chapter p prerequisites€¦ · © 2018 cengage learning. all rights reserved. may not be scanned,...

© 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

C H A P T E R P Prerequisites

Section P.1 Review of Real Numbers and Their Properties..................................... 2

Section P.2 Solving Equations ................................................................................... 5

Section P.3 The Cartesian Plane and Graphs of Equations .................................... 17

Section P.4 Linear Equations in Two Variables ..................................................... 29

Section P.5 Functions ............................................................................................... 42

Section P.6 Analyzing Graphs of Functions ........................................................... 51

Section P.7 A Library of Parent Functions ............................................................. 61

Section P.8 Transformations of Functions .............................................................. 66

Section P.9 Combinations of Functions: Composite Functions ............................ 76

Section P.10 Inverse Functions .................................................................................. 85

Review Exercises .......................................................................................................... 98

Problem Solving ......................................................................................................... 112

Practice Test ............................................................................................................. 118

Solution Manual for Trigonometry 10th Edition by Larson

Full file at https://TestbankDirect.eu/Solution-Manual-for-Trigonometry-10th-Edition-by-Larson


https://TestbankDirect.eu/Solution-Manual-for-Trigonometry-10th-Edition-by-Larson

2 © 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

C H A P T E R P Prerequisites

Section P.1 Review of Real Numbers and Their Properties

1. irrational

2. origin

3. absolute value

4. composite

5. terms

6. Zero-Factor Property

7. 7 22 3

9, , 5, , 2, 0, 1, 4, 2, 11− − − −

(a) Natural numbers: 5, 1, 2

(b) Whole numbers: 0, 5, 1, 2

(c) Integers: 9, 5, 0, 1, 4, 2, 11− − −

(d) Rational numbers: 7 22 3

9, , 5, , 0, 1, 4, 2, 11− − − −

(e) Irrational numbers: 2

8. 7 53 4

5, 7, , 0, 3.14, , 3, 12, 5− − −

(a) Natural numbers: 12, 5

(b) Whole numbers: 0, 12, 5

(c) Integers: 7, 0, 3, 12, 5− −

(d) Rational numbers: 7 53 4

7, , 0, 3.14, , 3, 12, 5− − −

(e) Irrational numbers: 5

9. 2.01, 0.6, 13, 0.010110111 . . ., 1, 6− −

(a) Natural numbers: 1

(b) Whole numbers: 1

(c) Integers: 13, 1, 6− −

(d) Rational numbers: 2.01, 0.6, 13, 1, 6− −

(e) Irrational numbers: 0.010110111 . . .

10. 12 15 2

25, 17, , 9, 3.12, , 7, 11.1, 13π− − −

(a) Natural numbers: 25, 9, 7, 13

(b) Whole numbers: 25, 9, 7, 13

(c) Integers: 25, 17, 9, 7, 13−

(d) Rational numbers:

125

25, 17, , 9, 3.12, 7, 11.1, 13− − −

(e) Irrational numbers: 12π

11. (a)

(b)

(c)

(d)

12. (a)

(b)

(c)

(d)

13. 4 8− > −

14. 163

1 <

15. 5 26 3

>

16. 8 37 7

− < −

17. (a) The inequality 5x ≤ denotes the set of all real numbers less than or equal to 5.

(b)

(c) The interval is unbounded.

40 1 2 3x

−2 −1

5

72

x40 1 2 3−1

x0 1−1−2−3

−

−4−5

52

x−5.2

−6−7 −1−2−3−4−5

x8.5

129 106 7 8 11

5

43

x40 1 2 3−1

x−4.75

−6−7 −1−2−3−4−5

x0 1−1−2−3

−

−4−5

83

−8 −7 −6 −5 −4x

0 1 2 3 4 5 6

163

x

0 1

32

65

x

−1 0−2

− 37− 8

7x

0 1 2 3 4 5 6

x






Section P.1 Review of Real Numbers and Their Properties 3


18. (a) The inequality 0x < denotes the set of all real numbers less than zero.

(b)


19. (a) The inequality 2 2x− < < denotes the set of all real numbers greater than 2− and less than 2.

(b)

(c) The interval is bounded.

20. (a) The inequality 0 6x< ≤ denotes the set of all real numbers greater than zero and less than or equal to 6.

(b)


21. (a) The interval [ )4, ∞ denotes the set of all real

numbers greater than or equal to 4.

(b)


22. (a) ( ), 2−∞ denotes the set of all real numbers less

than 2.

(b)


23. (a) The interval [ )5, 2− denotes the set of all real

numbers greater than or equal to 5− and less than 2.

(b)


24. (a) The interval ( ]1, 2− denotes the set of all real

numbers greater than 1− and less than or equal to 2.

(b)


25. [ )0; 0,y ≥ ∞

26. ( ]25; , 25y ≤ −∞

27. [ ]10 22; 10, 22t≤ ≤

28. [ )3 5; 3, 5k− ≤ < −

29. ( )10 10 10− = − − =

30. 0 0=

31. ( )3 8 5 5 5− = − = − − =

32. 6 2 4 4− = =

33. 1 2 1 2 1− − − = − = −

34. ( )3 3 3 3 6− − − = − − = −

35. ( )5 5 5 5 25− = =

36. ( )4 4 4 4 16− − = − = −

37. If 2,x < − then 2x + is negative.

So, ( )2 2

1.2 2

x x

x x

+ − += = −

+ +

38. If 1,x > then 1x − is positive.

So, 1 1

1.1 1

x x

x x

− −= =− −

39. 4 4− = because 4 4− = and 4 4.=

40. 5 5− = − because 5 5.− = −

41. 6 6− − < − because 6 6− = and

( )6 6 6.− − = − = −

42. 2 2− − = − because 2 2.− = −

43. ( )126, 75 75 126 51d = − =

44. ( ) ( )20, 30 30 20 50 50d − = − − = =

45. ( ) ( )5 5 52 2 2, 0 0d − = − − =

46. ( ) ( ) 5 51 11 11 14 4 4 4 2 2,d − − = − − − = − =

47. ( ), 5 5d x x= − and ( ), 5 3,d x ≤ so 5 3.x − ≤

48. ( ), 10 10 ,d x x− = + and ( ), 10 6,d x − ≥ so

10 6.x + ≥

−2 −1 0 1 2

x

−2 −1 0 1 2

x

x0 1 2 3 4 5 6

1 2 3 4 5 6 7

x

x0 1 2 3 4

1 3

x

−1−3−5

x210−2 −1






4 Chapter P Prerequisites


Receipts, R Expenditures, E −R E

49. $2524.0 billion $2982.5 billion 2524.0 2982.5 $458.5 billion− =




53. 7 4x +

Terms: 7 , 4x

Coefficient: 7

54. 2 3x −

Terms: 2 , 3x −

Coefficient: 2

55. 36 5x x−

Terms: 36 , 5x x−

Coefficients: 6, 5−

56. 34 0.5 5x x+ −

Terms: 34 , 0.5 , 5x x −

Coefficients: 4, 0.5

57. 23 3 1x +

Terms: 23 3 , 1x

Coefficient: 3 3

58. 22 2 3x −

Terms: 22 2 , 3x −

Coefficient: 2 2

59. 4 6x −

(a) ( )4 1 6 4 6 10− − = − − = −

(b) ( )4 0 6 0 6 6− = − = −

60. 9 7x−

(a) ( )9 7 3 9 21 30− − = + =

(b) ( )9 7 3 9 21 12− = − = −

61. 2 3 2x x− +

(a) ( ) ( )20 3 0 2 2− + =

(b) ( ) ( )21 3 1 2 1 3 2 6− − − + = + + =

62. 2 5 4x x− + −

(a) ( ) ( )21 5 1 4 1 5 4 10− − + − − = − − − = −

(b) ( ) ( )21 5 1 4 1 5 4 0− + − = − + − =

63. 1

1

x

x

+−

(a) 1 1 2

1 1 0

+ =−

Division by zero is undefined.

(b) 1 1 0

01 1 2

− + = =− − −

64. 2

2

x

x

−+

(a) 2 2 0

02 2 4

− = =+

(b) 2 2 4

2 2 0

− − −=− +

Division by zero is undefined.

65. ( )( )1

6 1, 66

h hh

+ = ≠ −+

Multiplicative Inverse Property

66. ( ) ( )3 3 0x x+ − + =

Additive Inverse Property

67. ( ) ( )( )

3 3 Associative Property of Multiplication

3 Commutative Property of Multiplication

x y x y

x y

= ⋅

=






Section P.2 Solving Equations 5


68. ( ) ( )1 17 7

7 12 7 12 Associative Property of Multiplication

1 12 Multiplicative Inverse Property

12 Multiplicative Identity Property

⋅ = ⋅

= ⋅=

69. 2 8 3 5

3 4 12 12 12

x x x x x− = − =

70. 3 15 4 19

4 5 20 20 20

x x x x x+ = + =

71. 3 5 1

10 6 2 2 4

x x x⋅ = ⋅ =

72. 2 6 2 7 7 7

3 7 3 6 3 3 9

x x x x÷ = ⋅ = ⋅ =

73. False. Because zero is nonnegative but not positive, not every nonnegative number is positive.

74. False. Two numbers with different signs will always have a product less than zero.

75. The product of two negative numbers is positive.

76. (a) Because the price can only be a positive rational number with at most two decimal places, the description matches graph (ii).

(b) Because the distance is a positive real number, the description matches graph (i).

A range of prices can only include zero and positive numbers with at most two decimal places. So, a range of prices can be represented by whole numbers and some noninteger positive fractions.

A range of lengths can only include positive numbers. So, a range of lengths can be represented by positive real numbers.

77. (a)

(b) (i) As n approaches 0, the value of 5 n increases

without bound (approaches infinity).

(ii) As n increases without bound (approaches infinity), the value of 5 n approaches 0.

Section P.2 Solving Equations

1. equation

2. 0+ =ax b

3. extraneous

4. factoring; square roots; completing; square; Quadratic Formula

5. 11 15

11 11 15 11

4

+ =+ − = −

=

x

x

x

6. 7 19

7 19

7 19

7 19 19 19

12

− =− + = +

= +− = + −− =

x

x x x

x

x

x

7. 7 2 25

7 7 2 25 7

2 18

2 18

2 2

9

− =− − = −

− =− =− −

= −

x

x

x

x

x

8. 7 2 23

7 2 2 23 2

7 21

7 21

7 73

x

x

x

x

x

+ =+ − = −

=

=

=

9. 3 5 2 7

3 2 5 2 2 7

5 7

5 5 7 5

12

x x

x x x x

x

x

x

− = +− − = − +

− =− + = +

=

n 0.0001 0.01 1 100 10,000

5 n 50,000 500 5 0.05 0.0005








10. 4 2 5 7 6

4 5 2 7 6

2 7 6

6 2 7 6 6

5 2 7

5 2 2 7 2

5 5

5 5

5 51

y y y

y y y

y y

y y y y

y

y

y

y

y

+ − = −− + = −

− + = −− + + = − +

+ =+ − = −

=

=

=

11. ( )3 2 3 8 5

6 9 8 5

5 9 8 5

5 5 9 8 5 5

9 8

x x x

x x x

x x

x x x x

− + = −

− − = −− − = −

− + − = − +− =/

Because 9 8− = is a contradiction, the equation has no

solution.

12. ( )9 10 5 2 2 5

9 10 5 4 10

9 10 9 10

x x x

x x x

x x

− = + −

− = + −− = −

Because the equation is an identity, the solution is the set of all real numbers.

13.

( ) ( ) ( )

3 44

8 33 4

24 24 24 48 3

9 32 96

23 96

96

23

− =

− =

− =− =

= −

x x

x x

x x

x

x

14.

( ) ( ) ( ) ( )

5 1 1

4 2 25 1 1

4 4 4 44 2 2

5 2 4 2

4

+ = −

+ = −

+ = −= −

xx

xx

x x

x

15.

( ) ( )

5 4 2

5 4 3

3 5 4 2 5 4

15 12 10 8

5 20

4

x

x

x x

x x

x

x

− =+− = +

− = +==

16.

( ) ( )

10 3 1

5 6 2

2 10 3 1 5 6

20 6 5 6

15 0

0

x

x

x x

x x

x

x

+ =++ = +

+ = +==

17. 13 5

10 4

10 13 4 5

10 13 4 5

6 18

3

x xx x

x xx x

x

x

− = +

− +=

− = +==

18.

( ) ( ) ( )

1 20

51 2

5 5 5 05

5 2 0

3 5 0

3 5

5

3

+ =−

− + − = −−

− + =− =

=

=

x x

x x x x x xx x

x x

x

x

x

19. 4

24 4

42

41 2

+ = −+ +

+ = −+

= −/

x

x xx

x

Because 1 2= − is a contradiction, the equation has no solution.








20. ( )( )

( ) ( ) ( )( )2 2

7 84 Multiply each term by 2 1 2 1 .

2 1 2 17 2 1 8 2 1 4 2 1 2 1

14 7 16 8 16 4

6 11

11

6

− = − + −+ −

− − + = − + −− − − = − +

=

=

xx x

x xx x x x x

x x x x

x

x

21. ( )( ) ( )( )

( ) ( )

2 1 2Multiply each term by 4 2 .

4 2 4 2

2 1 2 2 4

2 2 2 8

2 3 10

12 3

4

= + − −− − − −

= − + −= − + −= −==

x xx x x x

x x

x x

x

x

x

A check reveals that 4x = yields a denominator of zero. So, 4=x is an extraneous solution, and the original equation has no real solution.

22. ( )( )

( )( )( )( ) ( )( ) ( )( )

( ) ( )

12 3 2

1 3 1 3

12 3 21 3 1 3 1 3

1 3 1 3

12 3 3 2 1

12 3 9 2 2

12 5 7

5 5

1

= +− + − +

− + = − + + − +− + − +

= + + −

= + + −= +==

x x x x

x x x x x xx x x x

x x

x x

x

x

x

Multiply each term by ( )( )1 3 .− +x x

A check reveals that 1=x yields a denominator of zero. So, 1=x is an extraneous solution, and the original equation has no real solution.

23.

( )( )( ) ( )

2

1 1 10

3 3 91 1 10

3 3 3 3

1 3 1 3 10

2 10

5

+ =− + −

+ =− + + −

+ + − =

==

x x x

x x x x

x x

x

x

Multiply each term by ( )( )3 3 .+ −x x

24.

( )( ) ( )( )

( ) ( )

2

1 3 4

2 3 61 3 4

Multiply both sides by 3 2 .2 3 3 2

3 3 2 4

3 3 6 4

4 3 4

4 7

7

4

x x x x

x xx x x x

x x

x x

x

x

x

+ =− + + −

+ = + −− + + −

+ + − =+ + − =

− ==

=








25.

( )

26 3 0

3 2 1 0

x x

x x

+ =

+ =

12

3 0 or 2 1 0

0 or

x x

x x

= + == = −

26.

( )2

14

8 2 0

2 4 1 0

8 0 or 4 1 0

0 or

− =− =

= − == =

x x

x x

x x

x x

27.

( )( )2 10 25 0

5 5 0

5 0

5

x x

x x

x

x

+ + =+ + =

+ == −

28.

( )( )

2 2 8 0

4 2 0

x x

x x

− − =

− + =

4 0 or 2 0

4 or 2

x x

x x

− = + == = −

29.

( )( )

23 5 2 0

3 1 2 0

x x

x x

+ − =

− + =

12

3 0 or 1 2 0

3 or

x x

x x

− = + == = −

30.

( )( )

24 12 9 0

2 3 2 3 0

x x

x x

+ + =

+ + =

32

2 3 0x x+ = = −

31.

( )( )2

34

34

16 9 0

4 3 4 3 0

4 3 0

4 3 0

x

x x

x x

x x

− =+ + =

+ = = −

− = =

32.

( )( ) ( )( )

( )( )

2

2

2

2

8 12

8 12 0

1 8 12 1 0

8 12 0

6 2 0

x x

x x

x x

x x

x x

− + =

− + − =

− − + − = −

− + =

− − =

6 0 6

2 0 2

x x

x x

− = =− = =

33.

( ) ( )

( )( )

2

2

2

34

34

8 20 0

4 8 20 4 0

3 32 80 0

3 20 4 0

x x

x x

x x

x x

+ + =

+ + =

+ + =+ + =

203

3 20 0 or 4 0

or 4

x x

x x

+ = + == − = −

34.

( )( )

2

2

18

16 0

8 128 0

16 8 0

x x

x x

x x

− − =

− − =− + =

16 0 16

8 0 8

x x

x x

− = =+ = = −

35. 2 49

7

x

x

== ±

36. 2 43

43

6.56

=

= ±≈

x

x

x

37. 2

2

3 81

27

3 3

5.20

=

=

= ±≈ ±

x

x

x

38. 2

2

9 36

4

4 2

x

x

x

=

=

= ± = ±

39. ( )24 49

4 7

4 7

11 or 3

− =

− = ±= ±= = −

x

x

x

x x

40. ( )29 24

9 24

9 2 6

4.10, 13.90

+ =

+ = ±

= − ±≈ − −

x

x

x

41. ( )22 1 18

2 1 18

2 1 3 2

1 3 2

22.62, 1.62

− =

− = ±

= ±

±=

≈ −

x

x

x

x








42. ( ) ( )( )

2 27 3

7 3

x x

x x

− = +

− = ± +

7 3 or 7 3

7 3 or 2 4

2

x x x x

x

x

− = + − = − −− = =/

=

The only solution of the equation is 2.x =

43.

( )

2

2

2 2 2

2

4 32 0

4 32

4 2 32 2

2 36

2 6

2 6

4 or 8

x x

x x

x x

x

x

x

x x

+ − =

+ =

+ + = +

+ =

+ = ±= − ±= = −

44.

( ) ( )( )

2

2

2 22

2

2 3 0

2 3

2 1 3 1

1 4

1 4

1 2

3 or 1

x x

x x

x x

x

x

x

x x

− − =

− =

− + − = +

− =

− = ±= ±= = −

45.

( )

2

2

2 2 2

2

4 2 0

4 2

4 2 2 2

2 2

2 2

2 2

x x

x x

x x

x

x

x

+ + =

+ = −

+ + = − +

+ =

+ = ±

= − ±

46.

( )

2

2

2 2

2

8 14 0

8 14

8 4 14 16

4 2

4 2

4 2

x x

x x

x x

x

x

x

+ + =

+ = −

+ + = − +

+ =

+ = ±

= − ±

47.

( )

2

2

2 2 2

2

6 12 3

12

21

2 1 12

11

2

11

2

11

2

21

2

− = −

− = −

− + = − +

− =

− = ±

= ±

= ±

x x

x x

x x

x

x

x

x

48. 2

2

2 22

2

4 4 1

1

4

1 1 1

2 4 2

1 1

2 2

1 2

2 2

1 2

2 2

1 2

2

x x

x x

x x

x

x

x

x

− =

− =

− + − = + −

− =

− = ±

= ±

±=

49. 2

2

2

2 22

2

2 5 8 0

2 5 8

54

2

5 5 54

2 4 4

5 89

4 16

5 89

4 4

5 89

4 4

5 89

4

x x

x x

x x

x x

x

x

x

x

+ − =

+ =

+ =

+ + = +

+ =

+ = ±

= − ±

− ±=








50. 2

2

2

2 22

2

3 4 7 0

3 4 7

4 7

3 3

4 2 7 2

3 3 3 3

2 25

3 9

2 5

3 32 5

3 37

1 or 3

x x

x x

x x

x x

x

x

x

x x

− − =

− =

− =

− + − = + −

− =

− = ±

= ±

= − =

51.

( )

2 2 2

2

1 1

2 5 2 1 51

1 4

=− + − + +

=− +

x x x x

x

52. 2 2 2 2

2

2

4 4

10 74 10 (5) (5) 74

4

10 25 49

4

( 5) 49

x x x x

x x

x

=+ + + + − +

=+ + +

=+ +

53. 2 2

2 2 2

2

2

1 1

3 2 1( 2 3)

1

1 2 (1) (1) 3

1

1( 2 1) 4

1

4 ( 1)

x x x x

x x

x x

x

=+ − − − −

= − − + − −

=− − + +

=− −

54. 2 2 2 2 2

22

1 1 1

12 4 1( 4 12) 1 4 (2) (2) 12

1 1

16 ( 2)1 ( 4 4) 16

= =+ − − − − − − + − −

= =− − − − + −

x x x x x x

xx x

55. 22 1 0x x+ − =

( )( )( )

2

2

4

2

1 1 4 2 1

2 2

1 3 1, 1

4 2

b b acx

a

− ± −=

− ± − −=

− ±= = −

56. 22 1 0x x− − =

( ) ( ) ( )( )( )

2

2

4

2

1 1 4 2 1

2 2

1 1 8

41 3 1

1,4 2

b b acx

a

− ± −=

− − ± − − −=

± +=

±= = −

57. 29 30 25 0x x+ + =

( )( )( )

2

2

4

2

30 30 4 9 25

2 9

30 0 5

18 3

b b acx

a

− ± −=

− ± −=

− ±= = −

58. 2

2

28 49 4

49 28 4 0

x x

x x

− =

− + − =

( )( )( )

2

2

4

2

28 28 4 49 4

2 49

28 0 2

98 7

b b acx

a

− ± −=

− ± − − −=

−

− ±= =−








59. 22 7 1 0− + =x x

( ) ( ) ( )( )( )

( )

2

2

4

2

7 7 4 2 1

2 2

7 49 8

2 2

7 41

4

7 41

4 4

− ± −=

− − ± − −=

± −=

±=

= ±

b b acx

a

60. 2

2

3 1 0

3 1 0

x x

x x

+ − =

+ − =

( )( )( )

2

2

4

2

3 3 4 1 1

2 1

3 13 3 13

2 2 2

b b acx

a

− ± −=

− ± − −=

− ±= = − ±

61. 2

2

12 9 3

9 12 3 0

x x

x x

− = −

− + + =

( )( )( )

2

2

4

2

12 12 4 9 3

2 9

12 6 7 2 7

18 3 3

b b acx

a

− ± −=

− ± − −=

−

− ±= = ±−

62. 2

2

9 37 6

9 6 37 0

x x

x x

− =

− − =

( ) ( )( )( )

2

2

4

2

6 6 4 9 37

2 9

6 6 38 1 38

18 3 3

b b acx

a

− ± −=

± − − −=

±= = ±

63. 2

2

2 2 0

2 2 0

x x

x x

+ − =

− + + =

( )( )( )

2

2

4

2

2 2 4 1 2

2 1

2 2 3

2

1 3

− ± −=

− ± − −=

−

− ±=−

= ±

b b acx

a

64. 2

2

10 8 0

8 10 0

x x

x x

+ + =

+ + =

( ) ( ) ( )( )( )

( )

( )

2

2

4

2

8 8 4 1 10

2 1

8 64 40

2 1

8 24

2

8 2 6

2

2 4 6

2

4 6

− ± −=

− ± −=

− ± −=

− ±=

− ±=

− ±=

= − ±

b b acx

a

65. 2

2

8 5 2

2 8 5 0

t t

t t

= +

− + − =

( )( )( )

2

2

4

2

8 8 4 2 5

2 2

8 2 6 62

4 2

b b act

a

− ± −=

− ± − − −=

−

− ±= = ±−








66. 225 80 61 0h h+ + =

( )( )( )

2

2

4

2

80 80 4 25 61

2 25

80 6400 6100

50

8 10 3

5 50

8 3

5 5

b b ach

a

− ± −=

− ± −=

− ± −=

= − ±

= − ±

67. ( )2

2

5 2

12 25 0

y y

y y

− =

− + =

( ) ( ) ( )( )( )

2

2

4

2

12 12 4 1 25

2 1

12 2 116 11

2

b b acy

a

− ± −=

− − ± − −=

±= = ±

68. ( )2

2

2

6 2

12 36 2

14 36 0

z z

z z z

z z

+ = −

+ + = −

+ + =

( )( )( )

2

2

4

2

14 14 4 1 36

2 1

14 527 13

2

b b acz

a

− ± −=

− ± −=

− ±= = − ±

69. 2

2

5.1 1.7 3.2

5.1 1.7 3.2 0

x x

x x

− =

− − =

( ) ( )( )

( )

21.7 1.7 4 5.1 3.2

2 5.1

0.976, 0.643

x± − − −

=

≈ −

70. 2

2

2 2.53 0.42

2 2.53 0.42 0

− =

− − =

x x

x x

( ) ( )( )( )

22.53 2.53 4 2 0.42

2 2

2.53 9.7609

41.414, 0.149

± − − −=

±=

≈ −

x

71. 20.005 0.101 0.193 0x x− + − =

( ) ( )( )( )

2

2

4

2

0.101 0.101 4 0.005 0.193

2 0.005

0.101 0.006341

0.01

2.137, 18.063

b b acx

a

− ± −=

− ± − − −=

−

− ±=−

≈

72. 23.22 0.08 28.651 0x x− − + =

( ) ( ) ( )( )( )

2

2

4

2

0.08 0.08 4 3.22 28.651

2 3.22

0.08 369.0312.995, 2.971

6.44

b b acx

a

− ± −=

− − ± − − −=

−

±= ≈ −−

73.

( )

2

2

2 2 2

2

2 1 0 Complete the square.

2 1

2 1 1 1

1 2

1 2

1 2

x x

x x

x x

x

x

x

− − =

− =

− + = +

− =

− = ±

= ±

74.

( )( )

214 42 0 Factor.

14 3 0

3 0

+ =+ =

+ =

x x

x x x

x x

0 or 3 0

3

= + == −

x x

x








75. ( )22 64 Extract square roots.

2 8

x

x

+ =+ = ±

2 8 or 2 8

6 or 10

+ = + = −= = −

x x

x x

76.

( )2

14 49 0 Extract square roots.

7 0

7 0

7

x x

x

x

x

2 − + =

− =

− ==

77.

( ) ( )( )

2

2

2 22

2

114

114

1 11 12 4 2

1 122 4

1 122 4

12

0 Complete the square.

3

x x

x x

x x

x

x

x

− − =

− =

− + = +

− =

− = ±

= ±

78.

( )( )

2

22

2

34

3 3 92 4 4

32

32

32

3 0 Complete the square.

3

3

3

3

x x

x x

x

x

x

+ − =

+ + = +

+ =

+ = ±

= − ±

79.

( ) ( ) ( )( )( )

2

2

2

3 4 2 7 Quadratic Formula

0 2 3 11

3 3 4 2 11

2 2

3 97

4

3 97

4 4

+ = −= − −

− − ± − − −=

±=

= ±

x x

x x

x

80. ( )( )

( )( )

( )

2 2

22

12

1 Extract square roots.

1

1

For 1 :

0 1 No solution

For 1 :

2 1

+ =

= +

= ± += + +=/= − += −

= −

x x

x x

x x

x x

x x

x

x

81.

( ) ( ) ( )( )

( )2

2

1 13

11 1

1 1 1 31

1 3 1

1 3 3

0 3 3 1

− =+

+ − + = ++

+ − = +

= += + −

x x

x x x x x xx x

x x x x

x x

x x

( ) ( )( )

( )

23 3 4 3 1 3 21

2 3 6x

− ± − − − ±= =

82.

( ) ( ) ( )( )

( )( )

2

2

4 31

1 24 2 3 1 1 2

4 8 3 3 2

2 3 0

1 3 0

1 0 1

3 0 3

− =+ +

+ − + = + +

+ − = + ++ − =

− + =− = =+ = = −

x xx x x x

x x x x

x x

x x

x x

x x

83.

( )( ) ( )( ) ( )( )

2

2

2

2

13

4 21

2 2 2 2 3 2 24 2

2 3 12

3 2 10 0

+ =− +

+ − + + − = + −− +

+ − = −− − =

x

x xx

x x x x x xx x

x x x

x x

( ) ( ) ( )( )

( )

22 2 4 3 10

2 3

2 124 2 2 31 1 31

6 6 3

x− − ± − − −

=

± ± ±= = =








84.

( ) ( )

( )( ) ( )

( )( )

2

2

1 10

3 21 1

3 2 3 2 03 2

2 1 3 1 0

3 2 3 3 0

1 0

1 1 0

1 0 1

1 0 1

x x

xx x

x xx

x x x

x x x

x

x x

x x

x x

+ +− =+

+ ++ − + =+

+ + − + =

+ + − − =

− =

+ − =

+ = = −− = =

85.

( )4 2

2 2

2

2

6 54 0

6 9 0

6 0 0

9 0 3

− =

− =

= =

− = = ±

x x

x x

x x

x x

86.

( )( )

3 2

2

2

5 3 45 0

5 6 9 0

5 3 0

5 0 0

3 0 3

x x x

x x x

x x

x x

x x

+ − + =

+ + =

+ =

= =+ = = −

87.

( ) ( )( )( )

3 2

3 2

2

2

2

2 8 16

2 8 16 0

2 8 2 0

2 8 0

2 0 2

8 0 8 2 2

+ − =

+ − − =

+ − + =

+ − =

+ = = −

− = ± = ±

x x x

x x x

x x x

x x

x x

x

88.

( ) ( )( )( )

( )( )( )

3 2

3 2

3 2

2

2

3 3

3 3

3 3 0

3 3 0

3 1 0

3 3 1 0

3 0 3

1 0 1

1 0 1

− − = −

− − = −

− − + =

− − − =

− − =

− + − =

− = =+ = = −− = =

x x x

x x x

x x x

x x x

x x

x x x

x x

x x

x x

89.

( ) ( )4 2

22 2

4 3 0

4 3 0

x x

x x

− + =

− + =

Let 2.u x=

( )( )

2 4 3 0

3 1 0

3 0 3

1 0 1

u u

u u

u u

u u

− + =

− − =

− = =− = =

2 2

1 3

1 3

1 3

u u

x x

x x

= == == ± = ±

90.

( ) ( )4 2

22 2

13 36 0

13 36 0

x x

x x

− + =

− + =

Let 2.u x=

( )( )

2 13 36 0

9 4 0

9 0 9

4 0 4

u u

u u

u u

u u

− + =

− − =

− = =− = =

2 2

9 4

9 4

3 2

u u

x x

x x

= == == ± = ±

91.

( ) ( )2 2

5 10 0

5 10

5 10

5 100

20

− =

=

=

==

x

x

x

x

x

92.

( ) ( )2 2

8 5 0

8 5

8 5

8 25

17

+ − =

+ =

+ =

+ ==

x

x

x

x

x

93.

( ) ( )

3

3

3 33

4 2 9 0

2 9 4

2 9 4

2 9 64

2 55

55

2

+ − =

− = −

− = −

− = −= −

= −

x

x

x

x

x

x








94.

( ) ( )

3

3

3 33

12 3 0

12 3

12 3

12 27

15

15

x

x

x

x

x

x

− − =

− =

− =

− =− =

= −

95.

( ) ( )

( )( )

2 2

2

2

2

8 2

8 2

8 4 4

0 3 4

3 4 0

4 1 0

4 0 4, extraneous

1 0 1

x x

x x

x x x

x x

x x

x x

x x

x x

+ = +

+ = +

+ = + += + −

+ − =+ − =

+ = = −− = =

96.

( ) ( )

( )( )

22

2

2

34

2 5 24 3

2 3 5 24

2 3 5 24

4 12 9 5 24

4 17 15 0

4 3 5 0

4 3 0

5 5, extraneous

x x

x x

x x

x x x

x x

x x

x x

x x

= − + −

+ = − +

+ = − +

+ + = − ++ − =− + =

− = =

+ = −

97.

( ) ( )

( ) ( )

2 2

22

3 1

3 1

3 1

3 2 1

4 2

2

2

4

− + =

− = −

− = −

− = − +

− = −

=

=

=

x x

x x

x x

x x x

x

x

x

x

98.

( ) ( )( )

( )( )

2 2

2

2

2 1 2 3 1

2 1 1 2 3

2 1 1 2 3

4 1 1 2 2 3 2 3

2 2 2 3

2 3

2 3

2 3 0

3 1 0

3 0 3

1 0 1, extraneous

x x

x x

x x

x x x

x x

x x

x x

x x

x x

x x

x x

+ − + =

+ = + +

+ = + +

+ = + + + +

= +

= +

= +

− − =

− + =

− = =+ = = −

99. ( )( )

3 2

3 2

3

5 8

5 8

5 64

5 4 9

x

x

x

x

− =

− =

− == + =

100. ( )

( ) ( ) ( )( )( )

3 22

2 2 3

2

2

2

22 27

22 27

22 9

31 0

1 1 4 1 31 1 125 1 5 5

2 1 2 2

x x

x x

x x

x x

x

− − =

− − =

− − =

− − =

− − ± − − − ± ±= = =

101. ( ) ( )( ) ( )

( ) ( )( )

1 2 3 2

1 2

1 2

1 2

25

3 1 2 1 0

1 3 2 1 0

1 5 2 0

1 0 1 0 1

5 2 0 , extraneous

x x x

x x x

x x

x x x

x x

− + − =

− + − =

− − =

− = − = =

− = =








102. ( ) ( )( ) ( )

( ) ( )( ) ( )

1 3 4 32

1 3 4 3

1 3

1 3

35

4 1 6 1 0

2 2 1 3 1 0

2 1 2 3 1 0

2 1 5 3 0

2 0 0

1 0 1

5 3 0

x x x x

x x x x

x x x x

x x x

x x

x x

x x

− + − =

− + − =

− + − =

− − =

= =− = =

− = =

103.

( )

2 5 11

2 5 11 8

2 5 11 3

x

x x

x x

− =

− = =

− − = = −

104.

( )

53

3 2 7

3 2 7

3 2 7

3 2 7 3

x

x x

x

x x

+ =

+ = =

− + =

− − = = −

105. 21 5x x+ = −

First equation: Second equation:

( )( )

2

2

1 5

6 0

3 2 0

3 0 3

2 0 2

x x

x x

x x

x x

x x

+ = −

− − =

− + =

− = =+ = = −

( ) 2

2

2

1 5

1 5

4 0

1 17

2

x x

x x

x x

x

− + = −

− − = −

+ − =

− +=

Only 3x = and 1 17

2x

− −= are solutions of the original equation. 2x = − and 1 17

2x

− += are extraneous.

106. 2 6 3 18x x x+ = +

First equation: Second equation:

( )( )

2

2

6 3 18

3 18 0

3 6 0

3 0 3

6 0 6

x x x

x x

x x

x x

x x

+ = +

+ − =

− + =

− = =+ = = −

( )

( )( )

2

2

6 3 18

0 9 18

0 3 6

0 3 3

6 6

x x x

x x

x x

x x

x x x

− + = +

= + +

= + +

= + = −= + = −

The solutions of the original equation are 3x = ± and 6.x = −

107. Let 18.=y

0.514 14.75

18 0.514 14.75

32.75 0.514

32.75

0.51463.7

= −= −=

=

=

y x

x

x

x

x So, the height of the female is about 63.7 inches.

108. Let 23.=y

0.532 17.03

23 0.532 17.03

40.03 0.532

40.03

0.53275.2

= −= −=

=

=

y x

x

x

x

x

The height of the missing man is about 75.2 inches.

Because 75.2 in.1 ft

6.27 ft12 in. =

is about 6 feet

3 inches, it is possible the femur belongs to the missing man.






Section P.3 The Cartesian Plane and Graphs of Equations 17


109. False.

( )

( )

( )

2

2

2 1 2 1

2 1 4 4 1 1

4 4 1

8 16 16 1

24 0

24 0

x x

x x x

x x

x x x

x x

x x

+ = − + +

+ = − + + +

− = − +

− + = +

− =

− =

0 1 2 1,x = ≠ − + 24 5 2 5x = ≠ − +

110. ( )2 3 1 2 5

2 6 1 2 5

2 5 2 5

x x

x x

x x

− + = −

− + = −− = −

False. The equation is an identity, so every real number is a solution.

111. 10 10 0

10 10

− − − =

− = −

x x

x x

False. The equation is an identity, so every real number is a solution.

112. (a) The formula for volume of the glass cube is V = Length × Width × Height. The volume of water in the cube is the length × width × height of the water.

So, the volume is ( ) ( )23 3x x x x x⋅ ⋅ − = − .

(b) Given the equation ( )2 3 320x x − = . The

dimensions of the glass cube can be found by solving for x. Then, substitute that value into the

expression 3x to find the volume of the cube.

113. 3 2

75

3 2 35 and 9 20

3 33 11

11

+ =

+ = + == ==

x

x x

x x

x

Yes, they are equivalent equations. They both have the solution 11.=x

Section P.3 The Cartesian Plane and Graphs of Equations

1. Cartesian

2. Distance Formula

3. Midpoint Formula

4. solution or solution point

5. graph

6. intercepts

7. y-axis

8. circle; ( ), ;h k r

9. ( ) ( ) ( ) ( ): 2, 6 , : 6, 2 , : 4, 4 , : 3, 2A B C D− − − −

10. ( ) ( ) ( ) ( )3 52 2

: , 4 , : 0, 2 , : 3, , : 6, 0A B C D− − − −

11.

12.

13. ( )3, 4−

14. ( )12, 0−

15. 0x > and 0y < in Quadrant IV.

16. 0x < and 0y < in Quadrant III.

17. 4x = − and 0y > in Quadrant II.

18. 0<x and 7=y in Quadrant II.

19. 0, 0, 0x y x y+ = ≠ ≠ means or .x y y x= − = −This occurs in Quadrant II or IV.

20. ( ), , 0x y xy > means x and y have the same signs.

This occurs in Quadrant I or III.

y

x−2−4−6−8 2 4 6 8

−4

−6

2

4

6

8

(2, 4)(−6, 2)

(−4, 0)

(3, −1)

(1.5, −3.5)

(−1, −8)

y

x−4−6−8 2 4 6 8

−2

−4

−6

−8

2

4

6

8

(−2, −7)

(−2, 4)

(1, −5)

(3, 3)

(0, 5) ( , )23

52








21.

22.

23. ( ) ( )

( )( ) ( )

( ) ( )

2 22 1 2 1

2 2

2 2

3 2 6 6

5 12

25 144

13 units

d x x y y= − + −

= − − + − −

= + −

= +=

24. ( ) ( )

( ) ( )

( ) ( )

2 22 1 2 1

2 2

2 2

0 8 20 5

8 15

64 225

289

17 units

d x x y y= − + −

= − + −

= − +

= +

==

25. ( ) ( )

( ) ( )

( ) ( )

2 22 1 2 1

2 2

2 2

5 1 1 4

6 5

36 25

61 units

d x x y y= − + −

= − − + − −

= − + −

= +

=

26. ( ) ( )

( ) ( )( )( ) ( )

2 22 1 2 1

22

2 2

3.9 9.5 8.2 2.6

13.4 10.8

179.56 116.64

296.2

17.21 units

d x x y y= − + −

= − − + − −

= − +

= +

=≈

27. (a) ( ) ( )1, 0 , 13, 5

( ) ( )2 2

2 2

Distance 13 1 5 0

12 5 169 13

= − + −

= + = =

( ) ( )13, 5 , 13, 0

Distance 5 0 5 5= − = =

( ) ( )1, 0 , 13, 0

Distance 1 13 12 12= − = − =

(b) 2 2 25 12 25 144 169 13+ = + = =

28. (a) The distance between ( )1, 1− and ( )9, 1 is 10.

The distance between ( )9, 1 and ( )9, 4 is 3.

The distance between ( )1, 1− and ( )9, 4 is

( )( ) ( )2 29 1 4 1 100 9 109.− − + − = + =

(b) ( )22 210 3 109 109+ = =

Year, x Number of Stores, y

2007 7276

2008 7720

2009 8416

2010 8970

2011 10,130

2012 10,773

2013 10,942

2014 11,453

y

x2011 2012 2013 20142007 2008 2009 2010

7000

8000

9000

10,000

7500

8500

9500

10,50011,00011,500

Year

Num

ber

of s

tore

s

Month, x Temperature, y

1 –39

2 –39

3 –29

4 –5

5 17

6 27

7 35

8 32

9 22

10 8

11 –23

12 –34

Tem

pera

ture

(in

°F)

Month (1 ↔ January)

x

−40

10

−30

−20

−10

20

30

2 6 8 10 12

40

0

y








29. ( ) ( )

( ) ( )

( ) ( )

2 21

2 22

2 23

4 2 0 1 4 1 5

4 1 0 5 25 25 50

2 1 1 5 9 36 45

d

d

d

= − + − = + =

= + + + = + =

= + + + = + =

( ) ( ) ( )2 2 25 45 50+ =

30. ( )( ) ( )

( ) ( )

( )( ) ( )

2 21

2 22

2 23

3 1 5 3 16 4 20

5 3 1 5 4 16 20

5 1 1 3 36 4 40

d

d

d

= − − + − = + =

= − + − = + =

= − − + − = + =

( ) ( ) ( )2 2 220 20 40+ =

31. ( ) ( )

( ) ( )

( ) ( )

2 21

2 22

2 23

1 2

1 3 3 2 4 25 29

3 2 2 4 25 4 29

1 2 3 4 9 49 58

d

d

d

d d

= − + − − = + =

= + + − = + =

= + + − − = + =

=

32. ( ) ( )

( ) ( )

( )( ) ( )

2 21

2 22

2 23

1 2

4 2 9 3 4 36 40

2 4 7 9 36 4 40

2 2 3 7 16 16 32

d

d

d

d d

= − + − = + =

= − − + − = + =

= − − + − = + =

=

33. (a)

(b) ( ) ( )2 29 1 7 1 64 36 10d = − + − = + =

(c) ( )9 1 7 1, 5, 4

2 2

+ + =

34. (a)

(b) 2 2(5 ( 3)) (6 6) 64 8d = − − + − = =

(c) 6 6 5 ( 3)

, (6, 1)2 2

+ + − =

35. (a)

(b) ( ) ( )2 25 1 4 2

36 4 2 10

d = + + −

= + =

(c) ( )1 5 2 4, 2, 3

2 2

− + + =

36. (a)

(b) 2 2

1 5 41

2 2 3

1 829

9 3

d = + + −

= + =

(c) ( ) ( ) ( )5 2 1 2 4 3 1 7

, 1,2 2 6

− + + = −

37. 2 2120 150

36,900

30 41

192.09

d = +

=

=≈

The plane flies about 192 kilometers.

x(1, 1)

(9, 7)

12

10

8

6

4

2

2−2 4 6 8 10

y

x

2

−2

−4

4

6

102 4 8

(6, 5)

(6, −3)

y

x

(5, 4)

(−1, 2)

1−1 2 3 4 5

3

−1

4

5

y

x

( )( ), 1

−−−

25

134

222

3

12

52

12

12

32

52

,−

y

−2 −1








38. ( ) ( )2 2

2 2

42 18 50 12

24 38

2020

2 505

45

d = − + −

= +

=

=≈

The pass is about 45 yards.

39.

( )

1 2 1 2midpoint , 2 2

2010 2014 35,123 45,998,

2 2

2012, 40,560.5

x x y y+ + =

+ + =

=

In 2012, the sales for the Coca-Cola Company were about $40,560.5 million.

40.

( )

1 2 1 2midpoint , 2 2

2013 2015 1.17 3.25,

2 2

2014, 2.21

x x y y+ + =

+ + =

=

In 2014, the revenue per share for Twitter, Inc. was approximately $2.21.

41. (a) ( )?

0, 2 : 2 0 4

2 2

= +=

Yes, the point is on the graph.

(b) ( )?

?

5, 3 : 3 5 4

3 9

3 3

= +

==


42. (a) ( )?

?

1, 5 : 5 4 1 2

5 4 1

5 3

= − −

= −≠

No, the point is not on the graph.

(b) ( )?

?

6, 0 : 0 4 6 2

0 4 4

0 0

= − −

= −=


43. (a) ( ) ( ) ( )?2

?

2, 0 : 2 3 2 2 0

4 6 2 0

0 0

− + =

− + ==


(b) ( ) ( ) ( )?2

?

2, 8 : 2 3 2 2 8

4 6 2 8

12 8

− − − − + =

+ + =≠


44. (a) ( ) ( )

( )

? 2

?

1, 1 : 1 3 2 1

1 3 2 1

1 1

− = − −

= −=


(b) ( ) ( )

( )

? 2

?

2, 11 : 11 3 2 2

11 3 2 4

11 5

− = − −

= −≠ −


45. (a) ( ) ( ) ( )?2 2

?

3, 2 : 3 2 20

9 4 20

13 20

− + − =

+ =≠


(b) ( ) ( ) ( )?2 2

?

4, 2 : 4 2 20

16 4 20

20 20

− − + =

+ ==


46. (a) ( ) ( ) ( )

( ) ( )

?2 2

?

?

6, 0 : 2 6 5 0 8

2 36 5 0 8

72 0 8

72 8

+ =

+ =

+ =≠


(b) ( ) ( ) ( )

( ) ( )

?2 2

?

?

0, 4 : 2 0 5 4 8

2 0 5 16 8

0 80 8

80 8

+ =

+ =

+ =≠









47. 2 5y x= − +

48. 34

1+ =y x

49. 23+ =y x x

50. 25y x= −

51. 5 6y x= −

x-intercept:

65

0 5 6

6 5

x

x

x

= −=

=

( )65, 0

y-intercept: ( )5 0 6 6y = − = −

( )0, 6−

52. 8 3y x= −

x-intercept:

83

0 8 3

3 8

x

x

x

= −=

=

( )83, 0

y-intercept: ( )8 3 0 8y = − =

( )0, 8

53. 4y x= +

x-intercept: 0 4

0 4

4

x

x

x

= += +

− =

( )4, 0−

y-intercept: 0 4 2y = + =

( )0, 2

x −1 0 1 2 52

y 7 5 3 1 0

(x, y) ( )1, 7− ( )0, 5 ( )1, 3 ( )2, 1 ( )52, 0

x −2 0 1 43

2

y 52

− –1 14

− 0 12

(x, y) ( )52

2,− − ( )0, 1− ( )14

1, − ( )43, 0 ( )1

22,

x −1 0 1 2 3

y 4 0 –2 –2 0

(x, y) ( )1, 4− ( )0, 0 ( )1, 2− ( )2, 2− ( )3, 0

x −2 –1 0 1 2

y 1 4 5 4 1

x, y ( )2, 1− ( )1, 4− ( )0, 5 ( )1, 4 ( )2, 1

x541−1−2−3

4

7

2

1

−1

2

3

5

y

−4 −3 −2 −1 2 3 4

−4

−3

−2

1

2

3

4

x

y

1 2

3

4

5

4 5x

−2 −1−1

−2

y

−4 −3 −1 1 3 4

−2

−1

1

2

3

6

4

x

y








54. 2 1y x= −

x-intercept:

12

0 2 1

2 1 0

x

x

x

= −− =

=

( )12, 0

y-intercept: ( )2 0 1

1

y = −

= −

There is no real solution. There is no y-intercept.

55. 3 7y x= −

x-intercept:

73

0 3 7

0 3 7

0

x

x

= −

= −

=

( )73, 0

y-intercept: ( )3 0 7 7y = − =

( )0, 7

56. 10y x= − +

x-intercept: 0 10

10 0

10

x

x

x

= − +

+ == −

( )10, 0−

y-intercept: 0 10

10 10

y = − +

= − = −

( )0, 10−

57. 3 22 4y x x= −

x-intercept:

( )

3 2

2

0 2 4

0 2 2

0 or 2

x x

x x

x x

= −

= −

= =

( ) ( )0, 0 , 2, 0

y-intercept: ( ) ( )3 22 0 4 0

0

y

y

= −=

( )0, 0

58. 4 25y x= −

x-intercepts: 4

4

4 2

0 25

25

5 5

x

x

x

= −

=

= ± = ±

( )5, 0±

y-intercept: ( )40 25 25y = − = −

( )0, 25−

59. 2 6y x= −

x-intercept: 0 6

6

x

x

= −=

( )6, 0

y-intercepts: 2 6 0

6

y

y

= −

= ±

( ) ( )0, 6 , 0, 6−

60. 2 1y x= +

x-intercept: 0 1

1

x

x

= += −

( )1, 0−

y-intercepts: 2 0 1

1

y

y

= += ±

( ) ( )0, 1 , 0, 1−

61. 2 0x y− =

( )( )

( ) ( )

2 2

2 2

2 2

0 0 -axis symmetry

0 0 No -axis symmetry

0 0 No origin symmetry

x y x y y

x y x y x

x y x y

− − = − =

− − = + =

− − − = + =








62. 2 0x y− =

( )( )

( ) ( )

2 2

2 2

2 2


0 0 -axis symmetry


x y x y y

x y x y x

x y x y

− − = − − =

− − = − =

− − − = − − =

63.

( )

( )

3

3 3

3 3

3 3 3

No -axis symmetry

No -axis symmetry

Origin symmetry

y x

y x y x y

y x y x x

y x y x y x

=

= − = −

− = = −

− = − − = − =

64.

( ) ( )

( ) ( )

4 2

4 2 4 2

4 2 4 2

4 2 4 2

3

3 3 -axis symmetry



y x x

y x x y x x y

y x x y x x x

y x x y x x

= − +

= − − − + = − +

− = − + = − + −

− = − − − + = − + −

65.

( )

( )

2

2 2

2 2

2 2 2

1

No -axis symmetry11

No -axis symmetry1 1

Origin symmetry1 11

xy

xx x

y y yxx

x xy y x

x xx x x

y y yx xx

=+− −= =

+− +

−− = = + +− −− = − = =

+ +− +

66.

( )

( )

2

2 2

2 2

2 2

1

11 1

-axis symmetry11

1 1No -axis symmetry

1 11 1

No origin symmetry11

=+

= = +− +

−− = = + +

−− = = +− +

yx

y y yxx

y y xx x

y yxx

67.

( )( )

( )( )

2

2 2

2 2

2 2

10 0

10 0 10 0 No -axis symmetry

10 0 10 0 -axis symmetry

10 0 10 0 No origin symmetry

xy

x y xy y

x y xy x

x y xy

+ =

− + = − + =

− + = + =

− − + = − + =

68.

( )( )

( )( )

4



4 4 Origin symmetry

xy

x y xy y

x y xy x

x y xy

=

− = = −

− = = −

− − = =

69.

−4 −3 −1 1 3 4

1

2

−2

3

4

x

y








70.

71.

72.

73. 3 1y x= − +

x-intercept: ( )13, 0

y-intercept: ( )0, 1

No symmetry

74. 2 3y x= −


y-intercept: ( )0, 3−

No symmetry

75. 2 2y x x= −

x-intercepts: ( ) ( )0, 0 , 2, 0


No symmetry

76. 2 1x y= −

x-intercept: ( )1, 0−

y-intercepts: ( ) ( )0, 1 , 0, 1−

x-axis symmetry

x –1 0 1 2 3

y 3 0 –1 0 3

x –1 0 3

y 0 ±1 ±2

1 2 3 6 7 8

−4

−3

−2

−1

1

2

3

4

x

y

−4 −3 −2 1 2 3 4

−4

−3

−2

1

2

3

4

x

y

−4 −3 −2 −1 1 2 3 4

−4

−3

−2

1

2

3

4

x

y

( (x

(0, 1) 13

y

−1−2−3−4 1 2 3 4−1

−2

−3

1

4

5

, 0

(0, −3)

32 , 0( )

y

x−1−2−3 1 2 3

−1

−2

−3

1

2

−1

2

3

4

−1 1 2 3 4x

−2

−2

(0, 0) (2, 0)

y

−2 1 2 3 4

−3

−2

2

3

x

(0, −1)

(−1, 0) (0, 1)

y








77. 3 3y x= +

x-intercept: ( )3 3, 0−


No symmetry

78. 3 1y x= −


y-intercept: ( )0, 1−

No symmetry

79. 3y x= −


y-intercept: none

No symmetry

80. 1y x= −



No symmetry

81. 6y x= −



No symmetry

82. 1y x= −

x-intercepts: ( ) ( )1, 0 , 1, 0−


y-axis symmetry

83. Center: ( )0, 0 ; Radius: 7

( ) ( )2 2 2

2 2

0 0 7

49

− + − =

+ =

x y

x y

x –2 –1 0 1 2

y –5 2 3 4 11

x 3 4 7 12

y 0 1 2 3

x –2 0 2 4 6 8 10

y 8 6 4 2 0 2 4

−4 −3 −2−1

1

2

4

5

6

7

1 2 3 4x

(0, 3)

−3, 03( (

y

−3 −2 −1 2 3

−4

−3

−2

1

2

x(1, 0)

(0, −1)

y

1 2 3 4 5 6–1

1

2

3

4

5

x(3, 0)

y

−4 −3 −2 −1 1 2−1

2

3

4

5

x(1, 0)

(0, 1)

y

2−2

2

−2

4

6

8

10

12

4 6 8 10 12x

(6, 0)

(0, 6)

y

x2 31−2−3

2

−1

−2

−3

1

3

(−1, 0) (1, 0)

(0, 1)

y








84. Center: ( )4, 5 ;− Radius: 2

( ) ( )( ) [ ]

( ) ( )

2 2 2

2 2 2

2 2

4 5 2

4 5 4

− + − =

− − + − =

+ + − =

x h y k r

x y

x y

85. Center: ( )3, 8 ; Solution point: ( )9, 13−

( ) ( )

( ) ( )

( ) ( )

2 2

2 2

2 2

9 3 13 8

12 5

144 25

169

13

= − + −

= − − + −

= − +

= +

==

r x h y k

( ) ( )( ) ( )( ) ( )

2 2 2

2 2 2

2 2

3 8 13

3 8 169

− + − =

− + − =

− + − =

x h y k r

x y

x y

86. Center: ( )2, 6 ;− − Solution point: ( )1, 10−

( ) ( )

( ) ( )

( ) ( )

2 2

2 2

2 2

1 2 10 6

3 4

9 16

25

5

= − + −

= − − + − − −

= + −

= +

==

r x h y k

( ) ( )( ) ( )

( ) ( )

2 2 2

2 2 2

2 2

2 6 5

2 6 25

− + − =

− − + − − =

+ + + =

x h y k r

x y

x y

87. Endpoints of a diameter: ( ) ( )3, 2 , 9, 8− −

( ) ( )

( ) ( )

( )

2 2

2 2

19 3 8 2

21

12 1021

144 10021 1

244 2 61 612 2

= − − + − −

= − + −

= +

= = =

r

( ) ( ) ( )

( )

3 9 2 8 6 6, :

2 2 2 2

3, 3

+ − + − − − ⋅ = ⋅

= − −

h k

( ) ( )

( ) ( ) ( )( ) ( )

2 2 2

22 2

2 2

3 3 61

3 3 61

− + − =

− − + − − =

+ + + =

x h y k r

x y

x y

88. Endpoints of a diameter: ( ) ( )11, 5 , 3, 15−

( ) ( )

( ) ( )

( )

22

2 2

13 11 15 5

21

8 2021

64 40021 1

464 4 29 2 292 2

= − + − −

= − +

= +

= = =

r

( ) ( )11 3 5 15 14 10

, : , 7, 52 2 2 2

+ − + ⋅ = =

h k

( ) ( )

( ) ( ) ( )( ) ( )

2 2 2

22 2

2 2

7 5 2 29

7 5 116

− + − =

− + − =

− + − =

x h y k r

x y

x y

89. 2 2 25x y+ =

Center: ( )0, 0 , Radius: 5

1−1−2

−2−3−4

−6

−3−4

1234

6

2 3 4 6x

y

(0, 0)








90. ( )22 1 1x y+ − =

Center: ( )0, 1 , Radius: 1

91. ( ) ( )2 2 91 12 2 4

x y− + − =

Center: ( )1 12 2, , Radius: 3

2

92. ( ) ( )2 2 169

2 3x y− + + =

Center: ( )2, 3 ,− Radius: 43

93. 1,200,000 80,000 , 0 10= − ≤ ≤y t t

94. 9500 1000 , 0 6= − ≤ ≤y t t

95. (a)

(b)

When 85.5,x = the resistance is about 1.4 ohms.

(c) When 85.5,x =

2

10,3701.4 ohms.

(85.5)y = =

(d) As the diameter of the copper wire increases, the resistance decreases.

x 5 10 20 30 40 50 60 70 80 90 100

y 414.8 103.7 25.9 11.5 6.5 4.1 2.9 2.1 1.6 1.3 1.0

−2 −1 1 2

−1

1

3

x

y

(0, 1)

−1 1 2 3

1

3

x

y

( )12

, 12

y

x−1 1 2 3 4

−1

−2

−3

−4

1

(2, −3)

y

t1 2

200,000

400,000

600,000

800,000

1,000,000

1,200,000

YearD

epre

ciat

ed v

alue

3 4 5 6 7 8 9 10

y

t1

1000

3000

5000

700080009000

2000

4000

6000

10,000

Year

Dep

reci

ated

val

ue

2 3 4 5 6

x

Diameter of wire (in mils)

Res

ista

nce

(in

ohm

s)

y

20 40 60 80 100

50100150200250300350400450








96. (a)

The model fits the data well.

(b) Graphically: The point ( )50, 74.7 represents a life

expectancy of 74.7 years in 1990.

Algebraically: ( )

( )63.6 0.97 50

1 0.01 50

112.1

1.574.7

+=

+

=

=

y

So, the life expectancy in 1990 was about 74.7 years.

(c) Graphically: The point ( )24.2, 70.1 represents a life

expectancy of 70.1 years during the year 1964.

Algebraically:

( )

63.6 0.97

1 0.0163.6 0.97

70.11 0.01

70.1 1 0.01 63.6 0.97

70.1 0.701 63.6 0.97

6.5 0.269

24.2

+=+

+=+

+ = +

+ = +==

ty

tt

t

t t

t t

t

t

When 70.1,=y 24.2=t which represents the year

1964.

(d) ( )

( )63.6 0.97 0

1 0.01 0

63.663.6

1

+=

+

= =

y

The y-intercept is ( )0, 63.6 . In 1940, the life

expectancy of a child (at birth) was 63.6 years.

(e) Answers will vary.

97. False, you would have to use the Midpoint Formula 15 times.

98. True. Two sides of the triangle have lengths 149 and

the third side has a length of 18.

99. False. The line =y x is symmetric with respect to the

origin.

100. True. Depending upon the center and radius, the graph of a circle could intersect one, both, or neither axis.

101. Answers will vary. Sample answer: When the x-values are much larger or smaller than the y-values, different scales for the coordinate axes should be used.

102. The y-coordinate of a point on the x-axis is 0. The x-coordinates of a point on the y-axis is 0.

103. Use the Midpoint Formula to prove the diagonals of the parallelogram bisect each other.

0, ,

2 2 2 2

0 0, ,

2 2 2 2

b a c a b c

a b c a b c

+ + + =

+ + + + =

104. (a) Because ( )0 0,x y lies in Quadrant II, ( )0 0,x y− must

lie in Quadrant III. Matches (ii).

(b) Because ( )0 0,x y lies in Quadrant II, ( )0 02 ,x y−

must lie in Quadrant I. Matches (iii).

(c) Because ( )0 0,x y lies in Quadrant II, ( )0 012

,x y must

lie in Quadrant II. Matches (iv).

(d) Because ( )0 0,x y lies in Quadrant II, ( )0 0,x y− −

must lie in Quadrant IV. Matches (i).

105. Because 1 2

2m

x xx

+= and 1 2

2m

y yy

+= we have:

1 2 1 2

1 2 1 2

2 2

2 2

m m

m m

x x x y y y

x x x y y y

= + = +− = − =

So, ( ) ( )2 2 1 1, 2 , 2 .m mx y x x y y= − −

(a) ( ) ( ) ( ) ( )( ) ( )2 2 1 1, 2 , 2 2 4 1, 2 1 2 7, 0m mx y x x y y= − − = ⋅ − − − − =

(b) ( ) ( ) ( )( ) ( )2 2 1 1, 2 , 2 2 2 5 , 2 4 11 9, 3m mx y x x y y= − − = ⋅ − − ⋅ − = −

00

100

100






Section P.4 Linear Equations in Two Variables 29


Section P.4 Linear Equations in Two Variables

1. linear

2. slope

3. point-slope

4. parallel

5. perpendicular

6. rate or rate of change

7. linear extrapolation

8. general

9. (a) 23.m = Because the slope is positive, the line rises.

Matches 2.L

(b) m is undefined. The line is vertical. Matches 3.L

(c) 2.m = − The line falls. Matches 1.L

10. (a) 0.m = The line is horizontal. Matches 2.L

(b) 34.m = − Because the slope is negative, the line

falls. Matches 1.L

(c) 1.m = Because the slope is positive, the line rises. Matches 3.L

11.

12.

13. Two points on the line: ( )0, 0 and ( )4, 6

2 1

2 1

6 3Slope

4 2

y y

x x

−= = =−

14. The line appears to go through ( )0, 7 and ( )7, 0 .

2 1

2 1

0 7Slope 1

7 0

y y

x x

− −= = = −− −

15. 5 3y x= +

Slope: 5m =


16.

( )Slope: 1

-intercept: 0, 10

m

y

= −

−

17.

( )

34

34

1

Slope:

-intercept: 0, 1

= − −

= −

−

y x

m

y

x

12

(−4, 1)

−6 −2

−2

2

4

undefined

m = −3m = 3

m =

y

x

3

2−1−2−3−4 31

4

5

(0, 3)

y

−2−4−6−10−12 2−2

−4

−6

−10

−12

2

x

y

(0, −10)

(0, −1)

y

x−2−4−6 2 4 6

−2

−4

−6

−8

2

4

x1 2

1

2

(2, 3)

m = 1

m = 2

m = 0

m = −3

y








18.

( )

23

23

2

Slope:

-intercept: 0, 2

= +

=

y x

m

y

19. 5 0

5

− ==

y

y

( )

Slope: 0

-intercept: 0, 5

m

y

=

20. 4 0

4

+ == −

x

x

( )Slope: undefined vertical line

-intercept: noney

21. 25

5 2 0

, vertical line

− =

=

x

x

Slope: undefined

-intercept: noney

22.

53

3 5 0

3 5

y

y

y

+ == −

= −

Slope: 0m =

y-intercept: ( )53

0, −

23.

76

7 6 30

6 7 30

5

x y

y x

y x

− =− = − +

= −

( )

76

Slope:

-intercept: 0, 5

m

y

=

−

(0, 2)

y

x−2−6 2 4

−2

−4

4

6

(0, 5)

y

x−2−4 2 4

−2

2

4

6

8

y

x−2−6 2

−2

−4

2

4

−1 1 2 3

−2

−1

1

2

x

y

x

53( )

−1−2 21

−1

−2

−3

1

0, −

y

x

y

−1 1 2 3 5 6 7−1

−2

−3

−4

−5

−7

1

(0, −5)






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