Chapter(9(–(Rational(Functions( Answer’Key 9.1 Direct ...

Chapter 9 – Rational Functions Answer Key

CK-‐12 Algebra II with Trigonometry Concepts 1

9.1 Direct Variation

Answers

1. 5 ; 60y x y= =

2.1 ; 43

y x y= − = −

3.2 ; 43

y x y= =

4.1 ; 26

= − = −y x y

5.4 5;5 2

y x x= =

6.1 ; 126

y x x= =

7.14 ;2

y x x= − = −

8.5 8;4 5

y x x= =

9. no

10. yes

11. yes

12.351.6 calories; 30 miles

13.844 miles

14.$5.00; 32 more cars

15.$25 per hour, $1000



9.2 Inverse Variation

Answers

1.12 4;

5y y

x= =

2.2 2;

15y y

x= =

3.6 2;

5y y

x= =

4.5 1;4 12

y yx

= =

5.4 ; 2y xx

= =

6.6 ; 3y xx

= =

7.36 ; 365

y xx

= =

8.3 ; 1.5y xx

= =

9. yes

10. no

11. yes

12.2 pieces; 3 pieces; 24yx

=

13. 78 mph

14.12 hours; 9 friends total

15.1.3 degrees Celsius



9.3 Joint Variation

Answers

1. kwxy

=

2. 2

krq

=

3. kxyzw

=

4. kbac d

=

5. kmlp

=

6.2 ; 83

z xy z= = −

7. 2 ; 7z xy z= − =

8.2 ; 43

z xy y= − =

9.1 ; 34

z xy x= = −

10. ; 26yz xx

= − = −

11.2 ; 4yz xx

= =

12. 180 cars

13.1 ;5

k = 1 hour 15 min

14. 9.8W mh= ; 2940 joules

15.2

4Id

= ; 0.4 watts per m2



9.4 Graphing Rational Functions in Standard Form

Answers

1.VA: 8x = − ; HA: 3y = − ; yes

2. no

3.VA: 4x = ; HA: 6y =

4. no

5.ans-0902-04

6.ans-0902-30

VA: x = 0, HA: y = 6;

Domain: all real numbers, 6≠x ;

Range: all real numbers, 0y ≠ ;

1 ,06

⎛ ⎞−⎜ ⎟

⎝ ⎠

VA: 0x = ; HA: 0y = ;

Domain: all real numbers, 0x ≠ ;


no intercepts



7.ans-0902-31

VA: x = 0, HA: y = 0;

Domain: all real numbers, 0≠x ;


no intercepts

8.ans-0902-32

VA: x = -3, HA: y = 0;

Domain: all real numbers, 3≠ −x ;


no intercepts



9.ans-0902-05

10.ans-0902-06

VA: 5x = − ; HA: 0y = ;

Domain: all real numbers, 5x ≠ − ;


20,5

⎛ ⎞⎜ ⎟⎝ ⎠

VA: 3x = ; HA: 4y = − ;


Range: all real numbers, 4y ≠ − ;

13 130, , ,03 4

⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠



11.ans-0902-07

12.ans-0902-08

VA: 4x = − ; HA: 3y = − ;


Range: all real numbers, 3y ≠ − ;

5 100, , ,02 3

−⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

VA: 0x = ; HA: 2y = ;



5 ,02

⎛ ⎞−⎜ ⎟⎝ ⎠



13.ans-0902-09

14.1 46

yx

= −−

15.131

yx

= −+

VA: 2x = − ; HA: 3y = ;



5 50, , ,02 3

−⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠



9.5 Graphing When the Degrees of the Numerator and Denominator are the Same

Answers

1.VA: x = -7, HA: y = 1

2. ° ; 7x ≠ −

3. ° ; 1y ≠

4.Yes; (2, 0).

5.Yes; 20,7

⎛ ⎞−⎜ ⎟

⎝ ⎠

6.ans-0902-10

7.ans-0902-11

VA: 5x = ; HA: 1y = ;



( )30, , 3,05

⎛ ⎞− −⎜ ⎟⎝ ⎠

VA: 4x = ; HA: 5y = ;



1 20, , ,02 5

⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠



8.ans-0902-12

9.ans-0902-13

10.ans-0902-14

VA: 5x = − ; HA: 12

y = − ;


Range: all real numbers, 12

y ≠ − ;

( )30, , 3,010

⎛ ⎞⎜ ⎟⎝ ⎠

VA: 6, 2x x= = ; HA: 1y = ;

Domain: all real numbers, 6,2x ≠ ;

Range:

( ) ( ), 9.493 0.007,−∞ − ∪ − ∞;

( ) ( )10, , 3,0 , 2,02

⎛ ⎞ − −⎜ ⎟⎝ ⎠

VA: 3 , 12

x x= − = ; HA: 12

y = ;

Domain: all real numbers, 3 ,12

x ≠ − ;


y ≠ ;

40,3

⎛ ⎞−⎜ ⎟⎝ ⎠, no x-intercepts



11.ans-0902-15

12.ans-0902-16

13.ans-0902-17

VA: 5x = − ; HA: 1y = ;

Domain: all real numbers, 5,2x ≠ − ;


( )20, , 2,05

⎛ ⎞ −⎜ ⎟⎝ ⎠; hole at

42,7

⎛ ⎞⎜ ⎟⎝ ⎠

VA: 43

x = − ; HA: 23

y = ;

Domain: all real numbers, 4 , 23

x ≠ − − ;


y ≠ ;

5 50, , ,04 2

⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠; hole at

92,2

⎛ ⎞−⎜ ⎟⎝ ⎠

VA: 1 1,2 2

x x= − = ; HA: 32

y = ;

Domain: all real numbers, 1 1,2 2

x ≠ − ;

Range: all real numbers;

( ) 1 30,3 , ,0 , ,03 2

⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠



14.ans-0902-18

15.ans-0902-19

VA: 2, 1x x= − = − ; HA: 1y = ;

Domain: all real numbers, 2, 1,2x ≠ − − ;

Range: ( ) ( ), 12.928 0.928,−∞ − ∪ ∞;

( )0,2 , no x-intercepts

These are actually the same because

1 3 532 2

xx x

−+ =− −



9.6 Graphing When the Degrees of the Numerator and Denominator Different

Answers

1.VA: x = -‐4, x = -‐2, HA: y = 0

2.VA: x = -‐5, SA: 5y x= −

3.VA: x = 3, SA: 3y x= +

4.(-‐2, 0) and (2, 0)

5. (0, 0)

6.ans-0902-20

7.ans-0902-21

VA: 3, 4x x= − =

HA: 0y =

Intercepts: 10,12

⎛ ⎞−⎜ ⎟⎝ ⎠, ( )1,0−

VA: 3x =

SA: 6y x= +

Intercepts: 100,3

⎛ ⎞⎜ ⎟⎝ ⎠

, ( )2,0 , ( )5,0−



8.ans-0902-22

9.ans-0902-23

VA: 32

x = −

HA: 0y =

Intercepts: 10,3

⎛ ⎞⎜ ⎟⎝ ⎠

, ( )7,0

Hole: 17,17

⎛ ⎞⎜ ⎟⎝ ⎠

VA: 53

x = −

SA: 2 103 9

y x= +

Intercepts: 20,5

⎛ ⎞−⎜ ⎟⎝ ⎠, ( )1,0 ,

( )1,0−



10.ans-0902-24

11.ans-0902-25

No asymptotes

Intercepts: ( )0, 5− , ( )5,0

Hole: ( )6, 11− −

VA: 1 , 1, 32

x x x= = − =

HA: 0y =

Intercepts: ( )0, 10− , ( )5,0 ,

( )6,0−



12.ans-0902-26

13.ans-0902-27

VA: 32

x = −

SA: 3y x= +

Intercepts: ( )0,0 , ( )1,0− , ( )3,0

Hole: ( )3,12

VA: 6, 1x x= − =

HA: 0y =

Intercepts: 50,6

⎛ ⎞−⎜ ⎟⎝ ⎠,

5 ,02

⎛ ⎞−⎜ ⎟⎝ ⎠



14.ans-0902-28

15.ans-0902-29

VA: 3x =

SA: 12

y x= −

Intercepts: 20,3

⎛ ⎞−⎜ ⎟⎝ ⎠, ( )1,0− ,

( )4,0

If we divide, 23 10 23 5x x xx− − = −+

this can also be found by factoring the numerator and denominator and cancelling the like factor of 3 5x + . We have seen before that

when factors cancel, a hole is formed, not an asymptote.



9.7 Simplifying Rational Expressions

Answers

1. No.

2. Yes.

3.Answers will vary.

4. 24

2 3xx +

5. 2

( 5)xx x

++

6. 34

xx−−

7. 7

( 7)xx x

+−

8. 2(2 1)2 3xx

− ++

9.2( 4)

3( 1)( 1)x

x x−

− +

10. 3 43 1xx++

11. 2xx+

12.(2 3)4

x x +

13. 2x −

14. 2

1 31x

x−+

15. 3

2 5xx++



9.8 Multiplying Rational Expressions

Answers

1.False, add the exponents.

2. True.

3. True.

4.False, 2 2( 2) 4 4x x x+ = + +

5. 5

3

12yx

6. 5 8x y

7. 56x

8. ( )3 22y +

9. ( )2 2 1

1xx

−+

10. 31

xx+−

11. 3−

12.( )2 54 3xx+−

13.( )51

xx

− ++

14.( )( )3 2 3

8x x

x− +

+

15. 2 3

3xx−−



9.9 Dividing Rational Expressions

Answers

1. ba

2. 3

6

2xy

3. 5

7yz

4. 4x

5. 34x +

6. ( )

( )2

3 2 5x xx

− +−

7. ( )23

x xx++

8. ( )18

x xx

++

9.( )( )22 5 3 9

7x x x

x+ − +

+

10.( )

( )( )5

2 1 7x

x x− +− −

11. 3

12. 13

13.No, from what we know about fractions, this should be 1 2 or 36 1⋅ , so it is the same as #11.

14.The reciprocal of the denominator, 1315

.

15.

13151315

, which is the same as one.

30521513

131513

⋅

15

12

= .



9.10 Adding and Subtracting Rational Expressions with Like Denominators

Answers



3. 7x

4. 6x

5. 9 25x

x−

6. 4xx+

7. 1

8. 52

xx+−

9. 4

10. 55x −

11. 32x +

12. 12x −

13.( )2

2 11

xx

++

14. 5

2 3x −

15. 34x +



9.11 Adding and Subtracting Rational Expressions Where One Denominator is the LCD

Answers

1. 6x

2. ( 1)x x +

3.( 2)( 4)x x+ −

4. 2( 1)x x −

5. 74x

6. 2

2

3 13

x xx x+ −+

7. 13

8. 52x +

9. 1

2 3x−+

10.2

10 183 10

xx x

−+ −

11.( )

32 2 3xx−−

12. 3

5 3xx+−

13.2

7 192 15x

x x++ −

14. 33x −

15. 43x +



9.12 Adding and Subtracting Rational Expressions with Unlike Denominators

Answers

1. 21x

2.( 2)(2 1)x x− −

3.( 3)( 3)( 2)x x x+ − +

4. 4 ( 6)x x −

5. 2( 2) ( 2)( 5)x x x+ − +

6. 23 106xx+

7. 2

2 77xx+

8.2 3 74 ( 2)x xx x− −

+

9.7 9

2( 3)( 1)x

x x−

+ −

10.2 14 6(2 1)( 6)x xx x x

− +− +

11. 1

5( 1)x +

12.22(2 5 6)

( 2)( 5)( 3)x x

x x x+ +

+ − +

13.2 3 23( 2)(3 5)x xx x x

+ +− +

14.220 42 7

5 ( 1)( 1)x xx x x

+ −− +

15.3 27 12 7 10(2 1)(3 2)( )x x xx x x− + −− +



9.13 Complex Fractions

Answers

1. 720x

2. 2

3 ( 3)x x −

3. 1

5( 3)x +

4. 3(2 5)2x −

5. 12x

−−

6.( 1)( 2)x x

x+ +

7.( 8)( 1)(3 7)x xx

− ++

8.2

2

( 3)(2 4 11)3 12 21

x x xx x

− − −+ −

9.2

2

2( 2 15)8 3 15x xx x

+ +− +

10.(7 2)( 1)( 3)( 4)x xx x+ −−− +

11.2

2

( 1)(4 5 8)( 2)( 6 1)x x xx x x+ + −+ − + +



12.12112233445

++

++

+

, 1211223344556

++

++

++

, 121122334455667

++

++

++

+

13.2.6, 2.72, 2.716981, 2.718447, 2.71863

14.This is the formula used to generate the irrational number e.

15.See last term in #12; 2.71828.



9.14 Solving Rational Equations Using Cross-Multiplication

Answers

1. no

2. 6, 2x = −

3. 5,2x = −

4. 6, 1x = −

5. 1, 1x = −

6. 1 , 42

x = −

7. 3x = −

8. 92,2

x = −

9. 528

x =

10. 40x = −

11. 8, 1x = −

12.0 < a < -6

13. a = x

14. a = 0

15. a = 6



9.15 Solving Rational Equations Using the LCD

Answers

1. no

2. no

3. ( 4)( 4)x x− + −

4.6 ( 2)( 3)( 3)x x x x− + −

5.( 3)( 2)( 2)x x x− + −

6. 2, 5x = −

7. 713

x = −

8. 14

x =

9. 23

x =

10. 5,2x = −

11.15 3 17

4x ±=

12. 0,28x =

13. 5, 3x = −

14. 14

x = −

15.7 52

x − ±=

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