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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
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 2
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
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 3
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
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 4
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 ≠ ;
Range: all real numbers, 0y ≠ ;
no intercepts
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 5
7.ans-0902-31
VA: x = 0, HA: y = 0;
Domain: all real numbers, 0≠x ;
Range: all real numbers, 0y ≠ ;
no intercepts
8.ans-0902-32
VA: x = -3, HA: y = 0;
Domain: all real numbers, 3≠ −x ;
Range: all real numbers, 0y ≠ ;
no intercepts
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 6
9.ans-0902-05
10.ans-0902-06
VA: 5x = − ; HA: 0y = ;
Domain: all real numbers, 5x ≠ − ;
Range: all real numbers, 0y ≠ ;
20,5
⎛ ⎞⎜ ⎟⎝ ⎠
VA: 3x = ; HA: 4y = − ;
Domain: all real numbers, 3x ≠ ;
Range: all real numbers, 4y ≠ − ;
13 130, , ,03 4
⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 7
11.ans-0902-07
12.ans-0902-08
VA: 4x = − ; HA: 3y = − ;
Domain: all real numbers, 4x ≠ ;
Range: all real numbers, 3y ≠ − ;
5 100, , ,02 3
−⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
VA: 0x = ; HA: 2y = ;
Domain: all real numbers, 0x ≠ ;
Range: all real numbers, 2y ≠ ;
5 ,02
⎛ ⎞−⎜ ⎟⎝ ⎠
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 8
13.ans-0902-09
14.1 46
yx
= −−
15.131
yx
= −+
VA: 2x = − ; HA: 3y = ;
Domain: all real numbers, 2x ≠ − ;
Range: all real numbers, 3y ≠ ;
5 50, , ,02 3
−⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 9
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 = ;
Domain: all real numbers, 5x ≠ ;
Range: all real numbers, 1y ≠ ;
( )30, , 3,05
⎛ ⎞− −⎜ ⎟⎝ ⎠
VA: 4x = ; HA: 5y = ;
Domain: all real numbers, 4x ≠ ;
Range: all real numbers, 5y ≠ ;
1 20, , ,02 5
⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 10
8.ans-0902-12
9.ans-0902-13
10.ans-0902-14
VA: 5x = − ; HA: 12
y = − ;
Domain: all real numbers, 5x ≠ − ;
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 ≠ − ;
Range: all real numbers, 12
y ≠ ;
40,3
⎛ ⎞−⎜ ⎟⎝ ⎠, no x-intercepts
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 11
11.ans-0902-15
12.ans-0902-16
13.ans-0902-17
VA: 5x = − ; HA: 1y = ;
Domain: all real numbers, 5,2x ≠ − ;
Range: all real numbers, 1y ≠ ;
( )20, , 2,05
⎛ ⎞ −⎜ ⎟⎝ ⎠; hole at
42,7
⎛ ⎞⎜ ⎟⎝ ⎠
VA: 43
x = − ; HA: 23
y = ;
Domain: all real numbers, 4 , 23
x ≠ − − ;
Range: all real numbers, 23
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
⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 12
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
−+ =− −
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 13
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−
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 14
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−
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 15
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−
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 16
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
⎛ ⎞−⎜ ⎟⎝ ⎠
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 17
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.
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 18
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++
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 19
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−−
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 20
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
= .
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 21
9.10 Adding and Subtracting Rational Expressions with Like Denominators
Answers
1.Answers will vary.
2.Answers will vary.
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 +
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 22
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 +
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 23
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− + −− +
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 24
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+ + −+ − + +
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 25
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.
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 26
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
Chapter 9 – Rational Functions Answer Key
CK-‐12 Algebra II with Trigonometry Concepts 27
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 − ±=