) ThemanufacturerofaCDplayerhasfoundthattherevenueR(indollars)isR(p)=–5p2+1720p,whenthe

unitpriceispdollars.Ifthemanufacturersetsthepriceptomaximizerevenue,whatisthemaximumrevenue

tothenearestwholedollar?

A) $147,920 B) $295,840 C) $591,680 D) $1,183,360

12) TheownerofavideostorehasdeterminedthattheprofitsPofthestoreareapproximatelygivenby

P(x)=–x2+50x+67,wherexisthenumberofvideosrenteddaily.Findthemaximumprofittothenearest

dollar.

A) $692 B) $625 C) $1317 D) $1250

13) TheownerofavideostorehasdeterminedthatthecostC,indollars,ofoperatingthestoreisapproximately

givenbyC(x)=2x2–20x+570,wherexisthenumberofvideosrenteddaily.Findthelowestcosttothe

nearestdollar.

A) $520 B) $370 C) $470 D) $620

14) ThedailyprofitindollarsofaspecialtycakeshopisdescribedbythefunctionP(x)=–5x2+210x–1600,where

xisthenumberofcakespreparedinoneday.Themaximumprofitforthecompanyoccursatthevertexofthe

parabola.Howmanycakesshouldbepreparedperdayinordertomaximizeprofit?

A) 21cakes B) 2205 cakes C) 441 cakes D) 42cakes

15) Amongallpairsofnumberswhosesumis42

,

findapairwhoseproductisaslargeaspossible.

A) 21and21 B) 10.5 and10.5 C) 23 and19 D) 41and1

16) Amongallpairsofnumberswhosedifferenceis50

,

findapairwhoseproductisassmallaspossible.

A) –25and25 B) 25and25 C) –75 and–25 D) 75and25

17) Anarrowisfiredintotheairwithaninitialvelocityof160 feetpersecond.Theheightinfeetofthearrowt

secondsafteritwasshotintotheairisgivenbythefunctionh(x)=–16t2+160t.Findthemaximumheightof

thearrow.

A) 400ft B) 80ft C) 1200 ft D) 720 ft

18) Apersonstandingclosetotheedgeontopofa288–footbuildingthrowsabaseballverticallyupward.The

quadraticfunctions(t)=–16t2+64t+288 modelstheballʹsheightabovetheground,s(t),infeet,tsecondsafter

itwasthrown.Afterhowmanysecondsdoestheballreachitsmaximumheight?Roundtothenearesttenthof

asecondifnecessary.

A) 2seconds B) 6.7seconds C) 352 seconds D) 1.5seconds

19) Aprilshootsanarrowupwardintotheairataspeedof32 feetpersecondfromaplatformthatis12 feethigh.

Theheightofthearrowisgivenbythefunctionh(t)=–16t2+32t+12,wheretisthetimeisseconds.Whatis

themaximumheightofthearrow?

A) 28ft B) 11ft C) 16 ft D) 12ft

20) Anobjectispropelledverticallyupwardfromthetopofa224–footbuilding.Thequadraticfunction

s(t)=–16t2+128t+224modelstheballʹsheightabovetheground,s(t),infeet,tsecondsafteritwasthrown.

Howmanysecondsdoesittakeuntiltheobjectfinallyhitstheground?Roundtothenearesttenthofasecond

ifnecessary.

A) 9.5seconds B) 1.5seconds C) 4 seconds D) 2seconds

Page21

3.2 PolynomialFunctionsandTheirGraphs

1 IdentifyPolynomialFunctions

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Determinewhetherthefunctionisapolynomialfunction.

1) f(x)=4x+2x2

A) Yes B) No

2) f(x)=7–x4

6

A) Yes B) No

3) f(x)=3–2

x3

A) No B) Yes

4) f(x)=x2–8

x5

A) No B) Yes

5) f(x)=2x3–x2–2

A) No B) Yes

6) f(x)=16x3+7x+3

x

A) No B) Yes

7) f(x)=πx5+4x4–2

A) Yes B) No

8) f(x)=x4

/

3–x6+9

A) No B) Yes

9) f(x)=5x7–x5+3

2x

A) Yes B) No

10) f(x)=3x3+4x2–3x–5+100

A) No B) Yes

Findthedegreeofthepolynomialfunction.

11) f(x)=–4x+7x5

A) 5 B) 1 C) –4D)7

12) f(x)=2–x5

6

A) 5 B) –1

6C) 0 D) 2

Page22

13) f(x)=πx4+9x3–2

A) 4 B) 3 C) πD) 1

14) f(x)=5x–x4+5

4

A) 4 B) 1 C) 5 D) –1

15) g(x)=17x6–8

A) 6 B) 7 C) 0 D) 17

16) h(x)=–6x+9

A) 1 B) 2C)0D)–6

17) –10x3–8x2+6x–5y4+5

A) 4 B) 3 C) 10 D) –10

18) f(x)=–16x3–7x2–1

A) 3 B) 6 C) –7D)

–16

2 RecognizeCharacteristicsofGraphsofPolynomialFunctions

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Determinewhetherthegraphshownisthegraphofapolynomialfunction.

1)

x

y

x

y

A) notapolynomialfunction B) polynomialfunction

2)

x

y

x

y

A) polynomialfunction B) notapolynomialfunction

Page23

3)

x

y

x

y

A) polynomialfunction B) notapolynomialfunction

4)

x

y

x

y

A) polynomialfunction B) notapolynomialfunction

5)

x

y

x

y

A) notapolynomialfunction B) polynomialfunction

Page24

6)

x

y

x

y

A) notapolynomialfunction B) polynomialfunction

Findthex–interceptsofthepolynomialfunction.Statewhetherthegraphcrossesthex–axis,ortouchesthex–axisand

turnsaround,ateachintercept.

7) f(x)=7x2–x3

A) 0,touchesthex–axisandturnsaround;

7,crossesthex–axis

B) 0,crossesthex–axis;

7,crossesthex–axis;

–7,crossesthex–axis

C) 0,touchesthex–axisandturnsaround;

7,crossesthex–axis;

–7,crossesthex–axis

D) 0,touchesthex–axisandturnsaround;

7,touchesthex–axisandturnsaround

8) f(x)=x4–100x2

A) 0,touchesthex–axisandturnsaround;

10,crossesthex–axis;

–10,crossesthex–axis

B) 0,crossesthex–axis;

10,crossesthex–axis;

–10,crossesthex–axis

C) 0,touchesthex–axisandturnsaround;

100,touchesthex–axisandturnsaround

D) 0,touchesthex–axisandturnsaround;

100,crossesthex–axis

9) x5–28x3+75x=0

A) 0,crossesthex–axis;

5,crossesthex–axis;

–5,crossesthex–axis;

3,crossesthex–axis;

–3,crossesthex–axis

B) 0,touchesthex–axisandturnsaround;

5,crossesthex–axis;

–5,crossesthex–axis;

3,crossesthex–axis;

–3,crossesthex–axis

C) 0,crossesthex–axis;

25,touchesthex–axisandturnsaround;

3,touchesthex–axisandturnsaround

D) 0,touchesthex–axisandturnsaround;

25,touchesthex–axisandturnsaround;

3,touchesthex–axisandturnsaround

10) x4+8x3–33x2=0

A) 0,touchesthex–axisandturnsaround;

–11,crossesthex–axis;

3,crossesthex–axis

B) 0,touchesthex–axisandturnsaround;

11,touchesthex–axisandturnsaround;

–3,touchesthex–axisandturnsaround

C) 0,crossesthex–axis;

–11,crossesthex–axis;

3,crossesthex–axis

D) 0,touchesthex–axisandturnsaround;

11,crossesthex–axis;

–3,crossesthex–axis

Page25

11) f(x)=x3+9x2+24x+20

A) –2

,

touchesthex–axisandturnsaround;

–5,crossesthex–axis.

B) –2

,

crossesthex–axis;

–5,touchesthex–axisandturnsaround

C) 2

,

crossesthex–axis;

–2,crossesthex–axis;

–5,crossesthex–axis.

D) 2

,

crossesthex–axis;

–2,touchesthex–axisandturnsaround;

–5,crossesthex–axis.

12) f(x)=(x+1)(x–6)(x–1)2

A) –1,crossesthex–axis;

6,crossesthex–axis;

1,touchesthex–axisandturnsaround

B) –1,crossesthex–axis;

6,crossesthex–axis;

1,crossesthex–axis

C) 1,crossesthex–axis;

–6,crossesthex–axis;

–1,touchesthex–axisandturnsaround

D) 1,crossesthex–axis;

–6,touchesthex–axisandturnsaround;

–1,touchesthex–axisandturnsaround

13) f(x)=–x2(x+3)(x2–1)

A) 0,touchesthex–axisandturnsaround;

–3,crossesthex–axis;

–1,crossesthex–axis;

1,crossesthex–axis

B) 0,crossesthex–axis;

–3,crossesthex–axis;

–1,crossesthex–axis;

1,crossesthex–axis

C) 0,touchesthex–axisandturnsaround;

–3,crossesthex–axis;

1,touchesthex–axisandturnsaround

D) 0,touchesthex–axisandturnsaround;

3,crossesthex–axis;

–1,touchesthex–axisandturnsaround;

1,touchesthex–axisandturnsaround

14) f(x)=–x2(x+6)(x2+1)

A) 0,touchesthex–axisandturnsaround;

–6,crossesthex–axis

B) 0,touchesthex–axisandturnsaround;

6,crossesthex–axis

C) 0,touchesthex–axisandturnsaround;

–6,crossesthex–axis;

–1,touchesthex–axisandturnsaround

D) 0,touchesthex–axisandturnsaround;

–6,crossesthex–axis;

–1,crossesthex–axis;

1,crossesthex–axis;

15) f(x)=x2(x–1)(x–3)

A) 0,touchesthex–axisandturnsaround;

1,crossesthex–axis;

3,crossesthex–axis

B) 0,touchesthex–axisandturnsaround;

–1,crossesthex–axis;

–3,crossesthex–axis

C) 0,crossesthex–axis;

1,crossesthex–axis;

3,crossesthex–axis

D) 0,crossesthex–axis;

1,touchesthex–axisandturnsaround;

3,touchesthex–axisandturnsaround

16) f(x)=–x3(x+2)2(x–8)

A) 0,crossesthex–axis;

–2,touchesthex–axisandturnsaround;

8,crossesthex–axis

B) 0,crossesthex–axis;

2,touchesthex–axisandturnsaround;

–8,crossesthex–axis

C) 0,touchesthex–axisandturnsaround;

–2,touchesthex–axisandturnsaround;

8,crossesthex–axis

D) 0,touchesthex–axisandturnsaround;

2,crossesthex–axis;

8,crossesthex–axis

Page26

17) f(x)=(x–2)2(x2–9)

A) 2

,

touchesthex–axisandturnsaround;

–3,crossesthex–axis;

3,crossesthex–axis

B) 2

,

touchesthex–axisandturnsaround;

–3,touchesthex–axisandturnsaround;

3,touchesthex–axisandturnsaround

C) 2

,

touchesthex–axisandturnsaround;

9,touchesthex–axisandturnsaround

D) –2

,

touchesthex–axisandturnsaround;

9,crossesthex–axis

Findthey–interceptofthepolynomialfunction.

18) f(x)=3x–x3

A) 0 B) 3 C) –1D)

–3

19) f(x)=–x2–2x+8

A) 8 B) –8C)0 D)

–1

20) f(x)=(x+1)(x–8)(x–1)2

A) –8 B) 8 C) 0 D) –1

21) f(x)=–x2(x+4)(x2–1)

A) 0 B) –1C)

–4D)4

22) f(x)=–x2(x+7)(x2+1)

A) 0 B) 1 C) 7 D) –7

23) f(x)=x2(x–1)(x–6)

A) 0 B) –6C)6 D)

–1

24) f(x)=–x2(x+2)(x–8)

A) 0 B) –8C)

–16 D) 16

25) f(x)=(x–3)2(x2–25)

A) –225 B) 225 C) –75 D) 75

Determinewhetherthegraphofthepolynomialhasy–axissymmetry,originsymmetry,orneither.

26) f(x)=8x2–x3

A) y–axissymmetry B) originsymmetry C) neither

27) f(x)=8–x4

A) y–axissymmetry B) originsymmetry C) neither

28) f(x)=x4–81x2

A) y–axissymmetry B) originsymmetry C) neither

29) f(x)=x3–5x

A) originsymmetry B) y–axissymmetry C) neither

30) f(x)=x3+x2–4

A) originsymmetry B) y–axissymmetry C) neither

31) f(x)=x(2–x2)

A) originsymmetry B) y–axissymmetry C) neither

Page27

32) x5–27x3+50x=0

A) originsymmetry B) y–axissymmetry C) neither

33) f(x)=x3+10x2+33x+36

A) originsymmetry B) y–axissymmetry C) neither

34) f(x)=(x+1)(x–4)(x–1)2

A) y–axissymmetry B) originsymmetry C) neither

35) f(x)=–x2(x+6)(x2–1)

A) originsymmetry B) y–axissymmetry C) neither

36) f(x)=–x3(x+4)2(x–6)

A) originsymmetry B) y–axissymmetry C) neither

37) f(x)=(x–2)2(x2–9)

A) originsymmetry B) y–axissymmetry C) neither

38)

x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

A) y–axissymmetry B) originsymmetry C) neither

39)

x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

A) originsymmetry B) y–axissymmetry C) neither

Page28

40)

x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

A) originsymmetry B) y–axissymmetry C) neither

41)

x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

A) originsymmetry B) y–axissymmetry C) neither

Page29

3 DetermineEndBehavior

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

UsetheLeadingCoefficientTesttodeterminetheendbehaviorofthepolynomialfunction.Thenusethisendbehavior

tomatchthefunctionwithitsgraph.

1) f(x)=3x2–2x+1

A) risestotheleftandrisestotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

B) fallstotheleftandfallstotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

C) fallstotheleftandrisestotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

D) risestotheleftandfallstotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

Page30

2) f(x)=–2x2–3x–3

A) fallstotheleftandfallstotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

B) risestotheleftandrisestotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

C) risestotheleftandfallstotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

D) fallstotheleftandrisestotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

Page31

3) f(x)=4x3–3x2–2x–2

A) fallstotheleftandrisestotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

B) fallstotheleftandfallstotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

C) risestotheleftandrisestotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

D) risestotheleftandfallstotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

Page32

4) f(x)=–4x3–2x2+2x+2

A) risestotheleftandfallstotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

B) risestotheleftandrisestotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

C) fallstotheleftandfallstotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

D) fallstotheleftandrisestotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

Page33

5) f(x)=4x4–2x2

A) risestotheleftandrisestotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

B) fallstotheleftandfallstotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

C) fallstotheleftandrisestotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

D) risestotheleftandfallstotheright

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

UsetheLeadingCoefficientTesttodeterminetheendbehaviorofthepolynomialfunction.

6) f(x)=2x4+4x3+2x2–5x–2

A) risestotheleftandrisestotheright B) risestotheleftandfallstotheright

C) fallstotheleftandrisestotheright D) fallstotheleftandfallstotheright

7) f(x)=–3x4–2x3+2x2+3x+5

A) fallstotheleftandfallstotheright B) risestotheleftandfallstotheright

C) fallstotheleftandrisestotheright D) risestotheleftandrisestotheright

8) f(x)=4x3–2x2+5x–5

A) fallstotheleftandrisestotheright B) risestotheleftandfallstotheright

C) fallstotheleftandfallstotheright D) risestotheleftandrisestotheright

9) f(x)=x3–5x2–2x+1

A) fallstotheleftandrisestotheright B) risestotheleftandfallstotheright

C) fallstotheleftandfallstotheright D) risestotheleftandrisestotheright

10) f(x)=–3x3+3x2+3x+3

A) risestotheleftandfallstotheright B) fallstotheleftandrisestotheright

C) fallstotheleftandfallstotheright D) risestotheleftandrisestotheright

11) f(x)=3x3–3x3–x5

A) risestotheleftandfallstotheright B) fallstotheleftandrisestotheright

C) fallstotheleftandfallstotheright D) risestotheleftandrisestotheright

Page34

12) f(x)=x+2x2–5x3

A) risestotheleftandfallstotheright B) fallstotheleftandrisestotheright

C) fallstotheleftandfallstotheright D) risestotheleftandrisestotheright

13) f(x)=(x+3)(x+4)(x+5)2

A) risestotheleftandrisestotheright B) fallstotheleftandrisestotheright

C) risestotheleftandfallstotheright D) fallstotheleftandfallstotheright

14) f(x)=(x+1)(x+3)(x+5)3

A) fallstotheleftandrisestotheright B) risestotheleftandrisestotheright

C) risestotheleftandfallstotheright D) fallstotheleftandfallstotheright

15) f(x)=–5(x2+1)(x+1)2

A) fallstotheleftandfallstotheright B) fallstotheleftandrisestotheright

C) risestotheleftandrisestotheright D) risestotheleftandfallstotheright

16) f(x)=x3(x+2)(x+5)2

A) risestotheleftandrisestotheright B) fallstotheleftandrisestotheright

C) risestotheleftandfallstotheright D) fallstotheleftandfallstotheright

17) f(x)=–x2(x–2)(x+1)

A) fallstotheleftandfallstotheright B) fallstotheleftandrisestotheright

C) risestotheleftandfallstotheright D) risestotheleftandrisestotheright

18) f(x)=–6x3(x–4)(x+5)2

A) fallstotheleftandfallstotheright B) fallstotheleftandrisestotheright

C) risestotheleftandfallstotheright D) risestotheleftandrisestotheright

Solvetheproblem.

19) Aherdof

b

isonisintroducedtoawildliferefuge.Thenumberof

b

ison

,

N(t),aftertyearsisdescribedbythe

polynomialfunctionN(t)=–t4+18t+160.UsetheLeadingCoefficientTesttodeterminethegraphʹsend

behavior.Whatdoesthismeanaboutwhatwilleventuallyhappentothebisonpopulation?

A) The

b

isonpopulationintherefugewilldieout.

B) The

b

isonpopulationintherefugewillgrowoutofcontrol.

C) The

b

isonpopulationintherefugewillreachaconstantamountgreaterthan0.

D) The

b

isonpopulationintherefugewillbedisplacedbyʺoilʺwells.

20) Thefollowingtableshowsthenumberoflarcenythefts inacountyfortheyears1994–1998,where1represents

1994,2represents1995,andsoon.

Year,xLarcenyThefts,T

1994,14652.48

1995,2 4698.24

1996,3 4741.94

1997,4 4775.04

1998,5 4823

Thisdatacanbeapproximatedusingthethird–degreepolynomial

T(x)=–0.59x3+0.51x2+55.36x+4597.2.

Usethisfunctiontopredictthenumberoflarcenytheftsin2007.Roundtothenearestwholenumber.

A) 3853 B) 3846 C) –734 D) 3078

Page35

21) ThefollowingtableshowsthenumberofDWIarrests inacountyfortheyears1994–1998,where1represents

1994,2represents1995,andsoon.

Year,xDWIarrests,T

1994,14958.98

1995,2 4997.8

1996,3 5053.64

1997,4 5082.48

1998,5 5122.3

Thisdatacanbeapproximatedusingthethird–degreepolynomial

T(x)=–0.67x3+0.53x2+56.92x+4902.2.

UsetheLeadingCoefficientTesttodeterminetheendbehaviortotherightforthegraphofT.Willthis

functionbeusefulinmodelingthenumberofDWIarrestsoveranextendedperiodoftime?Explainyour

answer.

A) ThegraphofTdecreaseswithoutboundtotheright.Thismeansthatasxincreases,thevaluesofTwill

becomemoreandmorenegativeandthefunctionwillnolongermodelthenumberofDWIarrests.

B) ThegraphofTincreaseswithoutboundtotheright.Thismeansthatasxincreases,thevaluesofTwill

becomelargeandpositiveand,sincethevaluesofTwillbecomesolarge,thefunctionwillnolonger

modelthenumberofDWIarrests.

C) ThegraphofTapproacheszeroforlargevaluesofx.ThismeansthatTwillnotbeusefulinmodelingthe

numberofDWIarrestsoveranextendedperiod.

D) ThegraphofTdecreaseswithoutboundtotheright.Sincethenumberoflarcenytheftswilleventually

decrease,thefunctionTwillbeusefulinmodelingthenumberofDWIarrestsoveranextendedperiodof

time.

4 UseFactoringtoFindZerosofPolynomialFunctions

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findthezerosofthepolynomialfunction.

1) f(x)=x3+x2–12x

A) x=0,x=–4

,

x=3B)x=–4

,

x=3C)x=2

,

x=3D)x=0,x=2

,

x=3

2) f(x)=x3+9x2–x–9

A) x=–1,x=1,x=–9B)x=1,x= – 9

,

x=9

C) x=–9

,

x=9D)x=81

3) f(x)=x3–10x2+25x

A) x=0,x=5B)x=0,x= –5C)x=1,x=5D)x=0,x= –5

,

x=5

4) f(x)=x3+2x2–9x–18

A) x=–2

,

x=–3

,

x=3B)x=2

,

x= –3

,

x=3

C) x=–3

,

x=3D)x= –2

,

x=9

5) f(x)=4(x+1)(x+5)4

A) x=–1

,

x=–5

,

B) x=1

,

x=4C)x= –1

,

x=4D)x=1

,

x=5

,

x=4

Page36

5 IdentifyZerosandTheirMultiplicities

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findthezerosforthepolynomialfunctionandgivethemultiplicityforeachzero.Statewhetherthegraphcrossesthe

x–axisortouchesthex–axisandturnsaround,ateachzero.

1) f(x)=2(x+1)(x+2)4

A) –1

,

multiplicity1,crossesx–axis;–2

,

multiplicity4

,

touchesx–axisandturnsaround

B) 1

,

multiplicity1,crossesx–axis;2

,

multiplicity4

,

touchesx–axisandturnsaround

C) –1

,

multiplicity1,touchesx–axisandturnsaround;–2

,

multiplicity4

,

crossesx–axis

D) 1

,

multiplicity1,touchesx–axisandturnsaround;2

,

multiplicity4

,

crossesx–axis

2) f(x)=4(x+6)(x+5)3

A) –6

,

multiplicity1,crossesx–axis;–5

,

multiplicity3,crossesx–axis

B) 6

,

multiplicity1,crossesx–axis;5

,

multiplicity3,crossesx–axis

C) –6

,

multiplicity1,crossesx–axis;–5

,

multiplicity3,touchesx–axisandturnsaround

D) 6

,

multiplicity1,touchesx–axis;5

,

multiplicity3,touchesx–axisandturnsaround

3) f(x)=4x+2(x–2)3

A) –2

,

multiplicity1,crossesx–axis;2

,

multiplicity3,crossesx–axis

B) 2

,

multiplicity1,crossesx–axis;–2

,

multiplicity3,crossesx–axis

C) –2

,

multiplicity1,touchesthex–axisandturnsaround;2

,

multiplicity3,touchesx–axisandturns

around

D) 2

,

multiplicity1,touchesthex–axisandturnsaround;–2

,

multiplicity3,touchesx–axisandturnsaround

4) f(x)=3(x2+2)(x–1)2

A) 1

,

multiplicity2,touchesthex–axisandturnsaround

B) –2

,

multiplicity1,crossesthex–axis;1

,

multiplicity2,touchesthex–axisandturnsaround.

C) –2

,

multiplicity1,crossesthex–axis;1

,

multiplicity2,crossesthex–axis

D) 1

,

multiplicity2,crossesthex–axis

5) f(x)=1

4x4(x2–3)(x–7)

A) 0,multiplicity4

,

touchesx–axisandturnsaround;

7,multiplicity1,crossesx–axis;

3,multiplicity1,crossesx–axis;

–3,multiplicity1,crossesx–axis

B) 0,multiplicity4

,

crossesx–axis;

7,multiplicity1,touchesx–axisandturnsaround;

3,multiplicity1,touchesx–axisandturnsaround;

–3,multiplicity1,touchesx–axisandturnsaround

C) 0,multiplicity4

,

touchesx–axisandturnsaround;

7,multiplicity1,crossesx–axis

D) 0,multiplicity4

,

touchesx–axisandturnsaround;

7,multiplicity1,crossesx–axis

3,multiplicity2,touchesx–axisandturnsaround

Page37

6) f(x)=x+1

5

2

(x–2)5

A) –1

5,multiplicity2,touchesthex–axisandturnsaround;

2,multiplicity5,crossesthex–axis.

B) –1

5,multiplicity2,crossesthex–axis;

2,multiplicity5,touchesthex–axisandturnsaround

C) 1

5,multiplicity2,touchesthex–axisandturnsaround;

–2,multiplicity5,crossesthex–axis.

D) 1

5,multiplicity2,crossesthex–axis;

–2,multiplicity5,touchesthex–axisandturnsaround

7) f(x)=x+1

4

(x2+1)4

A) –1

4,multiplicity4,touchesthex–axisandturnsaround.

B) –1

4,multiplicity4,touchesthex–axisandturnsaround;

–1,multiplicity4,crossesthex–axis

C) 1

4,multiplicity4,touchesthex–axisandturnsaround;

1,multiplicity4,crossesthex–axis

D) –1

4,multiplicity4,crossesthex–axis.

8) f(x)=x3+x2–12x

A) 0,multiplicity1,crossesthex–axis

–4,multiplicity1,crossesthex–axis

3,multiplicity1,crossesthex–axis

B) –4

,

multiplicity2,touchesthex–axisandturnsaround

3,multiplicity1,crossesthex–axis

C) 0,multiplicity1,crossesthex–axis

4,multiplicity1,crossesthex–axis

–3,multiplicity1,crossesthex–axis

D) 0,multiplicity1,touchesthex–axisandturnsaround;

–4,multiplicity1,touchesthex–axisandturnsaround;

3,multiplicity1,touchesthex–axisandturnsaround

Page38

9) f(x)=x3+10x2+33x+36

A) –3

,

multiplicity2,touchesthex–axisandturnsaround;

–4,multiplicity1,crossesthex–axis.

B) –3

,

multiplicity2,crossesthex–axis;

–4,multiplicity1,touchesthex–axisandturnsaround

C) 3

,

multiplicity1,crossesthex–axis;

–3,multiplicity1,crossesthex–axis;

–4,multiplicity1,crossesthex–axis.

D) 3

,

multiplicity1,crossesthex–axis;

–3,multiplicity2,touchesthex–axisandturnsaround;

–4,multiplicity1,crossesthex–axis.

10) f(x)=x3+6x2–x–6

A) –1

,

multiplicity1,crossesthex–axis;

1,multiplicity1,crossesthex–axis;

–6,multiplicity1,crossesthex–axis.

B) 6

,

multiplicity1,crossesthex–axis;

1,multiplicity1,crossesthex–axis;

–6,multiplicity1,crossesthex–axis.

C) 1

,

multiplicity2,touchesthex–axisandturnsaround;

–6,multiplicity1,crossesthex–axis.

D) –1

,

multiplicity1,touchesthex–axisandturnsaround;

1,multiplicity1,touchesthex–axisandturnsaround;

–6,multiplicity1,touchesthex–axisandturnsaround

Writetheequationofapolynomialfunctionwiththegivencharacteristics.Usealeadingcoefficientof1or–1andmake

thedegreeofthefunctionassmallaspossible.

11) Crossesthex–axisat–1

,

0,and3;liesabovethex–axisbetween–1 and0;liesbelowthex–axisbetween0and

3.

A) f(x)=x3–2x2–3x B) f(x)=x3+2x2–3x

C) f(x)=–x3+2x2+3x D) f(x)=–x3–2x2+3x

12) Crossesthex–axisat–1

,

0,and4;liesbelowthex–axisbetween–1 and0;liesabovethex–axisbetween0and

4.

A) f(x)=–x3+3x2+4x B) f(x)=–x3–3x2+4x

C) f(x)=x3–3x2–4x D) f(x)=x3+3x2–4x

13) Touchesthex–axisat0andcrossesthex–axisat2;liesbelowthex–axisbetween0and2.

A) f(x)=x3–2x2B) f(x)=x3+2x2C) f(x)=–x3+2x2D) f(x)=–x3–2x2

14) Touchesthex–axisat0andcrossesthex–axisat4;liesabovethex–axisbetween0and4.

A) f(x)=–x3+4x2B) f(x)=x3+4x2C) f(x)=x3–4x2D) f(x)=–x3–4x2

6 UsetheIntermediateValueTheorem

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

UsetheIntermediateValueTheoremtodeterminewhetherthepolynomialfunctionhasarealzerobetweenthegiven

integers.

1) f(x)=10x3+10x2+4x+5;between–2and–1

A) f(–2)=–43andf(–1)=1;yes B) f(–2)=43 andf(–1)=1;no

C) f(–2)=–43andf(–1)=–1;no D) f(–2)=43 andf(–1)=–1;yes

Page39

2) f(x)=6x5–7x3+6x2–2;between–2and–1

A) f(–2)=–114andf(–1)=5;yes B) f(–2)=114 andf(–1)=5;no

C) f(–2)=–114andf(–1)=–5;no D) f(–2)=114 andf(–1)=–5;yes

3) f(x)=8x4–7x2–2;between1and2

A) f(1)=–1andf(2)=98;yes B) f(1)=1 andf(2)=99;no

C) f(1)=–1andf(2)=–98;no D) f(1)=1 andf(2)=–98;yes

4) f(x)=6x4–3x3+4x–3;between–1and0

A) f(–1)=2andf(0)=–3;yes B) f(–1)=2 andf(0)=3;no

C) f(–1)=–2andf(0)=–3;no D) f(–1)= –2 andf(0)=3;yes

5) f(x)=5x3–7x+7;between–2and–1

A) f(–2)=–19andf(–1)=9;yes B) f(–2)= –19 andf(–1)=–9;no

C) f(–2)=19andf(–1)=9;no D) f(–2)=19 andf(–1)=–9;yes

7 UnderstandtheRelationshipBetweenDegreeandTurningPoints

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Determinethemaximumpossiblenumberofturningpointsforthegraphofthefunction.

1) f(x)=–x2–8x+15

A) 1 B) 2 C) 0 D) 3

2) f(x)=5x8–8x7–9x–25

A) 7 B) 0 C) 5 D) 8

3) f(x)=x7+3x8

A) 7 B) 8 C) 3 D) 1

4) g(x)=4x+4

A) 0 B) 2 C) 1 D) 3

5) f(x)=(x+7)(x+1)(5x+4)

A) 2 B) 5 C) 3 D) 0

6) f(x)=x4(x4+2)(6x+4)

A) 8 B) 9 C) 48 D) 4

7) f(x)=(2x+3)2(x2–1)(x+1)

A) 4 B) 5 C) 10 D) 2

8) f(x)=(x+1)(x–2)(x–7)(x–4)

A) 3 B) 4 C) 0 D) 1

Page40

Solve.

9) Supposethatapolynomialfunctionisusedtomodelthedatashowninthegraphbelow.

Forwhatintervalsisthefunctionincreasing?

A) 0through10and25through40 B) 0through40

C) 0through10and20through50 D) 10through25and40through50

10) Supposethatapolynomialfunctionisusedtomodelthedatashowninthegraphbelow.

Forwhatintervalsisthefunctionincreasing?

A) 0through10and30through50 B) 0through50

C) 0through20and30through50 D) 0through10and40through50

11) Supposethatapolynomialfunctionisusedtomodelthedatashowninthegraphbelow.

Forwhatintervalsisthefunctiondecreasing?

A) 10through25and40through50 B) 10through50

C) 10through25and40through45 D) 0through10and25through40

Page41

12) Supposethatapolynomialfunctionisusedtomodelthedatashowninthegraphbelow.

Forwhatintervalsisthefunctiondecreasing?

A) 10through30 B) 0through30

C) 10through20and30through50 D) 0through10and30through50

13) Supposethatapolynomialfunctionisusedtomodelthedatashowninthegraphbelow.

Determinethedegreeofthepolynomialfunctionofbestfitandthesignoftheleadingcoefficient.

A) Degree4;negativeleadingcoefficient. B) Degree5;positiveleadingcoefficient.

C) Degree5;negativeleadingcoefficient. D) Degree4;positiveleadingcoefficient.

14) Supposethatapolynomialfunctionisusedtomodelthedatashowninthegraphbelow.

Determinethedegreeofthepolynomialfunctionofbestfitandthesignoftheleadingcoefficient.

A) Degree3;positiveleadingcoefficient. B) Degree4;negativeleadingcoefficient.

C) Degree3;negativeleadingcoefficient. D) Degree4;positiveleadingcoefficient.

Page42

15) Theprofits(inmillions)foracompanyfor8yearswereasfollows:

Year,xProfits,P

1993,1

1994,2

1995,3

1996,4

1997,5

1998,6

1999,7

2000,8

1.1

1.7

2.0

1.4

1.3

1.5

1.8

2.1

Whichofthefollowingpolynomialsisthebestmodelforthisdata?

A) P(x)=0.05x2–0.8x+6 B) P(x)=–0.08x3+7x2+1.3x–0.18

C) P(x)=0.03x3–0.3x2+1.3x+0.17 D) P(x)=–0.03x4–0.3x2+1.3x+0.17

Page43

8 GraphPolynomialFunctions

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Graphthepolynomialfunction.

1) f(x)=x4–4x2

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A)

x

–8–6–4–2 2468

y

20

16

12

8

4

-4

-8

-12

-16

-20

x

–8–6–4–2 2468

y

20

16

12

8

4

-4

-8

-12

-16

-20

B)

x

–8–6–4–2 2468

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–8–6–4–2 2468

y

10

8

6

4

2

-2

-4

-6

-8

-10

C)

x

–8–6–4–2 2468

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–8–6–4–2 2468

y

10

8

6

4

2

-2

-4

-6

-8

-10

D)

x

–10–8-6-4-2 2 4 6 810

y

800

640

480

320

160

-160

-320

-480

-640

-800

x

–10–8-6-4-2 2 4 6 810

y

800

640

480

320

160

-160

-320

-480

-640

-800

Page44

2) f(x)=3x2–x3

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page45

3) f(x)=1

2–1

2x4

x

-5 5

y

5

-5

x

-5 5

y

5

-5

A)

x

-5 5

y

5

-5

x

-5 5

y

5

-5

B)

x

-5 5

y

5

-5

x

-5 5

y

5

-5

C)

x

-5 5

y

5

-5

x

-5 5

y

5

-5

D)

x

-5 5

y

5

-5

x

-5 5

y

5

-5

Page46

4) f(x)=x3+7x2–x–7

x

y

x

y

A)

x

–10–8-6-4-2 2 4 6 810

y

100

80

60

40

20

-20

-40

-60

-80

-100

x

–10–8-6-4-2 2 4 6 810

y

100

80

60

40

20

-20

-40

-60

-80

-100

B)

x

–10–8-6-4-2 2 4 6 810

y

100

80

60

40

20

-20

-40

-60

-80

-100

x

–10–8-6-4-2 2 4 6 810

y

100

80

60

40

20

-20

-40

-60

-80

-100

C)

x

–10–8-6-4-2 2 4 6 810

y

500

400

300

200

100

-100

-200

-300

-400

-500

x

–10–8-6-4-2 2 4 6 810

y

500

400

300

200

100

-100

-200

-300

-400

-500

D)

x

–10–8-6-4-2 2 4 6 810

y

500

400

300

200

100

-100

-200

-300

-400

-500

x

–10–8-6-4-2 2 4 6 810

y

500

400

300

200

100

-100

-200

-300

-400

-500

Page47

5) f(x)=x3–2x2–5x+6

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page48

6) f(x)=5x–x3–x5

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page49

7) f(x)=6x4+9x3

x

-5 5

y

5

-5

x

-5 5

y

5

-5

A)

x

-5 5

y

5

-5

x

-5 5

y

5

-5

B)

x

-5 5

y

5

-5

x

-5 5

y

5

-5

C)

x

-5 5

y

5

-5

x

-5 5

y

5

-5

D)

x

-5 5

y

5

-5

x

-5 5

y

5

-5

Page50

8) f(x)=6x3–5x–x5

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page51

9) f(x)=x4+16x3+64x2

x

y

x

y

A)

x

–10–8-6-4-2 2 4 6 810

y

300

240

180

120

60

-60

-120

-180

-240

-300

x

–10–8-6-4-2 2 4 6 810

y

300

240

180

120

60

-60

-120

-180

-240

-300

B)

x

–10–8-6-4-2 2 4 6 810

y

300

240

180

120

60

-60

-120

-180

-240

-300

x

–10–8-6-4-2 2 4 6 810

y

300

240

180

120

60

-60

-120

-180

-240

-300

C)

x

-12-10 -8 -6 -4 -2 2 4 6 8 10 12

y

1000

800

600

400

200

-200

-400

-600

-800

-1000

x

-12-10 -8 -6 -4 -2 2 4 6 8 10 12

y

1000

800

600

400

200

-200

-400

-600

-800

-1000

D)

x

-12-10 -8 -6 –4 -2 2 4 6 8 10 12

y

250

200

150

100

50

-50

-100

-150

-200

-250

x

-12-10 -8 -6 –4 -2 2 4 6 8 10 12

y

250

200

150

100

50

-50

-100

-150

-200

-250

Page52

10) f(x)=x5–6x3–16x

x

–5–4–3–2–1 12345

y

150

120

90

60

30

-30

-60

-90

-120

-150

x

–5–4–3–2–1 12345

y

150

120

90

60

30

-30

-60

-90

-120

-150

A)

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

150

120

90

60

30

-30

-60

-90

-120

-150

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

150

120

90

60

30

-30

-60

-90

-120

-150

B)

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

150

120

90

60

30

-30

-60

-90

-120

-150

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

150

120

90

60

30

-30

-60

-90

-120

-150

C)

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

150

120

90

60

30

-30

-60

-90

-120

-150

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

150

120

90

60

30

-30

-60

-90

-120

-150

D)

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

150

120

90

60

30

-30

-60

-90

-120

-150

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

150

120

90

60

30

-30

-60

-90

-120

-150

Page53

11) f(x)=x4–2x3–x2+2

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

B)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

C)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

D)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Page54

12) f(x)=x4–4x3+4x2

x

-12-10 -8 -6 -4 -2 2 4 6 8 10

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

-12-10 -8 -6 -4 -2 2 4 6 8 10

y

10

8

6

4

2

-2

-4

-6

-8

-10

A)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

B)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

C)

x

–10–8-6-4-2 2 4 6 810

y

20

16

12

8

4

-4

-8

-12

-16

-20

x

–10–8-6-4-2 2 4 6 810

y

20

16

12

8

4

-4

-8

-12

-16

-20

D)

x

–10–8-6-4-2 2 4 6 810

y

800

640

480

320

160

-160

-320

-480

-640

-800

x

–10–8-6-4-2 2 4 6 810

y

800

640

480

320

160

-160

-320

-480

-640

-800

Page55

13) f(x)=3x(x+2)3

x

-5 5

y

10

5

-5

-10

x

-5 5

y

10

5

-5

-10

A)

x

-5 5

y

10

5

-5

-10

x

-5 5

y

10

5

-5

-10

B)

x

-5 5

y

10

5

-5

-10

x

-5 5

y

10

5

-5

-10

C)

x

-5 5

y

10

5

-5

-10

x

-5 5

y

10

5

-5

-10

D)

x

-5 5

y

10

5

-5

-10

x

-5 5

y

10

5

-5

-10

Page56

14) f(x)=x(x–2)(x–1)

x

–6–5–4–3–2–1 123456

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

–6–5–4–3–2–1 123456

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

A)

x

–6–5–4–3–2–1 123456

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

–6–5–4–3–2–1 123456

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

B)

x

–6–5–4–3–2–1 123456

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

–6–5–4–3–2–1 123456

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

C)

x

–6–5–4–3–2–1 123456

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

–6–5–4–3–2–1 123456

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

D)

x

–6–5–4–3–2–1 123456

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

–6–5–4–3–2–1 123456

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

Page57

15) f(x)=–x2(x+1)(x+3)

x

–4–3–2–1 1234

y

20

15

10

5

-5

-10

-15

-20

x

–4–3–2–1 1234

y

20

15

10

5

-5

-10

-15

-20

A)

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

B)

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

C)

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

D)

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

16) f(x)=(x+1)2(x2–25)

x

y

x

y

Page58

A)

x

–10–8-6-4-2 2 4 6 810

y

250

200

150

100

50

-50

-100

-150

-200

-250

x

–10–8-6-4-2 2 4 6 810

y

250

200

150

100

50

-50

-100

-150

-200

-250

B)

x

–10–8-6-4-2 2 4 6 810

y

250

200

150

100

50

-50

-100

-150

-200

-250

x

–10–8-6-4-2 2 4 6 810

y

250

200

150

100

50

-50

-100

-150

-200

-250

C)

x

–10–8-6-4-2 2 4 6 810

y

250

200

150

100

50

-50

-100

-150

-200

-250

x

–10–8-6-4-2 2 4 6 810

y

250

200

150

100

50

-50

-100

-150

-200

-250

D)

x

-25 -20 -15 -10 -5 5 10 15 20 25

y

2500

2000

1500

1000

500

-500

-1000

-1500

-2000

-2500

x

-25 -20 -15 -10 -5 5 10 15 20 25

y

2500

2000

1500

1000

500

-500

-1000

-1500

-2000

-2500

Page59

17) f(x)=–x2(x–4)(x–1)

x

–4–3–2–1 1234

y

20

15

10

5

-5

-10

-15

-20

x

–4–3–2–1 1234

y

20

15

10

5

-5

-10

-15

-20

A)

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

B)

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

C)

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

D)

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

Page60

18) f(x)=–2x3(x–3)2(x+1)

x

y

x

y

A)

x

-4 -3 -2 -1 1 2 3 4

y

160

120

80

40

-40

-80

–120

–160

x

-4 -3 -2 -1 1 2 3 4

y

160

120

80

40

-40

-80

–120

–160

B)

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

C)

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

x

-4 -3 -2 -1 1 2 3 4

y

20

15

10

5

-5

-10

-15

-20

D)

x

-4 -3 -2 -1 1 2 3 4

y

160

120

80

40

-40

-80

–120

–160

x

-4 -3 -2 -1 1 2 3 4

y

160

120

80

40

-40

-80

–120

–160

Page61

19) f(x)=(x+1)(x+3)(x+5)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

A)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

B)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

C)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

D)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

Page62

20) f(x)=(x+1)(x+3)(x+5)2

x

-6 -4 -2 2 4 6

y

12

8

4

-4

-8

-12

x

-6 -4 -2 2 4 6

y

12

8

4

-4

-8

-12

A)

x

-6 -4 -2 2 4 6

y

12

8

4

-4

-8

-12

x

-6 -4 -2 2 4 6

y

12

8

4

-4

-8

-12

B)

x

-6 -4 -2 2 4 6

y

12

8

4

-4

-8

-12

x

-6 -4 -2 2 4 6

y

12

8

4

-4

-8

-12

C)

x

-6 -4 -2 2 4 6

y

12

8

4

-4

-8

-12

x

-6 -4 -2 2 4 6

y

12

8

4

-4

-8

-12

D)

x

-6 -4 -2 2 4 6

y

12

8

4

-4

-8

-12

x

-6 -4 -2 2 4 6

y

12

8

4

-4

-8

-12

Page63

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Completethefollowing:

(a) UsetheLeadingCoefficientTesttodeterminethegraphʹsendbehavior.

(b) Findthex–intercepts.Statewhetherthegraphcrossesthex–axisortouchesthex–axisandturnsaroundateach

intercept.

(c) Findthey–intercept.

(d) Graphthefunction.

21) f(x)=x2(x+2)

x

y

x

y

22) f(x)=(x+2)(x–1)2

x

y

x

y

23) f(x)=–2(x–3)(x+2)3

x

y

x

y

Page64

3.3 DividingPolynomials;RemainderandFactorTheorems

1 UseLongDivisiontoDividePolynomials

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Divideusinglongdivision.

1) (x2–12x+35)÷(x–7)

A) x–5B)x–12 C) x2–5D)x

2–12

2) (9x2–62x–7)÷(x–7)

A) 9x+1B)9x–1C)x–62 D) 9x2+62

3) (28x2–27x+5)÷(–4x+1)

A) –7x+5B)28x+5C)x+5D)5x+1

4) 4m3+21m2–42m+49

m+7

A) 4m2–7m+7B)4m

2+7m+7C)m

2+8m+9D)m

2+7m+4

5) 5r3–23r2–5r–25

r–5

A) 5r2+2r+5B)5r

2–2r–5C)5r

2+2r+5

r–5D) r2+5r+2

6) (–6x3+4x2+17x+5)÷(3x+1)

A) –2x2+2x+5B)

–2x2+5C)x

2+2x+5D)x

2–2x–5

7) 4x3–47x–33

x+3

A) 4x2–12x–11 B) 4x2–59x+144

x+3C) 4x2+59x+144

x+3D) 4x2+12x–11

8) (15x3–3)÷(5x–1)

A) 3x2+3

5x+3

25 –72

25(5x–1) B) 3x2+3

5x+3

25 +72

25(5x–1)

C) 3x2+3

5x+3

25 D) 3x2–3

5x+3

25

9) –15x3+37x2–8x–6

3x–5

A) –5x2+4x+4+14

3x–5B) –5x2+4x+4

C) –5x2+4x+4+17

3x–5D) x2+4+4

3x–5

Page65

10) x4+16

x–2

A) x3+2x2+4x+8+32

x–2B) x3+2x2+4x+8+16

x–2

C) x3+2x2+4x+8D)x

3–2x2+4x–8+32

x–2

11) (2x4–7x2+14x3–49x)÷(2x+14)

A) x3–7

2xB)x

3+7

2xC)x

3–14x+4x

2x+14 D) x3–7

2x–98x

2x+14

12) 8u4+12u3–2u

2u2+u

A) 4u2+4u–2B)4u

2+8u+4+2u

2u2+u

C) 4u2+4u–6u

2u2+uD) 4u2+6u–2u

2u2+u

13) (15x3+x2–30x–2)÷(5x2–10)

A) 3x+1

5B) 3x+5C)3x+–2

5x2–10 D) 3x+2

5x2–10

14) (5x4–32x3–20x2–13x+42)÷(7–x)

A) –5x3–3x2–x+6B)

–5x3–3x2–x–6

C) –5x3–3x2–x–6+84

7–xD) –5x3–3x2+x–6

15) (4x5–x3+5x2–89x–25)÷(x2–5)

A) 4x3+19x+5+6x

x2–5B) 4x3+19x+5–6x

x2–5

C) 4x3+19x+5+6x–50

x2–5D) 4x3+19x–5+6x

x2–5

16) x4–2x3–10x2+5x+36

x2–3x–4

A) x2+x–3+24

x2–3x–4B) x2+x–3

C) x2–6x+4+8x–28

x2–3x–4D) x2–6x+4

17) –4t4+18t3+8t2–60t–40

2t2–4t–4

A) –2t2+5t+10 B) –2t2–5t+10 C) –2t2+5t–10 D) –2t2+6t+10

Page66

Solvetheproblem.

18) Arectanglewithwidth2x+1incheshasanareaof2x4+5x3–16x2–45x–18squareinches.Writea

polynomialthatrepresentsitslength.

A) x3+2x2–9x–18inches B) x3–9x2+2x–18inches

C) x3+6x2–10x–18inches D) x3–10x2+6x–18inches

19) Thewidthofarectangleisx–3

4feetanditsareais4x3+21x2+14x–24squarefeet.Writeapolynomialthat

representsthelengthoftherectangle.

A) 4x2+24x+32ft B) 4x2–24x+32ft C) 4x2+18x+1

2ft D) 4x2+24x–32ft

20) Twopeopleare31yearsoldand25yearsold,respectively.Inxyearsfromnow,theiragescanberepresented

byx+31andx+25.Uselongdivisiontofindtheratiooftheolderpersonʹsagetotheyoungerpersonʹsagein

xyears.

A) 1+6

x+25 B) 1+56

x+25 C) 1.2400 D) 1+56

x+31

2 UseSyntheticDivisiontoDividePolynomials

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Divideusingsyntheticdivision.

1) (x2+14x+45)÷(x+5)

A) x+9B)x–40 C) x2+9D)x

3–40

2) (x2+16x+61)÷(x+7)

A) x+9–2

x+7B) x+9+2

x+7C) x+9

x+7D) x+10

3) 4x2–33x+54

x–6

A) 4x–9B)x–9C)

–9x–6D)

–4x+9

4) 6x3–26x2+6x+8

x–4

A) 6x2–2x–2B)

–6x2+4x–2C)

3

2x2–13

2x+3

2D) –6x2–4x+2

5) –2x3–10x2–5x+12

x+4

A) –2x2–2x+3B)2x

2–4x+3C)

–1

2x2–5

2x–5

4D) –2x2x–5

2+3

6) x5+x3–5

x–2

A) x4+2x3+5x2+10x+20+35

x–2B) x4+2x3+4x2+9x+18+31

x–2

C) x4+3x2+1

x–2D) x4+3+1

x–2

Page67

7) x4–3x3+x2+4x–5

x–1

A) x3–2x2–x+3–2

x–1B) x3–2x2+x+5+4

x–1

C) x3+2x2–x+5–2

x–1D) x3–2x2+x+3+4

x–1

8) (x4+16)÷(x–2)

A) x3+2x2+4x+8+32

x–2B) x3+2x2+4x+8+16

x–2

C) x3+2x2+4x+8D)x

3–2x2+4x–8+32

x–2

9) (x5–4x4–6x3+x2–x+46)÷(x+2)

A) x4–6x3+6x2–11x+21+4

x+2B) x4–6x3+6x2–11x–21+4

x+2

C) x4–6x3+6x2–12x+21+10

x+2D) x4–6x3+6x2–12x–22+10

x+2

10) (5x5+12x4–7x3+x2–x+50)÷(x+3)

A) 5x4–3x3+2x2+5x+14+8

x+3B) 5x4–3x3+2x2–5x–15+8

x+3

C) 5x4–3x3+2x2–6x+15+14

x+3D) 5x4–3x3+2x2–6x–15+14

x+3

3 EvaluateaPolynomialUsingtheRemainderTheorem

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

UsesyntheticdivisionandtheRemainderTheoremtofindtheindicatedfunctionvalue.

1) f(x)=x4–9x3–8x2–9x–2;f(–3)

A) 277 B) –831 C) –277 D) 196

2) f(x)=2x3–7x2–5x+11;f(–3)

A) –91 B) –17 C) 57 D) –121

3) f(x)=6x4+2x3+3x2–4x+40;f(3)

A) 595 B) 377 C) 813 D) 1567

4) f(x)=x5–9x4+4x3+2;f(–3)

A) –1078 B) 1078 C) –118 D) –835

5) f(x)=x4+8x3–2x2+4x–5;f–1

4

A) –1599

256 B) 1599

1024 C) 1599

256 D) –25

4

Page68

4 UsetheFactorTheoremtoSolveaPolynomialEquation

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

1) Usesyntheticdivisiontodividef(x)=x3–1x2–52x+160byx+8.Usetheresulttofindallzerosoff.

A) {–8

,

4

,

5} B) {–8,–4

,

–5} C) {8

,

–4

,

–5} D) {8

,

4

,

5}

2) Solvetheequation3x3–31x2+82x–24=0giventhat4isazerooff(x)=3x3–31x2+82x–24.

A) 4,6,1

3B) 4,–6,–1

3C) 4

,

1

,

2 D) 4

,

–1

,

–2

3) Solvetheequation8x3–34x2+5x+12=0giventhat–1

2isaroot.

A) –1

2,3

4,4 B) –1

2,–3

4,–4 C) –1

2,1,3 D) –1

2,–1,–3

Usesyntheticdivisiontoshowthatthenumbergiventotherightoftheequationisasolutionoftheequation,then

solvethepolynomialequation.

4) x3+6x2+5x–12=0;–3

A) {1

,

–4

,

–3} B) {–1

,

–4

,

–3} C) {1

,

4

,

–3} D) {–1

,

4

,

–3}

5) 2x3–5x2–21x+36=0;4

A) 3

2,–3,4 B) –3

2,–3,4 C) 3

2,3,4 D) –3

2,3,4

6) 2x3–13x2+17x+12=0;3

A) –1

2,4,3 B) 1

2,4,3 C) –1

2,–4,3 D) 2

,

–1

,

3

7) 6x3+11x2–92x+15=0;3

A) 1

6,–5,3 B) –1

6,–5,3 C) 1

6,5,3 D) –5

6,1,3

Page69

Usethegraphortabletodetermineasolutionoftheequation.Usesyntheticdivisiontoverifythatthisnumberisa

solutionoftheequation.Thensolvethepolynomialequation.

8) x3+6x2+11x+6=0

x

–1 12345

y

5

4

3

2

1

-1

-2

-3

-4

x

–1 12345

y

5

4

3

2

1

-1

-2

-3

-4

A) –1;Theremainderiszero;–1,–2,and–3,or{–3,–2,–1}

B) –1;Theremainderiszero;1,–2,and–3,or{–3,–2,1}

C) –1;Theremainderiszero;–1,2,and–3,or{–3,–1,2}

D) –1;Theremainderiszero;–1,–2,and3,or{–2,–1,3}

9) x3+9x2+26x+24=0

x

–2–1 1234

y

5

4

3

2

1

-1

-2

-3

-4

x

–2–1 1234

y

5

4

3

2

1

-1

-2

-3

-4

A) –2;Theremainderiszero;–2,–3,and–4,or{–4,–3,–2}

B) –2;Theremainderiszero;2,–3,and–4,or{–4,–3,2}

C) –2;Theremainderiszero;–2,3,and–4,or{–4,–2,3}

D) –2;Theremainderiszero;–2,–3,and4,or{–3,–2,4}

Page70

10) 2x3+11x2+17x+6=0

xy1

–20

–1–2

06

136

2 100

3 210

A) –2;Theremainderiszero;–3,–2,and–1

2,or–3,–2,–1

2

B) –2;Theremainderiszero;3,–2,and–1

2,or–2,–1

2,3

C) –2;Theremainderiszero;–3,2,and–1

2,or–3,–1

2,2

D) –2;Theremainderiszero;–3,–2,and1

2,or–3,–2, 1

2

3.4 ZerosofPolynomialFunctions

1 UsetheRationalZeroTheoremtoFindPossibleRationalZeros

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

UsetheRationalZeroTheoremtolistallpossiblerationalzerosforthegivenfunction.

1) f(x)=x5–6x2+3x+3

A) ±1,±3B)

±1,±1

3C) ±1

6,±1

2,±3D)

±3,±1

3

2) f(x)=x5–6x2+4x+21

A) ±1,±7

,

±3

,

±21 B) ±1,±1

7,±1

3,±1

21

C) ±1,±1

7,±1

3,±1

21 ,±7,±3,±21 D) ±1,±7

,

±3

3) f(x)=x4+3x3–6x2+5x–12

A) ±1,±2,±3,±4,±6,±12

B) ±1,±1

2,±1

3,±1

4,±1

6,±1

12

C) ±1

2,±1

3,±1

4,±1

6,±1

12 ,±1,±2,±3,±4,±6,±12

D) ±1

12 ,±1,±12

4) f(x)=–2x3+3x2–4x+8

A) ±1

2,±1,±2,±4,±8B)

±1

4,±1

2,±1,±2,±4,±8

C) ±1

8,±1

4,±1

2,±1,±2,±4,±8D)

±1

2,±1,±2,±4

Page71

5) f(x)=7x3–x2+2

A) ±1

7,±2

7,±1,±2B)

±1

2,±7

2,±1,±7

C) ±1

7,±2

7,±1,±2,±7D)

±1

7,±1

2,±1,±2,±7

6) f(x)=6x4+3x3–2x2+2

A) ±1

6,±1

3,±1

2,±2

3,±1,±2B)

±1

6,±1

3,±1

2,±2

3,±1,±2,±3

C) ±1

6,±1

3,±1

2,±1,±2D)

±1

2,±3

2,±1,±2,±3,±6

7) f(x)=–4x4+3x2–2x+6

A) ±1

4,±1

2,±3

4,±3

2,±1,±2,±3,±6B)±1

6,±1

2,±1

3,±2

3,±4

3,±1,±2,±4

C) ±1

4,±1

2,±3

4,±3

2,±1,±2,±3,±4,±6D)±1

4,±1

2,±2

3,±3

4,±3

2,±1,±2,±3,±6

8) f(x)=7x5–4x2+5x–1

A) ±1,±1

7B) ±1,±7C)

±1,±7,±1

7D) ±7,±1

7

9) f(x)=6x4+3x3–4x2+3x–5

A) ±1,±5,±1

2,±5

2,±1

3,±5

3,±1

6,±5

6B) ±1,±2,±3,±6,±1

5,±2

5,±3

5,±6

5

C) ±1,±2,±3,±6,±1

2,±5

2,±1

3,±5

3,±1

6,±5

6D) ±1,±5,±1

5,±2

5,±3

5,±6

5

10) f(x)=3x4+7x3–5x2+5x–12

A) ±1,±2,±3,±4,±6,±12,±1

3,±2

3,±4

3

B) ±1,±3,±1

2,±3

2,±1

3,±1

4,±3

4,±1

6,±1

12

C) ±1,±2,±3,±4,±6,±12,±1

2,±3

2,±1

3,±1

4,±3

4,±1

6,±1

12

D) ±1,±2,±3,±6,±12,±1

3,±2

3,±3

4

2 FindZerosofaPolynomialFunction

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findarationalzeroofthepolynomialfunctionanduseittofindallthezerosofthefunction.

1) f(x)=x3+2x2–9x–18

A) {–3

,

–2

,

3} B) {–3

,

2

,

3} C) {–3} D) {–2}

2) f(x)=3x3–17x2+18x+8

A) –1

3,2,4 B) 1

3,2,–4 C) 4

3,–1,2 D) –4

3,–1,–2

Page72

3) f(x)=x3–8x2–x+8

A) {1,–1,8} B) {–1,2

,

–4} C) {1,2

,

4} D) {1,–1,–8}

4) f(x)=x3+8x2+14x+4

A) {–2,–3+7,–3–7} B) {1,–1,–4}

C) {2,–6+7,–6–7}D){

–2,–6+4,–6–4}

5) f(x)=x3+6x2+21x+26

A) {–2

,

–2+3i,–2–3i} B) {–2

,

3+2i,3–2i}

C) {2,–2+5,–4–5}D){

–2,3+5,3–5}

6) f(x)=3x3–x2–18x+6

A) { 1

3,6,–6}B){

–1

3,6,–6}C){3,6,–6}D){

–3,6,–6}

7) f(x)=x4+4x3–11x2–26x–12

A) {–1,3,–3+5,–3–5} B) {1,–3,–3+5,–3–5}

C) {–1,4,–3+2,–3–2}D){

–1,–3,–3+2,–3–2}

8) f(x)=x4–2x3+17x2+18x–234

A) {–3

,

3

,

1+5i,1–5i} B) {3

,

–3

,

1+5i,1–5i}

C) {–3

,

3

,

1+6i,1–6i} D) {3,–3,1+5,1–5}

9) f(x)=2x4–19x3+71x2–109x+39

A) {3,1

2,3+2i,3–2i} B) {–3,–1

2,2+3i,2–3i}

C) {3,–1

2,2+3i,2–3i} D) {–3,1

2,3+2i,3–2i}

10) f(x)=3x4+29x3+111x2+179x+78

A) {–3,–2

3,–3+2i,–3–2i} B) {3,+2

3,–2+3i,–2–3i}

C) {–3,+2

3,–2+3i,–2–3i} D) {3,–2

3,–3+2i,–3–2i}

3 SolvePolynomialEquations

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethepolynomialequation.Inordertoobtainthefirstroot,usesyntheticdivisiontotestthepossiblerationalroots.

1) x3+2x2–9x–18=0

A) {–3

,

–2

,

3} B) {–3

,

2

,

3} C) {–3} D) {–2}

2) 2x3–13x2+22x–8=0

A) 1

2,2,4 B) –1

2,2,–4 C) 2

,

1

,

2 D) –2

,

1

,

–2

3) x3–3x2–x+3=0

A) {1,–1,3} B) {–1,1

,

–3} C) {1,1

,

3} D) {1,–1,–3}

Page73

4) x3–6x2+7x+2=0

A) {2,2+5,2–5} B) {1,–1,–2}

C) {–2,4+5,4–5}D){2,4+2,4–2}

5) x3–5x2+17x–13=0

A) {1

,

2+3i,2–3i} B) {1

,

3+2i,3–2i}

C) {–1,2+5,4–5}D){1,3+5,3–5}

6) x3+6x2–14x+16=0

A) {1+i,1–i,–8} B) {1+i,1–i,8} C) {–8

,

8} D) {1+i,1–i,8i}

7) 3x3–x2–21x+7=0

A) { 1

3,7,–7}B){

–1

3,7,–7}C){3,7,–7}D){

–3,7,–7}

8) x4+3x3–15x2–45x–28=0

A) {–1,4,–3+2,–3–2} B) {1,–4,–3+2,–3–2}

C) {–1,5,–3+3,–3–3}D){

–1,–4,–3+3,–3–3}

9) x4–3x3+2x2+16x–16=0

A) {–2

,

1

,

2+2i,2–2i} B) {2

,

–1

,

2+2i,2–2i}

C) {–2

,

1

,

2+3i,2–3i} D) {2,–1,2+2,2–2}

10) 2x4–13x3+49x2–77x+39=0

A) {1,3

2,2+3i,2–3i} B) {–1,–3

2,3+2i,3–2i}

C) {1,–3

2,3+2i,3–2i} D) {–1,3

2,2+3i,2–3i}

11) 3x4+23x3+71x2+77x+26=0

A) {–1,–2

3,–3+2i,–3–2i} B) {1,+2

3,–2+3i,–2–3i}

C) {–1,+2

3,–2+3i,–2–3i} D) {1,–2

3,–3+2i,–3–2i}

Solvetheproblem.

12) Theconcentration,inpartspermillion,ofaparticulardruginapatientʹsbloodxhoursafterthedrugis

administeredisgivenbythefunction

f(x)=–x4+11x3–41x2+55x

Howmanyhoursafterthedrugisadministeredwillitbeeliminatedfromthebloodstream.

A) 5hours B) 11hours C) 4 hours D) 16hours

Page74

13) Aboxwithanopentopisformedbycuttingsquaresoutofthecornersofarectangularpieceofcardboardand

thenfoldingupthesides.Ifxrepresentsthelengthofthesideofthesquarecutfromeachcorner,andifthe

originalpieceofcardboardis20inchesby14inches,whatsizesquaremustbecutifthevolumeoftheboxisto

be288cubicinches?

A) 4in.by4in.square B) 3 in.by3 in.square

C) 12in.by12in.square D) 6 in.by6 in.square

14) Thepolynomialfunction

H(x)=–0.001183x4+0.05495x3–0.8523x2+9.054x+6.748

modelstheageinhumanyears,H(x),ofadogthatisxyearsold,wherex≥1.Usingthegraphofthisfunction

shownbelow,whatistheapproximatelyequivalentdogageforapersonwhois 60?

A) 11years B) 9years C) 8.5 years D) 12.5 years

4 UsetheLinearFactorizationTheoremtoFindPolynomialswithGivenZeros

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findannthdegreepolynomialfunctionwithrealcoefficientssatisfyingthegivenconditions.

1) n=3;3andiarezeros;f(2)=15

A) f(x)=–3x3+9x2–3x+9 B) f(x)=3x3–9x2+3x–9

C) f(x)=3x3–9x2–3x+9 D) f(x)=–3x3+9x2+3x–9

2) n=3;–5andiarezeros;f(–3)=60

A) f(x)=3x3+15x2+3x+15 B) f(x)=3x3+15x2–3x–15

C) f(x)=–3x3–15x2–3x–15 D) f(x)=–3x3–15x2+3x+15

3) n=3;–1and3+2iarezeros;leadingcoefficientis1

A) f(x)=x3–5x2+7x+13 B) f(x)=x3–4x2+7x+13

C) f(x)=x3–5x2+15x+13 D) f(x)=x3+5x2+7x–14

Page75

4) n=4;3,1

2,and3+2iarezeros;f(1)=32

A) f(x)=–4x4+38x3–142x2+218x–78 B) f(x)=2x4–19x3+71x2+218x–78

C) f(x)=–2x4+38x3–142x2+218x–78 D) f(x)=–6x4+57x3–213x2+327x–117

5) n=4;2i,3

,

and–3arezeros;leadingcoefficientis1

A) f(x)=x4–5x2–36 B) f(x)=x4+4x3–5x2–36

C) f(x)=x4+4x2–36 D) f(x)=x4+4x2–3x–36

5 UseDescartesʹsRuleofSigns

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

UseDescartesʹsRuleofSignstodeterminethepossiblenumberofpositiveandnegativerealzerosforthegiven

function.

1) f(x)=–6x9+x5–x2+8

A) 3or1positivezeros,2or0negativezeros B) 3or1positivezeros,3or1negativezeros

C) 2or0positivezeros,2or0negativezeros D) 2or0positivezeros,3or1negativezeros

2) f(x)=7x3–6x2+x+3.5

A) 2or0positivezeros,1negativezero B) 3or1positivezeros,1negativezero

C) 2or0positivezeros,nonegativezeros D) 3or1positivezeros,2or0negativezeros

3) f(x)=5x7–3x2+x+7

A) 2or0positivezeros,1negativezero B) 3or1positivezeros,3or1negativezeros

C) 2or0positivezeros,1or0negativezeros D) 2or0positivezeros,2or0negativezeros

4) f(x)=x7+x6+x2+x+4

A) 0positivezeros,3or1negativezeros B) 0positivezeros,0negativezeros

C) 0positivezeros,2or0negativezeros D) 0positivezeros,1negativezero

5) f(x)=x5–2.1x4–14.44x3+3x2+41.67x–15.216

A) 3or1positivezeros,2or0negativezeros B) 2or0positivezeros,2or0negativezeros

C) 3or1positivezeros,3or1negativezeros D) 2or0positivezeros,3or1negativezeros

6) f(x)=x2–14

A) 1positivezero,1negativezero B) 1positivezero,0negativezeros

C) 0positivezeros,0negativezeros D) 0positivezeros,1negativezero

7) f(x)=6x6–10x5+x4–3x3+20

A) 4,2or0positivezeros,nonegativezeros B) 4or2positivezeros,nonegativezeros

C) 4,2or0positivezeros,1negativezeros D) 4positivezeros,nonegativezeros

8) f(x)=–4x5–10x4–5x3+3x2+x+20

A) 1positivezero,4,3or1negativezeros B) 1positivezero,2or0negativezeros

C) 1positivezero,4or2negativezeros D) 1positivezero,3or1negativezeros

Page76

3.5 RationalFunctionsandTheirGraphs

1 FindtheDomainsofRationalFunctions

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findthedomainoftherationalfunction.

1) g(x)=2x

x–1

A) {x|x≠1} B) {x|x≠–1} C) {x|x≠0} D) allrealnumbers

2) f(x)=9x

(x+5)(x+3)

A) {x|x≠–5

,

x≠–3} B) {x|x≠5

,

x≠3}

C) {x|x≠–5

,

x≠–3

,

x≠–9} D) allrealnumbers

3) h(x)=x+9

x2–25

A) {x|x≠–5

,

x≠5} B) {x|x≠–5

,

x≠5

,

x≠–9}

C) {x|x≠0,x≠25} D) allrealnumbers

4) h(x)=x+7

x2+64

A) allrealnumbers B) {x|x≠–8

,

x≠8

,

x≠–7}

C) {x|x≠0,x≠–64} D) {x|x≠–8

,

x≠8}

5) f(x)=x+8

x2–4x

A) {x|x≠0,x≠4} B) {x|x≠–2

,

x≠2

,

x≠–8}

C) allrealnumbers D) {x|x≠–2

,

x≠2}

2 UseArrowNotation

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Usethegraphoftherationalfunctionshowntocompletethestatement.

1)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Asx→–2–,f(x)→?

A) –∞B) +∞C) 0 D) 2

Page77

2)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Asx→3+,f(x)→?

A) –∞B) +∞C) 0 D) 3

3)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Asx→–∞,f(x)→?

A) 0 B) +∞C) –∞D) –1

4)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Asx→3–,f(x)→?

A) –∞B) +∞C) 0 D) –3

Page78

5)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Asx→–2–,f(x)→?

A) +∞B) –∞C) 0 D) 2

6)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Asx→0–,f(x)→?

A) –∞B) +∞C) 1 D) –0

7)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Asx→3–,f(x)→?

A) –∞B) +∞C) 2 D) –3

Page79

8)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Asx→+∞,f(x)→?

A) 1 B) +∞C) –∞D) –1

9)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Asx→0+,f(x)→?

A) +∞B) –∞C) –1D)1

3 IdentifyVerticalAsymptotes

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findtheverticalasymptotes,ifany,ofthegraphoftherationalfunction.

1) g(x)=x

x+3

A) x=–3B)x=0andx= –3

C) x=0andx=3D)noverticalasymptote

2) g(x)=x+1

x(x–1)

A) x=0andx=1B)x=1

C) x=–1andx=1D)noverticalasymptote

3) h(x)=x

x(x+1)

A) x=–1B)x=0andx= –1

C) x=0andx=1D)noverticalasymptote

Page80

4) f(x)=x

x2+7

A) x=–7B)x= –7

,

x=7

C) x=7D)noverticalasymptote

5) g(x)=x

x2–25

A) x=5

,

x=–5B)x=5

,

x= –5

,

x=0

C) x=5D)noverticalasymptote

6) h(x)=x+2

x2–4

A) x=2B)x= –2

C) x=2

,

x=–2D)noverticalasymptote

7) x–9

x2–10x+24

A) x=6

,

x=4B)x=–6

,

x= –4C)x=6

,

x=4

,

x= – 9D)x= – 9

4 IdentifyHorizontalAsymptotes

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findthehorizontalasymptote,ifany,ofthegraphoftherationalfunction.

1) f(x)=4x

2x2+1

A) y=0B)y=2

C) y=1

2D) nohorizontalasymptote

2) g(x)=12x2

3x2+1

A) y=4B)y=0

C) y=1

4D) nohorizontalasymptote

3) h(x)=15x3

3x2+1

A) y=5B)y=0

C) y=1

5D) nohorizontalasymptote

4) f(x)=8x

8x+8

A) y=1B)y= – 1

C) y=0D)nohorizontalasymptote

Page81

5) f(x)=–4x–7

5x+6

A) y=–4

5B) y=–7

6

C) y=–4D)nohorizontalasymptote

6) g(x)=8x2–2x–3

9x2–5x+3

A) y=8

9B) y=0

C) y=2

5D) nohorizontalasymptote

7) f(x)=–20x

5x3+x2+1

A) y=0B)y= –4

C) y=–1

4D) nohorizontalasymptote

Page82

5 UseTransformationstoGraphRationalFunctions

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Usetransformationsoff(x)=1

xorf(x)=1

x2tographtherationalfunction.

1) f(x)=1

x–5

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

B)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

C)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

D)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Page83

2) f(x)=1

x–4

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

A)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

B)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

C)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

D)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

Page84

3) f(x)=1

x–5+3

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page85

4) f(x)=1

(x–5)2

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

B)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

C)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

D)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Page86

5) f(x)=1

x2–2

x

-8 -4 4 8

y

8

4

-4

-8

x

-8 -4 4 8

y

8

4

-4

-8

A)

x

-8 -4 4 8

y

8

4

-4

-8

x

-8 -4 4 8

y

8

4

-4

-8

B)

x

-8 -4 4 8

y

8

4

-4

-8

x

-8 -4 4 8

y

8

4

-4

-8

C)

x

-8 -4 4 8

y

8

4

-4

-8

x

-8 -4 4 8

y

8

4

-4

-8

D)

x

-8 -4 4 8

y

8

4

-4

-8

x

-8 -4 4 8

y

8

4

-4

-8

Page87

6) f(x)=1

(x–4)2+3

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page88

6 GraphRationalFunctions

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Graphtherationalfunction.

1) f(x)=2x

x+3

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page89

2) f(x)=4x

x2–36

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

–20 -10 10 20

y

20

10

-10

-20

x

–20 -10 10 20

y

20

10

-10

-20

Page90

3) f(x)=2x2

x2–25

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page91

4) f(x)=–4x

x+3

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page92

5) f(x)=–3

x2–9

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page93

6) f(x)=6

x2+4x+4

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page94

7) f(x)=3x2

x2+4

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page95

8) f(x)=x2+3x–4

x2–1



x

y

x

y

A)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

B)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

C)

x

-30 -20 -10 10 20 30

y

60

40

20

-20

-40

-60

x

-30 -20 -10 10 20 30

y

60

40

20

-20

-40

-60

D)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

Page96

9) f(x)=x4

x2+25

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

5

-5

x

-10 -5 5 10

y

5

-5

Page97

10) f(x)=x–2

x2–x–56

x

-12-10 -8 -6 -4 -2 2 4 6 8 10 12

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

-12-10 -8 -6 -4 -2 2 4 6 8 10 12

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

A)

x

-12-10 -8 -6 -4 -2 2 4 6 8 10 12

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

-12-10 -8 -6 -4 -2 2 4 6 8 10 12

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

B)

x

-12-10 -8 -6 –4 -2 2 4 6 8 10 12

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

-12-10 -8 -6 –4 -2 2 4 6 8 10 12

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

C)

x

-12-10 -8 -6 -4 -2 2 4 6 8 10 12

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

-12-10 -8 -6 -4 -2 2 4 6 8 10 12

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

D)

x

-12-10 -8 -6 –4 -2 2 4 6 8 10 12

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

-12-10 -8 -6 –4 -2 2 4 6 8 10 12

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

Page98

11) f(x)=x2

x2–x–56

x

y

x

y

A)

x

-16 -8 8 16

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

-16 -8 8 16

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

B)

x

-20 -16 -12 -8 -4 4 8 12 16 20

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

-20 -16 -12 -8 -4 4 8 12 16 20

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

C)

x

-12-10 -8 -6 -4 -2 2 4 6 8 10 12

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

-12-10 -8 -6 -4 -2 2 4 6 8 10 12

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

D)

x

-20 -16 -12 -8 -4 4 8 12 16 20

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

-20 -16 -12 -8 -4 4 8 12 16 20

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

Page99

12) f(x)=x2–x–56

x2–1

x

y

x

y

A)

x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

90

60

30

-30

-60

-90

x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

90

60

30

-30

-60

-90

B)

x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

90

60

30

-30

-60

-90

x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

90

60

30

-30

-60

-90

C)

x

-12-10 -8 -6 -4 -2 2 4 6 8 10 12

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

-12-10 -8 -6 -4 -2 2 4 6 8 10 12

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

D)

x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

Page100

13) f(x)=x2–3x

(x–2)2

x

y

x

y

A)

x

-12 -8 -4 4 8 12

y

12

8

4

-4

-8

-12

x

-12 -8 -4 4 8 12

y

12

8

4

-4

-8

-12

B)

x

-12 -8 -4 4 8 12

y

12

8

4

-4

-8

-12

x

-12 -8 -4 4 8 12

y

12

8

4

-4

-8

-12

C)

x

-24 -16 -8 8 16 24

y

30

20

10

-10

-20

-30

x

-24 -16 -8 8 16 24

y

30

20

10

-10

-20

-30

D)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

Page101

14) f(x)=x2–2x+1

(x–5)2

x

-12 -8 -4 4 8 12

y

12

8

4

-4

-8

-12

x

-12 -8 -4 4 8 12

y

12

8

4

-4

-8

-12

A)

x

-12 -8 -4 4 8 12

y

12

8

4

-4

-8

-12

x

-12 -8 -4 4 8 12

y

12

8

4

-4

-8

-12

B)

x

-12 -8 -4 4 8 12

y

12

8

4

-4

-8

-12

x

-12 -8 -4 4 8 12

y

12

8

4

-4

-8

-12

C)

x

-30 -20 -10 10 20 30

y

30

20

10

-10

-20

-30

x

-30 -20 -10 10 20 30

y

30

20

10

-10

-20

-30

D)

x

-12 -8 -4 4 8 12

y

12

8

4

-4

-8

-12

x

-12 -8 -4 4 8 12

y

12

8

4

-4

-8

-12

Findtheindicatedintercept(s)ofthegraphofthefunction.

15) x–interceptsoff(x)=x–9

x2+7x–3

A) (9

,

0) B) (3

,

0) C) (7

,

0) D) none

16) x–interceptsoff(x)=x2+3

x2+9x+3

A) (3

,

0) B) ( 3,0),(–3,0) C) (3

,

0) D) none

Page102

17) x–interceptsoff(x)=x+5

x2+5x–3

A) (–5

,

0) B) (5

,

0) C) 5

3,0 D) none

18) x–interceptsoff(x)=x2+3x

x2+3x–9

A) (0,0)and(–3

,

0) B) (–3

,

0) C) (0,0)and(3

,

0) D) (3

,

0)

19) x–interceptsoff(x)=(x–7)(2x+5)

x2+2x–5

A) (7,0)and–5

2,0 B) (–7,0)and5

2,0 C) (7

,

0)and(–5

,

0) D) none

20) y–interceptoff(x)=x–3

x2+3x–2

A) 0,3

2B) (0,3) C) 0,–2

3D) none

21) y–interceptoff(x)=x2–2x

x2+6x–7

A) (0,0) B) 0,2

7C) (0,2) D) 0,–7

2

22) y–interceptoff(x)=x2–15

x2+8x–14

A) 0,15

14 B) (0,15) C) 0,–14

15 D) none

23) y–interceptoff(x)=x2–7x+7

10x

A) 0,7

10 B) (0,7) C) 0,–10

7D) none

Solvetheproblem.

24) Istherey–axissymmetryfortherationalfunctionf(x)=8x2

5x4–5

?

A) Yes B) No

25) Istherey–axissymmetryfortherationalfunctionf(x)=–6x2

–2x3–19

?

A) Yes B) No

26) Istherey–axissymmetryfortherationalfunctionf(x)=–8x2–8x–12

6x+14 ?

A) Yes B) No

Page103

27) Isthereoriginsymmetryfortherationalfunctionf(x)=6x

9x2+10

?

A) Yes B) No

28) Isthereoriginsymmetryfortherationalfunctionf(x)=4x2+2

–9x ?

A) Yes B) No

29) Isthereoriginsymmetryfortherationalfunctionf(x)=9x2–2

–8x2+1

?

A) Yes B) No

7 IdentifySlantAsymptotes

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findtheslantasymptote,ifany,ofthegraphoftherationalfunction.

1) f(x)= x2+16

x

A) y=xB)y=x+16

C) x=0D)noslantasymptote

2) f(x)=x2+4x–4

x–4

A) y=x+8B)y=x+4

C) y=xD)noslantasymptote

3) f(x)=8x2

4x2+6

A) y=8x B) y=x+8

C) y=x+2D)noslantasymptote

4) f(x)=x2–6x+7

x+6

A) y=x–12 B) y=x+13

C) x=y+6D)noslantasymptote

5) g(x)=x3+4

x2–25

A) y=xB)y=x+4

C) y=x–25 D) noslantasymptote

6) f(x)=x3+5

x2+9x

A) y=x–9B)y=x+9C)y=x+5D)y=x

Page104

Graphthefunction.

7) f(x)= x2+9

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page105

8) f(x)=x2+4x–6

x–9

x

y

x

y

A)

x

–25-20-15-10 –5 5 10 15 20 25

y

60

50

40

30

20

10

-10

-20

-30

-40

-50

-60

x

–25-20-15-10 –5 5 10 15 20 25

y

60

50

40

30

20

10

-10

-20

-30

-40

-50

-60

B)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

C)

x

–25-20-15-10 –5 5 10 15 20 25

y

50

40

30

20

10

-10

-20

-30

-40

-50

x

–25-20-15-10 –5 5 10 15 20 25

y

50

40

30

20

10

-10

-20

-30

-40

-50

D)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Page106

9) f(x)=x3+4

x2–2x



x

y

x

y

A)

x

-16 -8 8 16

y

24

16

8

-8

-16

-24

x

-16 -8 8 16

y

24

16

8

-8

-16

-24

B)

x

-8 -4 4 8

y

8

4

-4

-8

x

-8 -4 4 8

y

8

4

-4

-8

C)

x

-8 -4 4 8

y

8

4

-4

-8

x

-8 -4 4 8

y

8

4

-4

-8

D)

x

-16 -8 8 16

y

16

8

-8

-16

x

-16 -8 8 16

y

16

8

-8

-16

Page107

8 SolveAppliedProblemsInvolvingRationalFunctions

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

1) AcompanythatproducesradioshascostsgivenbythefunctionC(x)=20x+15,000,wherexisthenumberof

radiosmanufacturedandC(x)ismeasuredindollars.Theaveragecosttomanufactureeachradioisgivenby

_

C (x)=20x+15,000

x.

Find

_

C (50).(Roundtothenearestdollar,ifnecessary.)

A) $320 B) $310 C) $50 D) $51

2) AcompanythatproducesinflatableraftshascostsgivenbythefunctionC(x)=15x+25,000

,

wherexisthe

numberofinflatableraftsmanufacturedandC(x)ismeasuredindollars.Theaveragecosttomanufactureeach

inflatableraftisgivenby

_

C (x)=15x+25,000

x.

Whatisthehorizontalasymptoteforthefunction

_

C?Describewhatthismeansinpracticalterms.

A) y=15;$15istheleastpossiblecostforproducingeachinflatableraft.

B) y=25,000;25,000isthemaximumnumberofinflatableraftsthecompanycanproduce.

C) y=15;15istheminimumnumberofinflatableraftsthecompanycanproduce.

D) y=25,000;$25,000istheleastpossiblecostforrunningthecompany.

3) Adrugisinjectedintoapatientandtheconcentrationofthedrugismonitored.Thedrugʹsconcentration,C(t),

inmilligramsafterthoursismodeledby

C(t)=5t

2t2+2.

Whatisthehorizontalasymptoteforthisfunction?Describewhatthismeansinpracticalterms.

A) y=0;0isthefinalamount,inmilligrams,ofthedrugthatwillbeleftinthepatientʹsbloodstream.

B) y=2.50;2.50isthefinalamount,inmilligrams,ofthedrugthatwillbeleftinthepatientʹsbloodstream.

C) y=1.25;After1.25hours,theconcentrationofthedrugisatitsgreatest.

D) y=2.50;After2.50hours,theconcentrationofthedrugisatitsgreatest.

4) Adrugisinjectedintoapatientandtheconcentrationofthedrugismonitored.Thedrugʹsconcentration,C(t),

inmilligramsperliterafterthoursismodeledby

C(t)=7t

2t2+2.

Estimatethedrugʹsconcentrationafter2hours.(Roundtothenearesthundredth.)

A) 1.40milligramsperliter B) 1.51 milligramsperliter

C) 2.33milligramsperliter D) 2.44 milligramsperliter

5) Therationalfunction

C(x)=125x

100–x,0≤x<100

describesthecost,C,inmillionsofdollars,toinoculatex%ofthepopulationagainstaparticularstrainofthe

flu.Determinethedifferenceincostbetweeninoculating75%ofthepopulationandinoculating50%ofthe

population.(Roundtothenearesttenth,ifnecessary.)

A) $250.0million B) $0.8 million C) $250.1 million D) $0.9 million

Page108

3.6 PolynomialandRationalInequalities

1 SolvePolynomialInequalities

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethepolynomialinequalityandgraphthesolutionsetonanumberline.Expressthesolutionsetininterval

notation.

1) (x–7)(x+4)>0

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

A) (–∞

,

–4)∪(7

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

B) (–∞

,

–7)∪(4

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

C) (–4

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

D) (–4

,

7)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

2) (x+7)(x–4)≤0

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

A) [–7

,

4]

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

B) (–7

,

4)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

C) (–∞

,

–7]∪[4

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

D) (–∞

,

–7)∪(4

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

3) x2–8x+12>0

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

A) (–∞

,

2)∪(6

,

∞)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

B) (2

,

6)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

C) (–∞

,

2)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

D) (6

,

∞)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

Page109

4) x2–3x–28<0

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

A) (–4

,

7)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

B) (–∞

,

–4)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

C) (7

,

∞)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

D) (–∞

,

–4)∪(7

,

∞)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

5) x2–4x–12≤0

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

A) [–2

,

6]

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

B) (–∞

,

–2]

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

C) [6

,

∞)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

D) (–∞

,

–2]∪[6

,

∞)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

Page110

6) x2+11x+30≥0

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

A) (–∞

,

–6]∪[–5

,

∞)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

B) [–6

,

–5]

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

C) (–∞

,

–6]

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

D) [–5

,

∞)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

7) x2+6x≤–8

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

A) [–4

,

–2]

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

B) (–∞

,

2]∪[4

,

∞)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

C) (2

,

4)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

D) [2

,

4]

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

Page111

8) x2+5x≥–6

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

A) (–∞

,

–3]∪[–2

,

∞)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

B) [–3

,

–2]

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

C) (–∞

,

–3]

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

D) [–2

,

∞)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

9) x2+6x+9>0

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

A) (–∞

,

–3)∪(–3

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

B) (–∞

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

C) (–3

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

D) (–∞

,

–3)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

10) 3x2+14x–24≤0

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

A) –6,4

3

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

B) (–∞,–6]∪

4

3,∞

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

C) –∞, 4

3

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

D) [–6

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

Page112

11) x2+9x≥0

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

A) (–∞

,

–9]∪[0,∞]

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

B) [–9

,

0]

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

C) (–∞

,

–9]

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

D) [0,∞]

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

12) 19x2–5x≤0

A) 0,5

19

-1 0 1-1 0 1

B) (–∞,0]∪ 5

19 ,∞

-1 0 1-1 0 1

C) –5

19 ,0

-1 0 1-1 0 1

D) 0,19

5

-7 -6 -5 -4 -3 -2 –1 0 1 2 3 4 5 6 7-7 -6 -5 –4 -3 -2 –1 0 1 2 3 4 5 6 7

13) (x+5)(x+2)(x–4)>0

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

A) (–5

,

–2)∪(4

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

B) (–∞

,

–5)∪(–2

,

4)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

C) (4

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

D) (–∞

,

–2)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

Page113

14) (x+6)(x+4)(x+2)<0

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

A) (–∞

,

–6)∪(–4

,

–2)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

B) (–2

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

C) (–∞

,

–4)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

D) (–6

,

–4)∪(–2

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

15) (4x–3)(x+5)≤0

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 –4 -3 -2 -1 0 1 2 3 4 5 6 7

A) –5,3

4

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

B) (–∞,–5]∪3

4,∞

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

C) –∞,3

4

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

D) [–5

,

∞)

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

16) (6z+1)(3z–8)>0

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

A) –∞,–1

6∪8

3,∞

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

B) 8

3,∞

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

C) –1

6,8

3

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

D) –1

6,8

3

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Page114

17) 3x2–4x≥7

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

A) (–∞,–1]∪7

3,∞

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

B) (–∞,–1)∪7

3,∞

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

C) –1,7

3

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

D) –1,7

3

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

18) 18x2<17x+1

-1 0 1-1 0 1

A) –1

18 ,1

-1 0 1-1 0 1

B) –1,1

18

-1 0 1-1 0 1

C) –∞,–1

18 ∪(1,∞)

-1 0 1-1 0 1

D) (–∞,–1)∪1

18 ,∞

-1 0 1-1 0 1

Page115

19) x<110–x2

–14–12–10–8–6–4–202468101214–14–12–10–8–6–4–202468101214

A) (–11

,

10)

–14 –12 –10 –8 –6 -4 –2 0 2 4 6 8 10 12 14–14 –12 –10 –8 –6 –4 –2 0 2 4 6 8 10 12 14

B) (–10

,

11)

–14 –12 –10 -8 –6 –4 –2 0 2 4 6 8 10 12 14–14 –12 –10 -8 –6 –4 –2 0 2 4 6 8 10 12 14

C) (–∞

,

–11)∪(10

,

∞)

–14 –12 –10 –8 –6 -4 –2 0 2 4 6 8 10 12 14–14 –12 –10 –8 –6 –4 –2 0 2 4 6 8 10 12 14

D) (–∞

,

10)∪(11

,

∞)

–1012345678910111213–1012345678910111213

20) x3+3x2–x–3>0

A) (–3

,

–1)∪(1,∞)

–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2 3–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2 3

B) (–∞

,

–3)∪(–1,1)

–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2 3–12–11–10–9-8-7-6-5-4-3-2-1 0 1 2 3

C) (–1,1)∪(3

,

∞)

–4–3–2–101234567891011–4–3–2–101234567891011

D) (–∞

,

–1)∪(1,3)

–4–3–2–101234567891011–4–3–2–101234567891011

Page116

21) 9x3+27x2–16x–48>0

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

A) –3,–4

3∪4

3,∞

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

B) 4

3,∞

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

C) (–∞,–3)∪–4

3,4

3

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

D) (–∞,–3]∪–4

3,4

3

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

2 SolveRationalInequalities

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetherationalinequalityandgraphthesolutionsetonarealnumberline.Expressthesolutionsetininterval

notation.

1) x–3

x+1<0

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

A) (–1

,

3)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

B) (–∞

,

–1)or(3

,

∞)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

C) (3

,

∞)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

D) (–∞

,

–1)

–9–8–7–6–5–4–3–2–10123456789–9–8–7–6–5–4–3–2–10123456789

Page117

2) x–2

x+5>0

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

A) (–∞

,

–5)or(2

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

B) (–5

,

2)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

C) (2

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

D) (–∞

,

–5)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

3) –x+4

x–2≥0

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

A) (2

,

4]

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

B) (–∞

,

2)or [4

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

C) [2

,

4]

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

D) (–∞

,

4]

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

4) –x–2

x+3≤0

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

A) (–∞

,

–3)or[–2

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

B) (–3

,

–2]

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

C) (–∞

,

–3]or[–2

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

D) [–2

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

Page118

5) 15–3x

6x+1≤0

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

A) –∞,–1

6or[5,∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

B) –1

6,5

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

C) –∞,–1

6or[5,∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

D) [5

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

6) 3x+5

4–2x ≥0

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

A) –5

3, 2

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

B) –∞,–5

3or(2,∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

C) –5

3,2

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

D) –5

3,∞

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

7) x

x+2>0

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

A) (–∞

,

–2)or(0,∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

B) (–2

,

0]

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

C) (–∞

,

–2]or[0,∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

D) (0,∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

Page119

8) (x+7)(x–5)

x–1≥0

–14–12–10–8–6–4–202468101214–14–12–10–8–6–4–202468101214

A) [–7

,

1)∪[5

,

∞)

–14 –12 –10 -8 –6 –4 –2 0 2 4 6 8 10 12 14–14 –12 –10 -8 –6 –4 –2 0 2 4 6 8 10 12 14

B) (–∞

,

–7]∪(1,5]

–14 –12 –10 -8 –6 –4 –2 0 2 4 6 8 10 12 14–14 –12 –10 -8 –6 –4 –2 0 2 4 6 8 10 12 14

C) (–∞

,

–7]∪[5

,

∞)

–14 –12 –10 -8 –6 –4 –2 0 2 4 6 8 10 12 14–14 –12 –10 -8 –6 –4 –2 0 2 4 6 8 10 12 14

D) [–7

,

1]∪[5

,

∞)

–14 –12 –10 -8 –6 –4 –2 0 2 4 6 8 10 12 14–14 –12 –10 -8 –6 –4 –2 0 2 4 6 8 10 12 14

9) (x–1)(3–x)

(x–2)2≤0

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

A) (–∞

,

1]∪[3,∞)

–12–10–8–6–4–2024681012–12–10–8–6–4–2024681012

B) (–∞

,

–3]∪(–2,–1)∪[1,∞)

–12–10–8–6–4–2024681012–12–10–8–6–4–2024681012

C) (–∞

,

–3)∪(–1,∞)

–12–10–8–6–4–2024681012–12–10–8–6–4–2024681012

D) (–∞

,

1)∪(3,∞)

–12–10–8–6–4–2024681012–12–10–8–6–4–2024681012

10) x+7

x+8<3

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

A) (–∞,–8)or(–17

2,∞)

-12-10 -8 -6 -4 –2 0 2 4 6 8 10 12-12–10 -8 –6 -4 -2 0 2 4 6 8 10 12

B) (–8,–17

2)

-12-10 -8 -6 -4 –2 0 2 4 6 8 10 12-12–10 -8 –6 -4 -2 0 2 4 6 8 10 12

C) (–∞,–17

2)or(8,∞)

-12-10 -8 -6 -4 –2 0 2 4 6 8 10 12-12–10 -8 –6 -4 -2 0 2 4 6 8 10 12

D) ∅

-12-10 -8 -6 -4 –2 0 2 4 6 8 10 12-12–10 -8 –6 -4 -2 0 2 4 6 8 10 12

Page120

11) 1

x–2<1

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

A) (–∞

,

2)or(3

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

B) (2

,

3)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

C) (–∞

,

2]or[3

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

D) (–∞

,

2)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

12) x

x+3≥2

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

A) [–6

,

–3)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

B) (–∞

,

–6]or (–3

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

C) (–∞

,

–3)or[0,∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

D) (–3

,

6]

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

13) 5x

x+7<x

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

A) (–7

,

–2)∪(0,∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

B) (–∞

,

–7)∪(–2

,

0)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

C) (–∞

,

–7)∪(0,∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

D) (–∞

,

2)∪(7

,

∞)

–10–9–8–7–6–5–4–3–2–1012345678910–10–9–8–7–6–5–4–3–2–1012345678910

3 SolveProblemsModeledbyPolynomialorRationalInequalities

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

1) Theaveragecostperunit,y,ofproducingxunitsofaproductismodeledby y=1,950,000+0.35x

x.Describe

thecompanyʹsproductionlevelsothattheaveragecostofproducingeachunitdoesnotexceed $6.85.

A) Atleast300,000units B) Notmorethan300,000units

C) Atleast400,000units D) Notmorethan400,000units

Page121

2) ThetotalprofitfunctionP(x)foracompanyproducingxthousandunitsisgivenbyP(x)=–2x2+28x–96.Find

thevaluesofxforwhichthecompanymakesaprofit.[Hint:ThecompanymakesaprofitwhenP(x)>0.]

A) xisbetween6thousandunitsand8 thousandunits

B) xisgreaterthan6thousandunits

C) xislessthan8thousandunits

D) xislessthan6thousandunitsorgreaterthan8 thousandunits

3) Anumberminustheproductof16anditsreciprocalislessthanzero.Findthenumberswhichsatisfythis

condition.

A) anynumberlessthan–4orbetween0and4 B) anynumberbetween0and4

C) anynumberbetween–4and4 D) anynumberlessthan4

4) Thesumof81timesanumberandthereciprocalofthenumberispositive.Findthenumberswhichsatisfythis

condition.

A) anynumbergreaterthan0 B) anynumbergreaterthan1

9

C) anynumberbetween–1

9and1

9D) anynumberbetween0and1

9

5) Anarrowisfiredstraightupfromthegroundwithaninitialvelocityof128 feet persecond.Itsheight,s(t), in

feetatanytimetisgivenbythefunctions(t)=–16t2+128t.Findtheintervaloftimeforwhichtheheightofthe

arrowisgreaterthan112feet.

A)

b

etween1and7sec B) after1 sec

C)

b

efore7sec D)

b

efore1 secorafter7sec

6) Aballisthrownverticallyupwardwithaninitialvelocityof192 feetpersecond.Thedistanceinfeetoftheball

fromthegroundaftertsecondsiss=192t–16t2.Forwhatintervaloftimeistheballmorethan432abovethe

ground?

A)

b

etween3and9seconds B)

b

etween2.5 and9.5seconds

C)

b

etween9and15seconds D)

b

etween5.5 and6.5seconds

7) Aballisthrownverticallyupwardwithaninitialvelocityof160 feetpersecond.Thedistanceinfeetoftheball

fromthegroundaftertsecondsiss=160t–16t2.Forwhatintervalsoftimeistheballlessthan384abovethe

ground(afteritistosseduntilitreturnstotheground)?

A)

b

etween0and4secondsandbetween6 and10 seconds

B)

b

etween4and6seconds

C)

b

etween0and3.5secondsandbetween6.5 and10 seconds

D)

b

etween0and4.5secondsandbetween5.5 and10 seconds

8) Therevenueachievedbysellingxgraphingcalculatorsisfiguredtobex(49 –0.2x)dollars.Thecostofeach

calculatoris$21.Howmanygraphingcalculatorsmustbesoldtomakeaprofit(revenue–cost)ofatleast

$975.00?

A)

b

etween65and75calculators B)

b

etween30 and40calculators

C)

b

etween66and64calculators D)

b

etween67 and73calculators

9) Therevenueachievedbysellingxgraphingcalculatorsisfiguredtobex(50 –0.5x)dollars.Thecostofeach

calculatoris$22.Howmanygraphingcalculatorsmustbesoldtomakeaprofit(revenue–cost)ofatleast

$379.50?

A)

b

etween23and33calculators B)

b

etween30 and40calculators

C)

b

etween24and32calculators D)

b

etween25 and31calculators

Page122

10) Youdrive98milesalongascenichighwayandthentakea22–milebikeride.Yourdrivingrateis3 timesyour

cyclingrate.Supposeyouhavenomorethanatotalof7hoursfordrivingandcycling.Letxrepresentyour

cyclingrateinmilesperhour.Writearationalinequalitythatcanbeusedtodeterminethepossiblevaluesofx.

Donotsimplifyanddonotsolvetheinequality.

A) 98

3x +22

x≤7B)

98

x+22

3x ≤7C)

3x

98 +x

22 ≤7D)

98

3x +22

x≥7

11) Youdrive120milesalongascenichighwayandthentakea22–milebikeride.Yourdrivingrateis4 timesyour

cyclingrate.Supposeyouhavenomorethanatotalof5hoursfordrivingandcycling.Letxrepresentyour

cyclingrateinmilesperhour.Usearationalinequalitytodeterminethepossiblevaluesofx.

A) x≥10.4mph B) x≤10.4 mph C) x≥25.1 mph D) x≤63.5 mph

12) Theperimeterofarectangleis54feet.Describethepossiblelengthsofasideiftheareaoftherectangleistobe

greaterthan152squarefeet.

A) Thelengthoftherectanglemustliebetween8 and19 ft

B) Thelengthoftherectanglemustbegreaterthan19 ft

C) Thelengthoftherectanglemustbegreaterthan19 ftorlessthan8 ft

D) Thelengthoftherectanglemustliebetween1and152 ft

3.7 ModelingUsingVariatio

n

1 SolveDirectVariationProblems

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writeanequationthatexpressestherelationship.Usekastheconstantofvariation.

1) gvariesdirectlyasv.

A) g=kv B) g=k

vC) k=gv D) v=k

g

2) svariesdirectlyasthesquareoft.

A) s=kt2B) s=k

t2C) s=kt D) s=k

t

Determinetheconstantofvariationforthestatedcondition.

3) gvariesdirectlyasf

,

andg=84whenf=6.

A) k=14 B) k=16 C) k=1

14 D) k=78

4) gvariesdirectlyasf

,

andg=5whenf=70.

A) k=1

14 B) k=15 C) k=14 D) k=65

5) gvariesdirectlyasf2,andg=45whenf=3.

A) k=5B)k=45 C) k=1

5D) k=42

Ifyvariesdirectlyasx,findthedirectvariationequationforthesituation.

6) y=9whenx=27

A) y=1

3xB)y=3x C) y=x+18 D) y=1

9x

Page123

7) y=12whenx=20

A) y=3

5xB)y=5

3xC)y=x–8D)y=4x

8) y=3whenx=1

5

A) y=15x B) y=1

15 xC)y=x+14

5D) y=1

3x

9) y=0.6whenx=0.3

A) y=2x B) y=0.3x C) y=x+0.3 D) y=0.5x

10) y=0.5whenx=4

A) y=0.125x B) y=0.5x C) y=x–3.5 D) y=8x

Solvetheproblem.

11) yvariesdirectlyaszandy=270whenz=15.Findywhenz=14.

A) 252 B) 196 C) 324 D) 225

12) Ifyvariesdirectlyasx,andy=2whenx=4

,

findywhenx=32.

A) 16 B) 1

4C) 64 D) 4

13) Ifyvariesdirectlyasx,andy=500whenx=150

,

findywhenx=60.

A) 200 B) 1250 C) 18 D) 1

18

14) yvariesdirectlyasz2andy=125whenz=5.Findywhenz=3.

A) 45 B) 25 C) 75 D) 15

15) Ifyvariesdirectlyasthesquareofx,andy=90 whenx=2

,

findywhenx=6.

A) 810 B) 270 C) 10 D) 30

16) Ifyvariesdirectlyasthecubeofx,andy=10 whenx=4

,

findywhenx=10.

A) 625

4B) 25 C) 4 D) 16

25

17) Ifyvariesdirectlyasthesquarerootofx,andy=10 whenx=25

,

findywhenx=16.

A) 8 B) 32

5C) 25

2D) 125

8

18) Theamountofwaterusedtotakeashowerisdirectlyproportionaltotheamountoftimethattheshowerisin

use.Ashowerlasting20minutesrequires8gallonsofwater.Findtheamountofwaterusedinashower

lasting5minutes.

A) 2gallons B) 32gallons C) 12.5 gallons D) 1.6 gallons

19) Iftheresistanceinanelectricalcircuitisheldconstant,theamountofcurrentflowingthroughthecircuitis

directlyproportionaltotheamountofvoltageappliedtothecircuit.When6voltsareappliedtoacircuit,

150milliamperesofcurrentflowthroughthecircuit.Findthenewcurrentifthevoltageisincreasedto8volts.

A) 200milliamperes B) 48milliamperes C) 192 milliamperes D) 225 milliamperes

Page124

20) Theamountofgasthatahelicopterusesisdirectlyproportionaltothenumberofhoursspentflying.The

helicopterfliesfor2hoursanduses14gallonsoffuel.Findthenumberofgallonsoffuelthatthehelicopter

usestoflyfor5hours.

A) 35gallons B) 10gallons C) 40 gallons D) 42gallons

21) Thedistancethatanobjectfallswhenitisdroppedisdirectlyproportionaltothesquareoftheamountoftime

sinceitwasdropped.Anobjectfalls88.2metersin3seconds.Findthedistancetheobjectfallsin5seconds.

A) 245meters B) 49meters C) 147 meters D) 15meters

22) Foraresistorinadirectcurrentcircuitthatdoesnotvaryitsresistance,thepowerthataresistormustdissipate

isdirectlyproportionaltothesquareofthevoltageacrosstheresistor.Theresistormustdissipate 1

16 wattof

powerwhenthevoltageacrosstheresistoris8volts.Findthepowerthattheresistormustdissipatewhenthe

voltageacrossitis16volts.

A) 1

4watt B) 1

8watt C) 4 watts D) 1

2watt

2 SolveInverseVariationProblems

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writeanequationthatexpressestherelationship.Usekastheconstantofvariation.

1) avariesinverselyasm.

A) a=k

mB) a=m

kC) a =km D) ka=m

2) dvariesinverselyasthesquareof

b

.

A) d=k

b2B) d=b2

kC) d=k

bD) d=b

k

Ifyvariesinverselyasx,findtheinversevariationequationforthesituation.

3) y=9whenx=2

A) y=18

xB) y=9

2xC)y=x

18 D) y=1

18x

4) y=20whenx=8

A) y=160

xB) y=5

2xC)y=x

160 D) y=1

160x

5) y=30whenx=1

6

A) y=5

xB) y=180x C) y=x

5D) y=1

5x

6) y=1

3whenx=15

A) y=5

xB) y=1

45 xC)y=x

5D) y=1

5x

7) y=0.2whenx=0.4

A) y=0.08

xB) y=0.5x C) y=12.5x D) y=12.5

x

Page125

Solvetheproblem.

8) xvariesinverselyasv,andx=21whenv=9.Findxwhenv=27.

A) x=7B)x=81 C) x=63 D) x=3

9) xvariesinverselyasy2,andx=6wheny=8.Findxwheny=2.

A) x=96 B) x=144 C) x=24 D) x=4

Solve.

10) Whenthetemperaturestaysthesame,thevolumeofagasisinverselyproportionaltothepressureofthegas.

Ifaballoonisfilledwith54cubicinchesofagasatapressureof14poundspersquareinch,findthenew

pressureofthegasifthevolumeisdecreasedto9cubicinches.

A) 84poundspersquareinch B) 9

14 poundspersquareinch

C) 70poundspersquareinch D) 78 poundspersquareinch

11) Theamountoftimeittakesaswimmertoswimaraceisinverselyproportionaltotheaveragespeedofthe

swimmer.Aswimmerfinishesaracein30secondswithanaveragespeedof5feetpersecond.Findthe

averagespeedoftheswimmerifittakes50secondstofinishtherace.

A) 3feetpersecond B) 4feetpersecond C) 5 feetpersecond D) 2feetpersecond

12) Iftheforceactingonanobjectstaysthesame,thentheaccelerationoftheobjectisinverselyproportionaltoits

mass.Ifanobjectwithamassof28kilogramsacceleratesatarateof6meterspersecondpersecondbyaforce,

findtherateofaccelerationofanobjectwithamassof4kilogramsthatispulledbythesameforce.

A) 42meterspersecondpersecond B) 6

7meterspersecondpersecond

C) 36meterspersecondpersecond D) 35 meterspersecondpersecond

13) Ifthevoltage,V,inanelectriccircuitisheldconstant,thecurrent,I,isinverselyproportionaltotheresistance,

R.Ifthecurrentis420milliampereswhentheresistanceis2ohms,findthecurrentwhentheresistanceis14

ohms.

A) 60milliamperes B) 2940 milliamperes C) 2933 milliamperes D) 120 milliamperes

14) Whiletravelingataconstantspeedinacar,thecentrifugalaccelerationpassengersfeelwhilethecaristurning

isinverselyproportionaltotheradiusoftheturn.Ifthepassengersfeelanaccelerationof8feetpersecondper

secondwhentheradiusoftheturnis80feet,findtheaccelerationthepassengersfeelwhentheradiusofthe

turnis160feet.

A) 4feetpersecondpersecond B) 5 feetpersecondpersecond

C) 6feetpersecondpersecond D) 7 feetpersecondpersecond

Writeanequationthatexpressestherelationship.Usekastheconstantofvariation.

15) TheintensityIoflightvariesinverselyasthesquareofthedistanceDfromthesource.Iftheintensityof

illuminationonascreen63ftfromalightis2.9foot–candles,findtheintensityonascreen90ftfromthelight.

A) 1.421foot–candles B) 2.03 foot–candles C) 4.14 foot–candles D) 5.92 foot–candles

16) Theweightofabodyabovethesurfaceoftheearthisinverselyproportionaltothesquareofitsdistancefrom

thecenteroftheearth.Whatistheeffectontheweightwhenthedistanceismultipliedby5?

A) Theweightisdividedby25 B) Theweightisdividedby5

C) Theweightismultipliedby25 D) Theweightismultipliedby5

Page126

3 SolveCombinedVariationProblems

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writeanequationthatexpressestherelationship.Usekfortheconstantofproportionality.

1) pvariesdirectlyasqandinverselyasr.

A) p=kq

rB) p=kr

qC) pqr =kD)p+q–r=k

2) xvariesdirectlyasyandinverselyasthesquareofz.

A) x=ky

z2B) x=kz2

yC) xyz2=kD)x+y–z2=k

3) svariesdirectlyasthesquareoftandinverselyasthecubeofu.

A) s=kt2

u3B) st2u3=kC)s=ku3

t2D) s+t2–u3=k

4) wvariesdirectlyasthesquareofxandinverselyasy.

A) w=kx2

yB) w=ky

x2C) w=k+x2–y2D) w=kx2y

5) pvariesjointlyasqandrandinverselyasthesquarerootofa.

A) p=kqr

aB) p=kq

ra C) p=k(q +r)

aD) p=qr

ka

6) rvariesdirectlyasaandinverselyasthedifferencebetweens andt.

A) r=ka

s–tB) r=a

k(s–t) C) r =ka(s –t) D) r=k

a(s–t)

Determinetheconstantofvariationforthestatedcondition.

7) zvariesdirectlyasxandinverselyasy

,

andz=4 whenx=52 andy=52.

A) k=4B)k=1

4C) k=1D)k=13

8) zvariesdirectlyasxandinverselyasy

,

andz=5 whenx=55 andy=25.

A) k=25

11 B) k=11

25 C) k=25 D) k=5

Findthevariationequationforthevariationstatement.

9) cvariesdirectlyasaandinverselyas

b

;c=2 whena=26 and

b

=78

A) c=6a

bB) c=a

6b C) c =6a

b

D) c=6

ab

Solvetheproblem.

10) yvariesdirectlyasxandinverselyasthesquareofz.y=72 whenx=72 andz=3.Findywhenx=43 andz=

10.

A) 3.87 B) 12.9 C) 477.78 D) 38.7

11) yvariesjointlyasaandbandinverselyasthesquarerootofc.y=10 whena=2

,

b=10

,

andc=36. Findy

whena=6,b=3,andc=4.

A) 27 B) 9 C) 13.5 D) 108

Page127

12) ThevolumeVofagivenmassofgasvariesdirectlyasthetemperatureTandinverselyasthepressureP.A

measuringdeviceiscalibratedtogiveV=325in3whenT=500°andP=20lb/in2.Whatisthevolumeonthis

devicewhenthetemperatureis170°andthepressureis25lb/in2?

A) V=88.4in3B) V=6.8in3C) V=108.4in3D) V=68.4in3

13) Thetimeinhoursittakesasatellitetocompleteanorbitaroundtheearthvariesdirectlyastheradiusofthe

orbit(fromthecenteroftheearth)andinverselyastheorbitalvelocity.Ifasatellitecompletesanorbit

710milesabovetheearthin12hoursatavelocityof22,000mph,howlongwouldittakeasatellitetocomplete

anorbitifitisat1300milesabovetheearthatavelocityof30,000mph?(Use3960milesastheradiusofthe

earth.)Roundyouranswertothenearesthundredthofanhour.

A) 9.91hours B) 16.11 hours C) 2.45 hours D) 99.12 hours

14) Thepressureofagasvariesjointlyastheamountofthegas(measuredinmoles)andthetemperatureand

inverselyasthevolumeofthegas.Ifthepressureis1395kPa(kiloPascals)whenthenumberofmolesis6,the

temperatureis310°Kelvin,andthevolumeis480cc,findthepressurewhenthenumberofmolesis7,the

temperatureis320°K,andthevolumeis420cc.

A) 1920 B) 1860 C) 960 D) 990

15) Body–massindex,orBMI,takesbothweightandheightintoaccountwhenassessingwhetheranindividualis

underweightoroverweight.BMIvariesdirectlyasoneʹsweight,inpounds,andinverselyasthesquareofoneʹs

height,ininches.Inadults,normalvaluesfortheBMIarebetween20and25.Apersonwhoweighs180

poundsandis72inchestallhasaBMIof24.41.WhatistheBMI,tothenearesttenth,forapersonwhoweighs

125poundsandwhois63inchestall?

A) 22.1 B) 22.5 C) 21.7 D) 21.4

4 SolveProblemsInvolvingJointVariation

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writeanequationthatexpressestherelationship.Usekastheconstantofvariation.

1) avariesjointlyasgandthesquareofz.

A) a=kgz2B) a=kg

z2C) a=kgkz2D) a=kz2

g

2) xvariesjointlyassandt.

A) x=kst B) x=ks

tC) x =kskt D) x=kt

s

3) wvariesjointlyasxandthecubeofy.

A) w=kxy3B) wxy3=kC)w=k+x+y3D) w+x+y3=k

4) rvariesjointlyasthesquareofsandthesquareoft.

A) r=ks2t2B) rs2t2=kC)r=k+s2+t2D) r+s2+t2=k

5) dvariesjointlyas

b

andthesumofpandc.

A) d=k

b

(p+c) B) d=k

b

(p+c) C) d =k

b

+p+cD)d=k(

b

p+c)

6) avariesjointlyasyandthedifferencebetweenpandz.

A) a=ky(p–z) B) a=ky

(p–z) C) a =ky +p–zD)a=k(yp–z)

Page128

Findthevariationequationforthevariationstatement.

7) zvariesjointlyasyandthecubeofx;z=384 whenx=4 andy= –3

A) y=–2x3yB)y=–2xy3C) y=2x3yD)y=2xy3

Determinetheconstantofvariationforthestatedcondition.

8) cvariesjointlyasaand

b

,

andc=48whena=24 and

b

=8.

A) k=1

4B) k=1

8C) k=4D)k=8

9) tvariesjointlyasrands

,

andt=1872whenr=36 ands=13.

A) k=4B)k=1

9C) k=1

4D) k=9

10) hvariesjointlyasfandg

,

andh=56whenf=35

,

andg=40.

A) k=1

25 B) k=1

40 C) k=25 D) k=40

Solvetheproblem.

11) hvariesjointlyasfandg.Findhwhenf=24

,

g=14

,

and k=4.

A) h=1344 B) h=336 C) h =84 D) h=7

3

12) yvariesjointlyasxandz.y=2.7whenx=45 andz=6.Findywhenx=30 andz=6.

A) 1.8 B) 180 C) 18 D) 3.6

13) fvariesjointlyasq2andh,andf=96whenq=4andh=3.Findfwhenq=2andh=6.

A) f=48 B) f=24 C) f=8D)f=12

14) fvariesjointlyasq2andh,andf=–54whenq=3andh=3.Findfwhenq=2andh=5.

A) f=–40 B) f=–20 C) f= –8D)f=–10

15) fvariesjointlyasq2andh,andf=54whenq=3andh=2.Findqwhenf=288andh=6.

A) q=4B)q=2C)q=3D)q=6

16) fvariesjointlyasq2andh,andf=54whenq=3andh=2.Findhwhenf=288andq=4.

A) h=6B)h=2C)h=3D)h=4

17) Theamountofpaintneededtocoverthewallsofaroomvariesjointlyastheperimeteroftheroomandthe

heightofthewall.Ifaroomwithaperimeterof75feetand8–footwallsrequires6quartsofpaint,findthe

amountofpaintneededtocoverthewallsofaroomwithaperimeterof45feetand6–footwalls.

A) 2.7quarts B) 270quarts C) 27 quarts D) 5.4 quarts

18) Thepowerthataresistormustdissipateisjointlyproportionaltothesquareofthecurrentflowingthroughthe

resistorandtheresistanceoftheresistor.Ifaresistorneedstodissipate108wattsofpowerwhen6amperesof

currentisflowingthroughtheresistorwhoseresistanceis3ohms,findthepowerthataresistorneedsto

dissipatewhen3amperesofcurrentareflowingthrougharesistorwhoseresistanceis9ohms.

A) 81watts B) 27watts C) 243 watts D) 162 watts

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19) Whiletravelinginacar,thecentrifugalforceapassengerexperiencesasthecardrivesinacirclevariesjointly

asthemassofthepassengerandthesquareofthespeedofthecar.Ifapassengerexperiencesaforceof32.4

newtonswhenthecarismovingataspeedof30kilometersperhourandthepassengerhasamassof40

kilograms,findtheforceapassengerexperienceswhenthecarismovingat50kilometersperhourandthe

passengerhasamassof50kilograms.

A) 112.5newtons B) 125newtons C) 100 newtons D) 150 newtons

20) Theamountofsimpleinterestearnedonaninvestmentoverafixedamountoftimeisjointlyproportionalto

theprincipleinvestedandtheinterestrate.Aprincipleinvestmentof$3200.00withaninterestrateof5%

earned$320.00insimpleinterest.Findtheamountofsimpleinterestearnediftheprincipleis$1800.00andthe

interestrateis2%.

A) $72.00 B) $7200.00 C) $180.00 D) $128.00

21) Thevoltageacrossaresistorisjointlyproportionaltotheresistanceoftheresistorandthecurrentflowing

throughtheresistor.Ifthevoltageacrossaresistoris12voltsforaresistorwhoseresistanceis2ohmsand

whenthecurrentflowingthroughtheresistoris6amperes,findthevoltageacrossaresistorwhoseresistance

is5ohmsandwhenthecurrentflowingthroughtheresistoris4amperes.

A) 20volts B) 8volts C) 24 volts D) 30volts

22) Thepressureofagasvariesjointlyastheamountofthegas(measuredinmoles)andthetemperatureand

inverselyasthevolumeofthegas.Ifthepressureis936kPa(kiloPascals)whenthenumberofmolesis8,the

temperatureis260°Kelvin,andthevolumeis960cc,findthepressurewhenthenumberofmolesis10,the

temperatureis270°K,andthevolumeis600cc.

A) 1944 B) 1872 C) 972 D) 1008

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Ch.3 PolynomialandRationalFunctions

AnswerKey

3.1 QuadraticFunctions

1 RecognizeCharacteristicsofParabolas

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2 GraphParabolas

3 DetermineaQuadraticFunctionʹsMinimumorMaximumValue

4 SolveProblemsInvolvingaQuadraticFunctionʹsMinimumorMaximumValue

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3.2 PolynomialFunctionsandTheirGraphs

1 IdentifyPolynomialFunctions

2 RecognizeCharacteristicsofGraphsofPolynomialFunctions

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3 DetermineEndBehavior

4 UseFactoringtoFindZerosofPolynomialFunctions

5 IdentifyZerosandTheirMultiplicities

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6 UsetheIntermediateValueTheorem

7 UnderstandtheRelationshipBetweenDegreeandTurningPoints

8 GraphPolynomialFunctions

Page135

Page136

3.3 DividingPolynomials;RemainderandFactorTheorems

1 UseLongDivisiontoDividePolynomials

2 UseSyntheticDivisiontoDividePolynomials

3 EvaluateaPolynomialUsingtheRemainderTheorem

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4 UsetheFactorTheoremtoSolveaPolynomialEquation

3.4 ZerosofPolynomialFunctions

1 UsetheRationalZeroTheoremtoFindPossibleRationalZeros

2 FindZerosofaPolynomialFunction

3 SolvePolynomialEquations

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4 UsetheLinearFactorizationTheoremtoFindPolynomialswithGivenZeros

5 UseDescartesʹsRuleofSigns

3.5 RationalFunctionsandTheirGraphs

1 FindtheDomainsofRationalFunctions

2 UseArrowNotation

3 IdentifyVerticalAsymptotes

7) A

4 IdentifyHorizontalAsymptotes

7) A

5 UseTransformationstoGraphRationalFunctions

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6 GraphRationalFunctions

7 IdentifySlantAsymptotes

8 SolveAppliedProblemsInvolvingRationalFunctions

3.6 PolynomialandRationalInequalities

1 SolvePolynomialInequalities

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21) A

2 SolveRationalInequalities

3 SolveProblemsModeledbyPolynomialorRationalInequalities

3.7 ModelingUsingVariatio

n

1 SolveDirectVariationProblems

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2 SolveInverseVariationProblems

3 SolveCombinedVariationProblems

4 SolveProblemsInvolvingJointVariation

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