College Algebra 10th Edition Sullivan Solutions Manual

College Algebra 10th Edition Sullivan SOLUTIONS MANUAL Full download at: https://testbankreal.com/download/college-algebra-10th-editionsullivansolutions-manual/ College Algebra 10th Edition Sullivan TEST BANK Full download at: https://testbankreal.com/download/college-algebra-10th-edition-sullivantestbank/

Chapter 2 Graphs (f) Quadrant IV 2. 3.

5 − ( −3) = 8 = 8 32 + 4 2 = 25 = 5

4. 112 + 60 2 = 121 + 3600 = 3721 = 612 Since the sum of the squares of two of the sides of the triangle equals the square of the third side, the triangle is a right triangle. 1 5. bh 2

12. True; M =

6. true 7. x-coordinate or abscissa; y-coordinate or ordinate 8. quadrants 9. midpoint 10. False; the distance between two points is never negative. 11. False; points that lie in Quadrant IV will have a positive x-coordinate and a negative y-coordinate. The point ( −1, 4 ) lies in Quadrant II.

147 147 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

x1+ x2 y1+ y2  ,

16. (a) Quadrant I (b) Quadrant III (c) Quadrant II (d) Quadrant I (e) y-axis (f) x-axis

2

2

 

17. The points will be on a vertical line that is two units to the right of the y-axis.

13. b 14. a 15. (a) (b) (c) (d) (e)

Quadrant II x-axis Quadrant III Quadrant I y-axis

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Chapter 2: Graphs

Section 2.1: The Distance and Midpoint Formulas

18. The points will be on a horizontal line that is three units above the x-axis.

28. d (P1 , P2 ) =

( 6 − (− 4) )

2

+ ( 2 − (−3) )

2

= 10 2 + 52 = 100 + 25 = 125 = 5 5 29. d (P1 , P2 ) = (0 − a) 2 + (0 − b) 2 = (−a) 2 + (−b) 2 = a 2 + b 2 30. d (P1 , P2 ) = (0 − a) 2 + (0 − a) 2 = (−a) 2 + (−a) 2 = a 2 + a 2 = 2a 2 = a

19. d (P1 , P2 ) = (2 − 0) 2 + (1− 0) 2 = 2 2 + 12 = 4 + 1 = 5

31. A = (−2, 5), B = (1, 3), C = (−1, 0) d ( A, B) =

20. d (P1 , P2 ) = (−2 − 0) 2 + (1 − 0) 2

21. d (P1 , P2 ) = (−2 −1) + (2 − 1)

d (B, C ) =

2

= (−3) 2 + 12 = 9 + 1 = 10 22. d (P1 , P2 ) =

(1 − (−2) )2 + (3 − 5) 2

= 32 + (−2) 2 = 9 + 4 = 13

= (−2) 2 + 12 = 4 + 1 = 5 2

2

( −1 − 1)2 + (0 − 3) 2

= (−2) 2 + (−3)2 = 4 + 9 = 13 d ( A, C ) =

( 2 − (−1) )2 + (2 −1) 2

( −1 − (−2) )2 + (0 − 5) 2

= 1 + (−5) 2

2

= 1 + 25 = 26

= 32 + 12 = 9 + 1 = 10 23. d (P1 , P2 ) = (5 − 3) 2 + ( 4 − ( −4 ) )

2

= 2 2 + ( 8 ) = 4 + 64 = 68 = 2 17 2

24. d (P1 , P2 ) = = 25. d (P1 , P2 ) =

2 ( 2 − ( −1) ) + ( 4 − 0 )2

( 3)2

+ 4 2 = 9 + 16 = 25 = 5

( 6 − (−3) )2 + (0 − 2)2

= 9 + (− 2) = 81 + 4 = 85 2

26. d (P , P ) = 1

2

( 4 − 2 )2 + ( 2 − (−3) )2

2

= 2 2 + 52 =

Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem:

[ d ( A, B)]2 + [ d (B, C )]2 = [ d ( A, C )]2

( ) ( ) ( 2

13

2

13

=

26

)

2

13 + 13 = 26

4 + 25 = 29

27. d (P1 , P2 ) = (6 − 4) 2 + ( 4 − (−3) )

+

26 = 26 2

= 2 2 + 7 2 = 4 + 49 = 53

The area of a triangle is A = problem,

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1 ⋅ bh . In this 2

Chapter 2: Graphs

Section 2.1: The Distance and Midpoint Formulas

A = 1 ⋅ [ d ( A, B) ] ⋅ [ d (B, C ) ] 2 1 = ⋅ 13 ⋅ 13 = 1 ⋅13 2 2 13 = 2 square units 32. A = (−2, 5), B = (12, 3), C = (10, −11)

(12 − (−2) )

d ( A, B) =

+ (3 − 5) 2

2

problem, 1 A = ⋅ [ d ( A, B) ] ⋅ [ d (B, C ) ] 2 1 = ⋅10 2 ⋅10 2 2 1 = ⋅100 ⋅ 2 = 100 square units 2 33. A = (− 5, 3), B = (6, 0), C = (5, 5) d ( A, B) =

= 14 2 + (−2) 2

( 6 − (− 5) )

2

+ (0 − 3) 2

= 196 + 4 = 200

= 112 + (− 3) 2 = 121 + 9

= 10 2

= 130

(10 − 12 )

d (B, C ) =

2

+ (−11 − 3) 2

d (B, C ) =

( 5 − 6 )2 + (5 − 0) 2

= (−2) 2 + (−14) 2

= (−1) 2 + 52 = 1 + 25

= 4 + 196 = 200

= 26 d ( A, C ) =

= 10 2

(10 − (−2) )2 + (−11 − 5) 2

d ( A, C ) =

( 5 − (− 5) )2 + (5 − 3) 2

= 10 2 + 2 2 = 100 + 4

= 12 2 + (−16) 2

= 104

= 144 + 256 = 400 = 20

= 2 26

Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem: Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem: [ d ( A, B)]2 + [ d (B, C )]2 = [ d ( A, C )]2

(10 2 ) + (10 2 ) 2

2

= ( 20 )

2

200 + 200 = 400 400 = 400 1 The area of a triangle is A = bh . In this 2

[ d ( A, C )]2 + [ d (B, C )]2 = [ d ( A, B)]2

( 104) + ( 2

26

) = ( 130 ) 2

2

104 + 26 = 130 130 = 130 1 The area of a triangle is A = bh . In this 2

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Chapter 2: Graphs

Section 2.1: The Distance and Midpoint Formulas

problem, 1 A = ⋅ [ d ( A, C ) ] ⋅ [ d ( B, C ) ] 2 1 = ⋅ 104 ⋅ 26 2 1 = ⋅ 2 26 ⋅ 26 2 1 = ⋅ 2 ⋅ 26 2 = 26 square units

problem, 1 A = ⋅ [ d ( A, C ) ] ⋅ [ d ( B, C ) ] 2 1 = ⋅ 29 ⋅ 2 29 2 1 = ⋅ 2 ⋅ 29 2 = 29 square units 35. A = (4, −3), B = (0, −3), C = (4, 2)

34. A = (−6, 3), B = (3, −5), C = (−1, 5)

( 3 − (−6) )2 + (−5 − 3) 2

d ( A, B) =

= 9 2 + (−8) 2 = 81 + 64 = 145

( −1 − 3)

d (B, C ) =

2

+ (5 − (−5))

2

= (−4) 2 + 10 2 = 16 + 100 = 116 = 2 29

( −1 − (− 6) )

d ( A, C ) =

2

+ (5 − 3)

2

2

= 16 =4 d (B, C ) =

( 4 − 0)

2

+ ( 2 − (−3) )

2

= 4 2 + 52 = 16 + 25 = 41 2 d ( A, C ) = (4 − 4) + ( 2 − (−3) )

2

=5

Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem:

[ d ( A, C )]2 + [ d (B, C )]2 = [ d ( A, B)]2 2

= (− 4) 2 + 0 2 = 16 + 0

= 25

= 29

( 29) + ( 2

2

= 0 2 + 52 = 0 + 25

= 5 + 2 = 25 + 4 2

d ( A, B) = (0 − 4)2 + ( −3 − (−3) )

29

) = ( 145) 2

Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem:

[ d ( A, B)]2 + [ d ( A, C )]2 = [ d (B, C )]2

2

4 2 + 52 =

29 + 4 ⋅ 29 = 145 29 + 116 = 145

( 41)

2

16 + 25 = 41

145 = 145 1 The area of a triangle is A = bh . In this 2

41 = 41 1 The area of a triangle is A = bh . In this 2

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Chapter 2: Graphs

Section 2.1: The Distance and Midpoint Formulas

problem, 1 A = ⋅ [ d ( A, B) ] ⋅ [ d ( A, C ) ] 2 1 = ⋅ 4⋅5 2 = 10 square units

The area of a triangle is A = 1 ⋅ [ d ( A, B) ] ⋅ [ d (B, C ) ] 2 1 = ⋅4⋅2 2 = 4 square units

A=

36. A = (4, −3), B = (4, 1), C = (2, 1) d ( A, B) = (4 − 4) 2 + (1 − (−3) )

37. The coordinates of the midpoint are: x+x y+y ( x, y) = 1 2 , 1 2 2 2  3+ 5 − 4+ 4 = , 2 2  8 0 = ,  2 2  = (4, 0)

2

= 02 + 42 = 0 + 16 = 16 =4 d (B, C ) =

1 bh . In this problem, 2

( 2 − 4 )2 + (1 − 1)2

= (−2) 2 + 02 = 4 + 0 = 4 =2 d ( A, C ) = (2 − 4) 2 + (1 − (−3) )

2

= (−2) 2 + 4 2 = 4 + 16 = 20 =2 5

38. The coordinates of the midpoint are: x+x y+y ( x, y) = 1 2 , 1 2  2 2  −2+ 2 0+ 4 = , 2 2  0 4 = , 2 2  = ( 0, 2 ) 39. The coordinates of the midpoint are: x+x y+y ( x, y) = 1 2 , 1 2  2 2  −3+ 6 2+ 0  = , 2 2  3 2 = ,  2 2  3  = ,1 2 

Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem:

40. The coordinates of the midpoint are: x+x y+y ( x, y) = 1 2 , 1 2   2 2 

[ d ( A, B)]2 + [ d (B, C )]2 = [ d ( A, C )]2

(

42 + 22 = 2 5 16 + 4 = 20 20 = 20

)

=

2

=

2+ 4 −3+ 2 , 2 2 

6 −1 , 2 2

1 = 3, −  2

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 

Chapter 2: Graphs

Section 2.1: The Distance and Midpoint Formulas

41. The coordinates of the midpoint are: x+x y+y  ( x, y) = 1 2 , 1 2 2 2  4+ 6 −3+ 1  = , 2 2   10 − 2 = , 2 2   = (5, −1)

42. The coordinates of the midpoint are: x+x y+y  ( x, y) = 21 2 , 12 2   − 4+ 2 −3+ 2 = , 2 2 − 2 −1 ,  2 2  1 = −1, − 2

52 + b 2 = 132 25 + b 2 = 169 b 2 = 144 b = 12 Thus the coordinates will have an y value of −1 − 12 = −13 and −1 + 12 = 11 . So the points are ( 3,11) and ( 3, −13) . Consider points of the form ( 3, y ) that are a

b.

distance of 13 units from the point ( −2, −1) . d=

( x2 − x1 )2 + ( y2 − y1 )2

=

( 3 − (−2)2 )

=

2

+ ( −1 − y )

2

= 25 + 1 + 2 y + y 2 

=

43. The coordinates of the midpoint are: x+x y+y  ( x, y) = 1 2 , 1 2

y 2 + 2 y + 26

(

y + 2 y + 26

13 = 2

y 2 + 2 y + 26

13 = 2

2

( 52)

=

+ ( −1 − y )

2

)

2



a + 0 b + 0 , 2 2  a b = ,  2 2

=

169 = 2y + 2 y + 26 0 = y 2 + 2 y −143

44. The coordinates of the midpoint are: x+x y+y  ( x, y) = 1 2 , 1 2  2 2  a+0 a+0 = , 2 2  a a = ,  2 2  45. The x coordinate would be 2 + 3 = 5 and the y coordinate would be 5 − 2 = 3 . Thus the new point would be ( 5, 3) .

0 = ( y − 11) ( y + 13 ) y − 11 = 0 or y + 13 = 0 y = 11 y = −13 Thus, the points ( 3,11) and ( 3, −13) are a distance of 13 units from the point ( −2, −1) . 47. a. If we use a right triangle to solve the problem, we know the hypotenuse is 13 units in length. One of the legs of the triangle will be 2+3=5. Thus the other leg will be:

46. The new x coordinate would be −1 − 2 = −3 and the new y coordinate would be 6 + 4 = 10 . Thus the new point would be ( −3,10 )

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Chapter 2: Graphs

Section 2.1: The Distance and Midpoint Formulas

48. a. If we use a right triangle to solve the problem, we know the hypotenuse is 17 units in length. One of the legs of the triangle will be 2+6=8. Thus the other leg will be: 82 + b 2 = 17 2 64 + b 2 = 289 b 2 = 225 b = 15 Thus the coordinates will have an x value of 1 −15 = −14 and 1 + 15 = 16 . So the points are ( −14, −6 ) and (16, −6 ) .

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Chapter 2: Graphs

Section 2.1: The Distance and Midpoint Formulas

Consider points of the form ( x, −6 ) that are

b.

a distance of 17 units from the point (1, 2 ) .

( x2 − x1 )

d= =

(1 − x)

2

2

+ ( y2 − y1 )

+ ( 2 − ( −6 ) )

= x − 2x + 1 + ( 8 ) 2

x = 4 + 3 3 or x = 4 − 3 3

(

)

(

)

Thus, the points 4 + 3 3, 0 and 4 − 3 3, 0 are on the x-axis and a distance of 6 units from the point ( 4, −3 ) .

2

2

50. Points on the y-axis have an x-coordinate of 0. Thus, we consider points of the form ( 0, y ) that

2

= x 2 − 2x + 1 + 64

are a distance of 6 units from the point ( 4, −3) .

= x 2 − 2x + 65

d=

17 = x 2 − 2x + 65

(

17 2 =

x 2 − 2x + 65

)

2

x = −14 x = 16 Thus, the points ( −14, −6 ) and (16, −6 ) are a distance of 13 units from the point (1, 2 ) .

49. Points on the x-axis have a y-coordinate of 0. Thus, we consider points of the form ( x, 0 ) that are a distance of 6 units from the point ( 4, −3 ) .

=

( 4 − x)

2

+ ( y 2 − y1 )

+ ( −3 − 0 )

2

2

= 16 − 8x + x 2 + ( −3 )

2

(

x 2 − 8x + 25

)

= 6= 62 =

y 2 + 6 y + 25

)

2

36 = y + 6 y + 25 0 = y2 + 6 y − 11 y=

(−6) ± (6)2 − 4(1)(−11) 2(1)

−6 ± 36 + 44 −6 ± 80 = 2 2 −6± 4 5 = = −3 ± 2 5 2 y = −3 + 2 5 or y = −3 − 2 5 =

)

(

)

are on the y-axis and a distance of 6 units from the point ( 4, −3 ) .

2

51. a.

To shift 3 units left and 4 units down, we subtract 3 from the x-coordinate and subtract 4 from the y-coordinate. (2 − 3, 5 − 4) = ( −1,1)

b.

To shift left 2 units and up 8 units, we subtract 2 from the x-coordinate and add 8 to the y-coordinate. 2

36 = x − 8x + 25 0 = x 2 − 8x − 11 −(−8) ± (−8) 2 − 4(1)(−11) 2(1)

8± 64+ 44 8± 108 = 2 2 8± 6 3 = = 4±3 3

(

y2 + 6 y + 25

2

2

x=

2

y 2 + 6 y + 25

(

= x 2 − 8x + 25

62 =

+ ( −3 − y )

2

Thus, the points 0, −3 + 2 5 and 0, −3 − 2 5

= 16 − 8x + x 2 + 9 6 = x 2 − 8x + 25

+ ( y2 − y1 )

= 16 + 9 + 6 y + y2

0 = ( x + 14 ) ( x −16 ) x + 14 = 0 or x −16 = 0

( x2 − x1 )

2

2

= 42 + 9 + 6 y + y 2

0 = x 2 − 2x − 224

d=

( 4 − 0)

=

289 = x 2 − 2x + 65

2

( x2 − x1 )

=

155 155 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

Chapter 2: Graphs

Section 2.1: The Distance and Midpoint Formulas

( 2 − 2, 5 + 8 ) = ( 0,13)

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Chapter 2: Graphs

Section 2.1: The Distance and Midpoint Formulas

52. Let the coordinates of point B be ( x, y ) . Using the midpoint formula, we can write −1 + x 8+ y ( 2, 3) = 2 , 2 .  This leads to two equations we can solve. −1 + x 8+ y =2 =3 2 2 −1 + x = 4 8+ y = 6 y = −2 x=5

1 = x2

d ( A, F ) = (2 − 0) 2 + (5 − 0) 2 = 2 2 + 52 = 4 + 25 = 29 56. Let P1 = (0, 0), P2 = (0, 4), P = (x, y) 2 2 + (4 − 0) d ( P1 , P2 ) = (0 − 0)

= 16 = 4

y +y y= 1 2 2 6+ y 2 4= 2 8 = 6 + y2

d ( P1 , P ) = ( x − 0) 2 + ( y − 0) 2 = x2 + y 2 = 4 →

= x 2 + ( y − 4) 2 = 4 → x 2 + ( y − 4) 2 = 16 Therefore,

x+x y+y 54. M = ( x, y ) = 1 2 , 1 2 . 2 2

 P2 = ( x2 , y2 ) = (7, −2) and ( x, y) = (5, −4) , so

3 = x1 Thus, P1 = (3, −6) .

x 2 + y 2 = 16

d ( P2 , P ) = ( x − 0)2 + ( y − 4) 2

2 = y2

and

2 + (2 − 0)

= 20 = 2 5

Thus, P2 = (1, 2) .

x +x x= 1 2 2 x 1+ 7 5= 2 10 = x1 + 7

+ (3 − 4) 2

= (− 4) 2 + 2 2 = 16 + 4

x1+ x2 y1+ y2 , . 2 2

and

( 2 − 62 )

d ( B, E) =

 P1 = ( x1 , y1 ) = (−3, 6) and ( x, y) = (−1, 4) , so

x +x x= 1 2 2 −3+ x2 −1 = 2 −2 = −3 + x2

2

= (− 4) 2 + (−1) 2 = 16 + 1 = 17

Point B has coordinates ( 5, −2 ) . 53. M = ( x, y ) =

( 0 − 4)

d (C, D) =

y 2 = ( y − 4)

2

y 2 = y 2 − 8 y + 16 8 y = 16

y +y y= 1 2 2 y1 + (−2) −4 = 2 −8 = y1 + (−2)

y=2 which gives x 2 + 2 2 = 16

−6 = y1

x 2 = 12

0+ 6 0+ 0  , 2 2 = ( 3, 0 )

55. The midpoint of AB is: D =

0+ 4 0+ 4  , The midpoint of AC is: E = 2 2  = ( 2, 2 )

x = ±2 3 Two triangles are possible. The third vertex is 

(− 2

) (

)

3, 2 or 2 3, 2 .

57. Let P1 = ( 0, 0 ) , P2 = ( 0, s ) , P3 = ( s, 0 ) , and P = ( s, s ) .

4

6+ 4 0+ 4  , 2 2  = ( 5, 2 )

The midpoint of BC is: F =

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Chapter 2: Graphs

Section 2.1: The Distance and Midpoint Formulas

y (0, s)

P4 = M P1P2 = 

(s, s)

P5 = M P2 P3 = (s, 0)

0+ a 0+ 0 a , = ,0 2 2 2 a 3a a+ 0 + = 3a , 2, 2 4 4 2 2

a 3a 0+ 0+ 2 2 = , = 2 2

P6 = M P1P3

d P ,P = 0+ s s+0 s s  , = ,  2 2 2 2 The midpoints of the diagonals are the same. Therefore, the diagonals of a square intersect at =

M

(

4

5

3a

−

a

4 =

their midpoints.

a

58. Let P1 = ( 0, 0 ) , P2 = ( a, 0 ) , and

=

a 3a  ,  . To show that these vertices 2 2

= d ( P2 , P3 ) =

( x2 − x1 )

( a − 0 ) 2 + ( 0 − 0 )2

= d ( P1 , P3 ) =

2

a −a +

2



6



3a − 0 

2

2

=

( x2 − x1 )

2

+ ( y2 − y1 )

2

 − 0

2

2

3a a =  − 0 +  2 2 2

=



4 

2



3a  + 4



2

a 2 3a 2 a + = 16 16 2 2

+ 

3a − 4

3a

2

4

2

a +0 2

2

a a2 = 4 2 Since the sides are the same length, the triangle is equilateral. =

2

2

a 2 3a 2 a + = 16 16 2

3a a d ( P5 , P6 ) =  − 4 4

2

a 3a 4a + = = a2 = a 4 4 4

3a 

2

= a2 = a



2

4

 a = −  4

2

2

3a 4 

a a 3a −0 − + 4 2 4

( x2 − x1 )2 + ( y2 − y1 )2 2

=

+ ( y2 − y1 )

4

,

 −0

3a

+

2

d (P , P ) =

d ( P1 , P2 ) =

2

a 4

2

4

2



2

+

form an equilateral triangle, we need to show that the distance between any pair of points is the same constant value.



)

2,3

P3 =

3a

x

(0, 0)

The points P1 and P4 are endpoints of one diagonal and the points P2 and P3 are the endpoints of the other diagonal. 0+ s 0+ s s s M 1,4 = , = , 2 2 2 2



2

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Chapter 2: Graphs

Section 2.1: The Distance and Midpoint Formulas

a 3a 4a + = = a2 = a 4 4 4 Since all three distances have the same constant value, the triangle is an equilateral triangle. Now find the midpoints: =

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Chapter 2: Graphs

Section 2.1: The Distance and Midpoint Formulas

59. d (P1 , P2 ) = (− 4 − 2) 2 + (1− 1) 2

61. d ( P1 , P2 ) =

( 0 − (− 2) )

+ ( 7 − (−1) )

2

2

= (− 6) 2 + 02

= 2 2 + 82 = 4 + 64 = 68

= 36

= 2 17

=6 d (P2 , P3 ) =

( − 4 − (− 4) )

= 0 + (− 4) 2

2

+ (−3 − 1)

2

+ (2 − 7) 2

= 32 + (− 5) 2 = 9 + 25

2

= 34

= 16 =4

( 3 − (−2) )

d (P1 , P3 ) =

2

+ ( 2 − (−1) )

2

= 5 2 + 3 2 = 25 + 9

d ( P1 , P3 ) = (− 4 − 2) 2 + (−3 −1) 2

= 34 Since d (P2 , P3 ) = d (P1 , P3 ) , the triangle is isosceles.

= (− 6) 2 + (− 4) 2 = 36 + 16 = 52

Since [ d (P1 , P3 ) ] + [ d (P2 , P3 )] = [ d (P1, P2 ) ] , 2

= 2 13 Since [ d ( P1 , P2 ) ] + [ d (P2 , P3 ) ] = [ d (P1 , P3 ) ] , 2

2

2

the triangle is a right triangle. 60. d ( P1 , P2 ) =

(3 − 0)

d (P2 , P3 ) =

2

( − 4 − 7 )2 + ( 0 − 2 )2

= 7 2 + (− 2) 2

= (−11) 2 + (− 2) 2

= 49 + 4

= 121+ 4 = d (P2 , P3 ) =

( 4 − 6 )2 + (−5 − 2) 2

= 100

= 4 + 49

= 10

= 53

( 4 − (−1) )

( 4 − (− 4) )2 + (6 − 0) 2

= 8 2 + 6 2 = 64 + 36

= (− 2)2 + (− 7) 2

d (P1 , P3 ) =

125

=5 5

= 53 d (P2 , P3 ) =

2

the triangle is also a right triangle. Therefore, the triangle is an isosceles right triangle. 62. d (P1 , P2 ) =

( 6 − (−1) )2 + (2 − 4) 2

2

2

+ (−5 − 4)

d (P1 , P3 ) =

2

= (−3) 2 + 4 2 = 9 + 16

= 52 + (− 9) 2

= 25

= 25 + 81

=5

= 106

Since [ d (P1 , P3 ) ] + [ d (P2 , P3 )] = [ d (P1, P2 ) ] , 2

Since [ d ( P1 , P2 ) ] + [ d (P2 , P3 )] = [ d (P1, P3 ) ] , 2

( 4 − 7 )2 + ( 6 − 2 ) 2

2

2

the triangle is a right triangle.

the triangle is a right triangle.

Since d ( P1 , P2 ) = d ( P2 , P3 ) , the triangle is isosceles. Therefore, the triangle is an isosceles right triangle.

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2

2

Chapter 2: Graphs

Section 2.1: The Distance and Midpoint Formulas

63. Using the Pythagorean Theorem: 90 2 + 90 2 = d 2

66. a.

First: (60, 0), Second: (60, 60) Third: (0, 60)

8100 + 8100 = d 2 16200 = d

y

2

(0,60)

(60,60)

d = 16200 = 90 2 ≈ 127.28 feet 90

90

x

d

(60,0)

(0,0)

b. Using the distance formula:

90

90

d = (180 − 60) 2 + (20 − 60) 2 = 120 2 + (− 40) 2 = 16000

64. Using the Pythagorean Theorem: 60 2 + 60 2 = d 2

= 40 10 ≈ 126.49 feet

3600 + 3600 = d 2 → 7200 = d 2 c.

d = 7200 = 60 2 ≈ 84.85 feet 60

Using the distance formula: d = (220 − 0) 2 + (220 − 60)2 = 220 2 + 160 2 = 74000

60

= 20 185 ≈ 272.03 feet

d 60

60

65. a.

First: (90, 0), Second: (90, 90), Third: (0, 90) Y

(0,90)

67. The Focus heading east moves a distance 30t after t hours. The truck heading south moves a distance 40t after t hours. Their distance apart after t hours is: d = (30t) 2 + (40t) 2 = 900t 2 + 1600t 2

(90,90)

= 2500t 2 = 50t miles 30t X (0,0)

40t

(90,0)

d

b. Using the distance formula: d = (310 − 90) 2 + (15 − 90) 2 = 220 2 + (−75)2 = 54025 = 5 2161 ≈ 232.43 feet c.

Using the distance formula: d = (300 − 0) 2 + (300 − 90)2

68.

15miles 5280ft 1hr ⋅ ⋅ = 22 ft/sec 1 hr 1 mile 3600 sec d = 1002 + ( 22t )

2

= 10000 + 484t 2

= 300 2 + 210 2 = 134100 = 30 149 ≈ 366.20 feet

161 161 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

feet

Chapter 2: Graphs

Section 2.1: The Distance and Midpoint Formulas

pair ( 2013, 23624 ) . The midpoint is

22t

( year, $) = d

100

=

2003+ 2013 18660+ 23624 , 2 2 4016 42284 , 2 2

= ( 2008, 21142 ) 69. a.

The shortest side is between P1 = (2.6, 1.5) and P2 = (2.7, 1.7) . The estimate for the desired intersection point is: x1+ x2 y1+ y2 2.6+ 2.7 1.5+ 1.7   , =  , 2 2 2 2 5.3 3.2  = , 2 2  = ( 2.65, 1.6 )

b.

Using the midpoint, we estimate the poverty level in 2008 to be $21,142. This is lower than the actual value.

Using the distance formula:

72. Answers will vary. 73. To find the domain, we know the denominator cannot be zero. 2x − 5 = 0 2x = 5 5 x= 2

d = (2.65 −1.4) 2 + (1.6 − 1.3) 2

So the domain is all real numbers not equal to

{

= (1.25) 2 + (0.3) 2

or x | x ≠

= 1.5625 + 0.09 74.

= 1.6525 ≈ 1.285 units 70. Let P1 = (2007, 345) and P2 = (2013, 466) . The midpoint is: x+x y+y ( x, y ) = 1 2 2 , 1 2 2 =

2007+ 2013 345+ 466 , 2 2

=

4020 811 , 2 2

= ( 2010, 405.5) The estimate for 2010 is $405.5 billion. The estimate net sales of Wal-Mart Stores, Inc. in 2010 is $0.5 billion off from the reported value of $405 billion. 71. For 2003 we have the ordered pair (2003,18660) and for 2013 we have the ordered

}

5 . 2

3x 2 − 7 x − 20 = 0 (3x + 5)( x − 4) = 0 (3x + 5) = 0 or ( x − 4) = 0 5 x = − or x = 4 3 5 So the solution set is: − ,4 3

75. (7 + 3i)(1 − 2 i) = 7 − 14i + 3i − 6i 2 = 7 − 11i − 6(−1) = 7 − 11i + 6 = 13 − 11i 76. 5( x − 3) + 2x ≥ 6(2 x − 3) − 7 5x − 15 + 2x ≥ 12 x − 18 − 7 7 x − 15 ≥ 12 x − 25 −5x ≥ −10 x≤2

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5 2

Chapter 2: Graphs

Section 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry

15. y 2 = x 2 + 9

Section 2.2

0 2 = 32 + 9 0 2 = (−3) 2 + 9 32 = 0 2 + 9 9=9 0 ≠ 18 0 ≠ 18 The point (0, 3) is on the graph of the equation.

1. 2 ( x + 3 ) − 1 = −7 2 ( x + 3) = −6 x + 3 = −3

16. y 3 = x + 1

x = −6 The solution set is {−6} .

23 = 1 + 1

13 = 0 + 1

03 = −1 + 1

8≠2 1=1 0=0 The points (0, 1) and (–1, 0) are on the graph of the equation.

2. x 2 − 9 = 0 x2 = 9 x = ± 9 = ±3 The solution set is {−3, 3} .

17. x 2 + y2 = 4

( 2) + ( 2 ) 2

2

(− 2)2 + 22 = 4

3. intercepts

02 + 2 2 = 4 4=4

4. y = 0

(0, 2) and

2 , 2 are on the graph of the

(

)

8≠4

=4

4=4

equation. 5. y-axis 18. x 2 + 4 y 2 = 4

6. 4 7.

0 2 + 4 ⋅12 = 4

( −3, 4 )

22 + 4 ⋅ 02 = 4

9. False; the y-coordinate of a point at which the graph crosses or touches the x-axis is always 0. The x-coordinate of such a point is an x-intercept. 10. False; a graph can be symmetric with respect to both coordinate axes (in such cases it will also be symmetric with respect to the origin). For example: x 2 + y2 = 1

19. y = x + 2 x-intercept: 0 = x+2 −2 = x

y-intercept: y = 0+2 y=2

The intercepts are ( −2, 0 ) and ( 0, 2 ) .

11. d 12. c 13. y = x 4 − x 4 = (2) 4 − 2

1 = 14 − 1

0=0 1≠ 0 4 ≠ 16 − 2 The point (0, 0) is on the graph of the equation. 14. y = x − 2 x 3

0=0 −2 0 3

2

4=4 4=4 5≠4 The points (0, 1) and (2, 0) are on the graph of the equation.

8. True

0 = 04 − 0

2 2 + 4 ( 12 ) = 4

1=1 − 2 1 3

−1 = 1 − 2 1 3

0=0 1 ≠ −1 −1 = −1 The points (0, 0) and (1, –1) are on the graph of

20. y = x − 6 x-intercept: 0 = x−6 6=x

163 163 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

y-intercept: y = 0−6 y = −6

Chapter 2: Graphs

Section 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry

the equation.

164 164 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

Chapter 2: Graphs

Section 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry

The intercepts are ( 6, 0 ) and ( 0, −6 ) .

23. y = x 2 − 1 x-intercepts: 0 = x2 − 1

y-intercept: y = 02 − 1 y = −1

x2 = 1

x = ±1 The intercepts are ( −1, 0 ) , (1, 0 ) , and ( 0, −1) .

21. y = 2 x + 8 x-intercept: 0 = 2x + 8

y-intercept: y = 2( 0 ) + 8

2 x = −8

y =8 x = −4 The intercepts are ( −4, 0 ) and ( 0, 8 ) .

24. y = x 2 − 9 x-intercepts: 0 = x2 − 9

y-intercept: y = 02 − 9 y = −9

x2 = 9

x = ±3 The intercepts are ( −3, 0 ) , ( 3, 0 ) , and ( 0, −9 ) .

22. y = 3x − 9 x-intercept: 0 = 3x − 9 3x = 9

y-intercept: y = 3(0) − 9 y = −9

x=3 The intercepts are ( 3, 0 ) and ( 0, −9 ) . 25. y = − x 2 + 4 x-intercepts: 0 = −x + 4 2

x2 = 4 x = ±2

165 165 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

y-intercepts:

y = − ( 0) + 4 2

y=4

Chapter 2: Graphs

Section 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry

The intercepts are ( −2, 0 ) , ( 2, 0 ) , and ( 0, 4 ) .

28. 5x + 2 y = 10 x-intercepts: 5x + 2 ( 0 ) = 10

y-intercept: 5 ( 0 ) + 2 y = 10

5x = 10

2 y = 10

x=2

y=5

The intercepts are ( 2, 0 ) and ( 0, 5 ) .

26. y = − x 2 + 1 x-intercepts: 0 = −x +1 2

y-intercept:

y = − ( 0) + 1 2

y =1 x2 = 1 x = ±1 The intercepts are ( −1, 0 ) , (1, 0 ) , and ( 0,1) .

29. 9x 2 + 4 y = 36 x-intercepts:

9x + 4 ( 0 ) = 36 2

y-intercept:

9 ( 0 ) + 4 y = 36 2

9x 2 = 36

4 y = 36

x =4

y=9

2

x = ±2 The intercepts are ( −2, 0 ) , ( 2, 0 ) , and ( 0, 9 ) .

27. 2 x + 3 y = 6 x-intercepts: 2x + 3(0) = 6

y-intercept: 2 ( 0) + 3y = 6

2x = 6

3y = 6

x=3

y=2

The intercepts are ( 3, 0 ) and ( 0, 2 ) .

2 30. 4 x + y = 4 x-intercepts:

y-intercept:

4x + 0 = 4

4 ( 0) + y = 4

4 x2 = 4

y=4

2

x2 = 1 x = ±1

166 166 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

2

Chapter 2: Graphs

Section 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry

The intercepts are ( −1, 0 ) , (1, 0 ) , and ( 0, 4 ) .

34.

35.

y 5

31. (c) = (−5, 2)

(a) = (5, 2)

−5

5

(5, −2)

(b) = (−5, −2) −5

36. 32. 33.

37. 38.

167 167 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

Chapter 2: Graphs

Section 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry

49. a.

x-intercept: [ −2,1] , y-intercept 0

b. Not symmetric to x-axis, y-axis, or origin. 50. a.

x-intercept: [ −1, 2] , y-intercept 0

b. Not symmetric to x-axis, y-axis, or origin. 51. a. b.

39. 40.

Intercepts: none Symmetric with respect to the origin.

52. a. Intercepts: none b. Symmetric with respect to the x-axis. 53.

41. a.

Intercepts: ( −1, 0 ) and (1, 0 )

b. Symmetric with respect to the x-axis, y-axis, and the origin. 42. a.

54.

Intercepts: ( 0,1)

b. Not symmetric to the x-axis, the y-axis, nor the origin 43. a.

(

)

(

Intercepts: − π , 0 , ( 0,1) , and π , 0 2 2

)

b. Symmetric with respect to the y-axis. 44. a.

Intercepts: ( −2, 0 ) , ( 0, −3 ) , and ( 2, 0 )

b. Symmetric with respect to the y-axis. 45. a.

Intercepts: ( 0, 0 )

b. Symmetric with respect to the x-axis. 46. a. b.

Intercepts: ( −2, 0 ) , ( 0, 2 ) , ( 0, −2 ) , and ( 2, 0 )

55. 56.

Symmetric with respect to the x-axis, y-axis, and the origin.

47. a.

Intercepts: ( −2, 0 ) , ( 0, 0 ) , and ( 2, 0 )

b.

Symmetric with respect to the origin.

48. a.

Intercepts: ( −4, 0 ) , ( 0, 0 ) , and ( 4, 0 )

b.

Symmetric with respect to the origin.

168 168 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

Chapter 2: Graphs

Section 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry

57. y 2 = x + 4 x-intercepts: 02 = x + 4 −4 = x

y-intercepts: y2 = 0 + 4 y2 = 4 y = ±2

The intercepts are ( −4, 0 ) , ( 0, −2 ) and ( 0, 2 ) . Testx-axissymmetry: Let y = − y

(− y)

2

= x+4

y 2 = x + 4 same Testy-axissymmetry: Let x = − x y 2 = − x + 4 different Testoriginsymmetry: Let x = − x and y = − y .

(− y)

= −x + 4 2 y = − x + 4 different 2

Testx-axissymmetry: Let y = − y

(− y)

− y = 3 −x = −3 x y = 3 x same Therefore, the graph will have origin symmetry. 60. y = 5 x x-intercepts: y-intercepts: 0= 3 x y= 50 =0 0=x The only intercept is ( 0, 0 ) . Testx-axissymmetry: Let y = − y Testy-axissymmetry: Let x = − x y = 5 − x = − 5 x different Testoriginsymmetry: Let x = − x and y = − y

y-intercepts: y2 = 0 + 9 y2 = 9 y = ±3

The intercepts are ( −9, 0 ) , ( 0, −3) and ( 0, 3 ) . 2

Testoriginsymmetry: Let x = − x and y = − y

− y = 5 x different

Therefore, the graph will have x-axis symmetry. 58. y 2 = x + 9 x-intercepts: (0) 2 = − x + 9 0 = −x + 9 x=9

Testy-axissymmetry: Let x = − x y = 3 − x = − 3 x different

= x+9

y = x + 9 same

− y = 5 −x = −5 x y = 5 x same Therefore, the graph will have origin symmetry. 2 61. x + y − 9 = 0 x-intercepts: x2 − 9 = 0

y-intercepts: 02 + y − 9 = 0

x2 = 9

2

y=9

Testy-axissymmetry: Let x = − x y 2 = − x + 9 different

x = ±3 The intercepts are ( −3, 0 ) , ( 3, 0 ) , and ( 0, 9 ) .

Testoriginsymmetry: Let x = − x and y = − y .

Testx-axissymmetry: Let y = − y

(− y)

x 2 − y − 9 = 0 different

2

= −x + 9

Testy-axissymmetry: Let x = − x

y 2 = − x + 9 different Therefore, the graph will have x-axis symmetry. 59. y = 3 x x-intercepts: 0= 3 x

( − x )2 + y − 9 = 0 x 2 + y − 9 = 0 same

y-intercepts: y= 30 =0

0=x The only intercept is ( 0, 0 ) .

Testoriginsymmetry: Let x = − x and y = − y

( − x )2 − y − 9 = 0 x 2 − y − 9 = 0 different Therefore, the graph will have y-axis symmetry.

Testx-axissymmetry: Let y = − y − y = 3 x different

169 169 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

Chapter 2: Graphs

Section 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry

62. x 2 − y − 4 = 0 x-intercepts: y-intercept: 2 x −0−4 = 0 02 − y − 4 = 0 2 −y = 4 x =4 y = −4 x = ±2 The intercepts are ( −2, 0 ) , ( 2, 0 ) , and ( 0, −4 ) . Testx-axissymmetry: Let y = − y x2 − ( − y ) − 4 = 0

4 x2 = 4

y2 = 4 y = ±2

x2 = 1 x = ±1 The intercepts are ( −1, 0 ) , (1, 0 ) , ( 0, −2 ) , and

2

4 x 2 + y2 = 4 same

x 2 − y − 4 = 0 same Testoriginsymmetry: Let x = − x and y = − y

Testy-axissymmetry: Let x = − x 4(−x) + y2 = 4 2

( − x )2 − ( − y ) − 4 = 0

4x 2 + y2 = 4 same

x 2 + y − 4 = 0 different Therefore, the graph will have y-axis symmetry.

Testoriginsymmetry: Let x = − x and y = − y 4 (−x) + (− y ) = 4 2

63. 9x 2 + 4 y2 = 36 x-intercepts:

9x 2 + 4 ( 0 ) = 36 2

y-intercepts:

9 ( 0 ) + 4 y 2 = 36 2

9 x = 36 4 y = 36 y2 = 9 x2 = 4 y = ±3 x = ±2 The intercepts are ( −2, 0 ) , ( 2, 0 ) , ( 0, −3 ) , and 2

2

( 0, 3) .

Testx-axissymmetry: Let y = − y 9x 2 + 4 ( − y ) = 36 2

9 x 2 + 4 y2 = 36 same

2

4 x2 + y 2 = 4 same Therefore, the graph will have x-axis, y-axis, and origin symmetry. 65. y = x 3 − 27 x-intercepts: 0 = x3 − 27

y-intercepts: y = 03 − 27 y = −27

x3 = 27

x=3 The intercepts are ( 3, 0 ) and ( 0, −27 ) . Testx-axissymmetry: Let y = − y

Testy-axissymmetry: Let x = − x 9 ( − x ) + 4 y = 36 2

9 x 2 + 4 y2 = 36 same Testoriginsymmetry: Let x = − x and y = − y 9 ( − x ) + 4 ( − y ) = 36 2

2

4 x2 + ( − y ) = 4

−y−4= 0

2

4 ( 0) + y 2 = 4

2

Testx-axissymmetry: Let y = − y

Testy-axissymmetry: Let x = − x 2

y-intercepts:

4x + 0 = 4 2

( 0, 2 ) .

x 2 + y − 4 = 0 different

(−x)

64. 4 x 2 + y 2 = 4 x-intercepts:

2

9 x 2 + 4 y 2 = 36 same Therefore, the graph will have x-axis, y-axis, and origin symmetry.

− y = x3 − 27 different Testy-axissymmetry: Let x = − x y = ( − x ) − 27 3

y = − x3 − 27 different Testoriginsymmetry: Let x = − x and y = − y − y = ( − x ) − 27 3

y = x3 + 27 different Therefore, the graph has none of the indicated symmetries.

170 170 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

Chapter 2: Graphs

66. y = x 4 − 1 x-intercepts: 0 = x4 − 1

Section 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry

Testx-axissymmetry: Let y = − y y-intercepts: y = 04 − 1 y = −1

x =1 4

Testx-axissymmetry: Let y = − y − y = x 4 − 1 different

y = (−x) + 4 y = x 2 + 4 same Testoriginsymmetry: Let x = − x and y = − y − y = (−x) + 4 2

− y = x 2 + 4 different

Testy-axissymmetry: Let x = − x

Therefore, the graph will have y-axis symmetry.

y = ( − x ) −1 4

y = x 4 − 1 same

69. y =

Testoriginsymmetry: Let x = − x and y = − y − y = (−x) −1 4

− y = x 4 − 1 different Therefore, the graph will have y-axis symmetry. 2

0 = ( x − 4 ) ( x + 1)

Testy-axissymmetry: Let x = − x 2

x = ±1 The intercepts are ( −1, 0 ) , (1, 0 ) , and ( 0, −1) .

67. y = x − 3x − 4 x-intercepts: 0 = x 2 − 3x − 4

− y = x 2 + 4 different

3x 2

x +9 x-intercepts: 3x 0= 2 x +9 3x = 0

y-intercepts: 3( 0) 0 y= 2 = =0 0 +9 9

x=0 The only intercept is ( 0, 0 ) .

y-intercepts: y = 02 − 3 ( 0 ) − 4

Testx-axissymmetry: Let y = − y

y = −4

different x2 + 9 Testy-axissymmetry: Let x = − x 3( − x) y= ( − x )2 + 9

x = 4 or x = −1 The intercepts are ( 4, 0 ) , ( −1, 0 ) , and ( 0, −4 ) . Testx-axissymmetry: Let y = − y − y = x 2 − 3x − 4 different Testy-axissymmetry: Let x = − x

−y =

3x

y=−

3x different x +9 2

y = (−x) − 3(−x) − 4

Testoriginsymmetry: Let x = − x and y = − y

y = x 2 + 3x − 4 different

−y =

2

Testoriginsymmetry: Let x = − x and y = − y − y = (−x) − 3(−x) − 4

( − x )2 + 9

−y = −

2

− y = x 2 + 3x − 4 different Therefore, the graph has none of the indicated symmetries. 68. y = x 2 + 4 x-intercepts: 0 = x2 + 4

3(−x)

y=

3x x2 + 9 3x

x2 + 9

same

Therefore, the graph has origin symmetry.

y-intercepts: y = 02 + 4 y=4

x 2 = −4 no real solution The only intercept is ( 0, 4 ) .

171 171 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

Chapter 2: Graphs

Section 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry

x2 − 4 2x x-intercepts: x2 − 4 0= 2x x2 − 4 = 0

Testy-axissymmetry: Let x = − x

70. y =

y-intercepts: 0 2 − 4 −4 y= = 2 (0) 0 undefined

x2 = 4 x = ±2 The intercepts are ( −2, 0 ) and ( 2, 0 ) . Testx-axissymmetry: Let y = − y −y =

− (− x)

3

x2 − 4 different 2x

y= y=

(−x) 3

x different x2 − 9

Testoriginsymmetry: Let x = − x and y = − y

−y = y=

(−x) − 4 2(−x)

x −4 different 2x Testoriginsymmetry: Let x = − x and y = − y 2

(−x) − 4 2(−x) 2

−y =

x2 − 4 −2 x x2 − 4 y= same 2x Therefore, the graph has origin symmetry. −y =

− x3 x2 − 9 x-intercepts:

71. y =

0=

− x3 x2 − 9

y-intercepts: y=

−03 0 = =0 0 2 − 9 −9

same

2

x −9

x4 + 1 2 x5 x-intercepts:

72. y =

y-intercepts: 0 4+ 1 1 y= = 5 0 2 ( 0)

x4 + 1 2 x5

0=

undefined x 4 = −1 no real solution There are no intercepts for the graph of this equation. Testx-axissymmetry: Let y = − y −y =

x4 +1

different 2 x5 Test y-axis symmetry: Let x = − x y= y=

( − x )4 + 1 5 2(−x) x4 + 1

different

5

Testoriginsymmetry: Let x = − x and y = − y

x=0 The only intercept is ( 0, 0 ) .

(−x) +1 5 2(−x) 4

Testx-axissymmetry: Let y = − y − x3 x2 − 9 x3 y= 2 different x −9

x3 x2 − 9 − x3

−2 x

− x3 = 0

−y =

( − x )2 − 9

Therefore, the graph has origin symmetry.

2

y=−

− (−x)

3

−y =

Testy-axissymmetry: Let x = − x y=

−9

2

−y = −y = y=

x4 + 1 −2 x5 x4 + 1 2x 5

same

Therefore, the graph has origin symmetry. 172 172 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

Chapter 2: Graphs

73. y = x 3

Section 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry

77. If the point ( a, 4 ) is on the graph of y = x 2 + 3x , then we have 4 = a 2 + 3a 0 = a 2 + 3a − 4

0 = ( a + 4 ) ( a − 1) a+4=0

or a − 1 = 0

a = −4 a =1 Thus, a = −4 or a = 1 . 78. If the point ( a, −5 ) is on the graph of 74. x = y

2

y = x 2 + 6 x , then we have −5 = a 2 + 6a 0 = a 2 + 6a + 5 0 = ( a + 5 ) ( a + 1) a+5 = 0

or a + 1 = 0

a = −5 a = −1 Thus, a = −5 or a = −1 . 79. For a graph with origin symmetry, if the point ( a, b ) is on the graph, then so is the point ( −a, 75. y = x

−b ) . Since the point (1, 2 ) is on the graph of an equation with origin symmetry, the point ( −1, −2 ) must also be on the graph.

80. For a graph with y-axis symmetry, if the point ( a, b ) is on the graph, then so is the point

( −a, b ) . Since 6 is an x-intercept in this case, the point ( 6, 0 ) is on the graph of the equation. Due to the y-axis symmetry, the point ( −6, 0 ) must also be on the graph. Therefore, −6 is another xintercept.

76. y =

1 x

81. For a graph with origin symmetry, if the point ( a, b ) is on the graph, then so is the point

( −a, −b ) . Since −4 is an x-intercept in this case, the point ( −4, 0 ) is on the graph of the equation. Due to the origin symmetry, the point ( 4, 0 ) must also be on the graph. Therefore, 4 is another x-intercept. 82. For a graph with x-axis symmetry, if the point ( a, b ) is on the graph, then so is the point

( a, −b )

. Since 2 is a y-intercept in this case, the

173 173 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

Chapter 2: Graphs

Section 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry

point ( 0, 2 ) is on the graph of the equation. Due

Testoriginsymmetry: Let x = − x and y = − y

(( − x )

to the x-axis symmetry, the point ( 0, −2 ) must also be on the graph. Therefore, −2 is another yintercept. 83. a.

(x

2

+ y2 − x

)

2

x-intercepts:

(x

2

+ ( 0) − x 2

(x

−x

2

)

84. a. = x2 + ( 0)

2

= x2

2

x 4 − 2x 3 = 0 x3 ( x − 2 ) = 0

(( 0)

+ y2 − 0

2

)

2

(y )

2

2

2

= y2

y − y2 = 0

b.

)

16 y 2 = 120 ( − x ) − 225

16 y 2 = −120 x − 225 different

and ( 0,1) .

Testoriginsymmetry: Let x = − x and y = − y 16 ( − y ) = 120 ( − x ) − 225 2

Testx-axissymmetry: Let y = − y 2

2

)

+y −x

)

+ (− y) − x

(x

2

Testx-axissymmetry: Let y = − y

Testy-axissymmetry: Let x = − x

y2 = 1

The intercepts are ( 0, 0 ) , ( 2, 0 ) , ( 0, −1) ,

(x

16 y 2 = 120x − 225 x-intercepts: 16 y 2 = 120 ( 0 ) − 225

16 y 2 = 120 x − 225 same

y = ±1

b.

2

2

2

= x2 + ( − y )

16 y 2 = −120x − 225 different

2

Thus, the graph will have x-axis symmetry. =x +y 2

2

same

85. a.

Testy-axissymmetry: Let x = − x

(( − x )

2

+ y2 − ( − x )

)

(x

)

2

different

2

y 2 −1 = 0

y=0

= x2 + y2

2

16 ( − y ) = 120 x − 225

y2 y2 −1 = 0 y 2 = 0 or

2

2

 15 ,0 . The only intercept is  8

2

4

(

)

= (−x) + (− y)

0 = 120 x − 225 −120x = −225 −225 15 x= = −120 8

= ( 0) + y2

y =y

+ y2 + x

2

y-intercepts: 2 16 ( 0 ) = 120 x − 225

x−2= 0 x=2

4

2

)

16 y 2 = −225 225 y2 = − 16 no real solution

x 4 − 2x 3 + x 2 = x 2

y-intercepts:

2

Thus, the graph will have x-axis symmetry.

2

x3 = 0 or x=0

+ (− y) − (−x)

(x

= x2 + y2

)

2

+ y2 + x

2

2 = (−x) + y

2

= x2 + y2

2

different

174 174 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

Chapter 2: Graphs

Section 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry

88. Answers will vary 89. Answers will vary 90. Answers will vary. Case 1: Graph has x-axis and y-axis symmetry, show origin symmetry. ( x, y ) on graph → ( x, − y ) on graph (from x-axis symmetry) ( x, − y ) on graph → ( − x, − y ) on graph

( from y-axis symmetry ) Since the point ( − x, − y )

is also on the graph, the graph has origin symmetry. Case 2: Graph has x-axis and origin symmetry, show y-axis symmetry. ( x, y ) on graph → ( x, − y ) on graph

b.

x 2 = x for all x , the graphs of

Since

y = x 2 and y = x are the same. c.

For y =

( x)

2

, the domain of the variable

x is x ≥ 0 ; for y = x , the domain of the variable x is all real numbers. Thus,

( x)

2

= x only for x ≥ 0.

d. For y = x , the range of the variable y is y ≥ 0 ; for y = x , the range of the variable 2

y is all real numbers. Also, if x ≥ 0 . Otherwise,

x 2 = x only

x2 = − x .

86. Answers will vary. A complete graph presents enough of the graph to the viewer so they can “see” the rest of the graph as an obvious continuation of what is shown. 87. Answers will vary. One example:

( from x-axis symmetry ) ( x, − y ) on graph → ( − x, y ) on graph ( from origin symmetry ) Since the point ( − x, y ) is also on the graph, the graph has y-axis symmetry. Case 3: Graph has y-axis and origin symmetry, show x-axis symmetry. ( x, y ) on graph → ( − x, y ) on graph

( from y-axis symmetry ) ( − x, y ) on graph → ( x, − y ) on graph ( from origin symmetry ) Since the point ( x, − y ) is also on the graph, the graph has x-axis symmetry. 91. Answers may vary. The graph must contain the points ( −2, 5 ) , ( −1, 3 ) , and ( 0, 2 ) . For the graph to be symmetric about the y-axis, the graph must also contain the points ( 2, 5 ) and (1, 3 ) (note that (0, 2) is on the y-axis).

y

For the graph to also be symmetric with respect to the x-axis, the graph must also contain the points ( −2, −5 ) , ( −1, −3) , ( 0, −2 ) , ( 2, −5 ) , and x

(1, −3) . Recall that a graph with two of the symmetries (x-axis, y-axis, origin) will necessarily have the third. Therefore, if the original graph with y-axis symmetry also has x-

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Chapter 2: Graphs

92.

Section 2.3: Lines

axis symmetry, then it will also have origin symmetry.

6. m1 = m2 ; y-intercepts; m1 ⋅ m2 = −1

6+ (−2) 4 1 = = 6 − (− 2) 8 2

7. 2 8. −

93. 3x 2 − 30 x + 75 =

1 2

9. False; perpendicular lines have slopes that are opposite-reciprocals of each other.

3( x 2 − 10x + 25) = 3( x − 5)( x − 5) = 3(x − 5) 2

10. d

94.

−196 = (−1)(196) = 14i

11. c

95.

x 2 − 8x + 4 = 0

12. b

x − 8x = −4 2

x − 8x + 16 = −4 + 16 2

( x − 4)2

13. a.

= 12

Slope =

b. If x increases by 2 units, y will increase by 1 unit.

x − 4 = ± 12 x = 4 ± 12 = 4± 2 3

14. a.

Slope =

1− 0 1 =− −2−0 2

b. If x increases by 2 units, y will decrease by 1 unit.

Section 2.3

=

1. undefined; 0 15. a. 2. 3; 2 x-intercept: 2 x + 3(0) = 6 2x = 6 x=3 y-intercept: 2(0) + 3 y = 6 3y = 6

1 =−

1 − (− 2)

= 16. a.

3

2− 1

1 =

Slope

2 − (−1) 3 b. If x increases by 3 units, y will increase by 1 unit.

3. True 4. False; the slope is 3 . 2 2 y = 3x + 5

Slope

1− 2

b. If x increases by 3 units, y will decrease by 1 unit.

y=2

y=

1− 0 1 = 2−0 2

17.

Slope =

y2 − y1 0− 3 3 = =− x2 − x1 4 − 2 2

3 5 x+ 2 2 ?

5. True; 2 (1) + ( 2 ) = 4 ?

2+ 2=4

176 176 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

Chapter 2: Graphs

Section 2.3: Lines

4=4 True

177 177 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

Chapter 2: Graphs

Section 2.3: Lines

18. Slope =

y2 − y1 4− 2 2 = −2 = = x2 − x1 3 − 4 −1

22. Slope =

y2 − y1 0 2− 2 = = =0 x2 − x1 −5 − 4 −9

19. Slope =

y2 − y1 1− 3 −2 1 = = =− x2 − x1 2 − (− 2) 4 2

23. Slope =

−2 2− −4 y2 − y1 = = undefined. x2 − x1 −1 − (−1) 0

20. Slope =

y2 − y1 3− 1 2 = = x2 − x1 2 − (−1) 3

24. Slope =

y2 − y1 2 −0 2 = = undefined. x2 − x1 2 − 2 0

21. Slope =

y2 − y1 −1 − (−1) 0 = = =0 x2 − x1 2 − (−3) 5

25. P = (1, 2 ) ; m = 3 ; y − 2 = 3( x − 1)

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Chapter 2: Graphs

Section 2.3: Lines

26. P = ( 2,1) ; m = 4 ; y − 1 = 4( x − 2)

27. P = ( 2, 4 ) ; m = −

3 3 ; y − 4 = − ( x − 2) 4 4

30. P = ( 2, −4 ) ; m = 0 ; y = −4

31. P = ( 0, 3 ) ; slope undefined ; x = 0

(note: the line is the y-axis) 28. P = (1, 3 ) ; m = −

2 2 ; y − 3 = − ( x − 1) 5 5

29. P = ( −1, 3 ) ; m = 0 ; y − 3 = 0

32. P = ( −2, 0 ) ; slope undefined x = −2

4 ; point: (1, 2 ) 1 If x increases by 1 unit, then y increases by 4 units. Answers will vary. Three possible points are: x = 1 + 1 = 2 and y = 2 + 4 = 6 ( 2, 6 )

33. Slope = 4 =

x = 2 + 1 = 3 and y = 6 + 4 = 10

( 3,10 ) x = 3 + 1 = 4 and y = 10 + 4 = 14

( 4,14 )

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Chapter 2: Graphs

Section 2.3: Lines

2 ; point: ( −2, 3) 1 If x increases by 1 unit, then y increases by 2 units. Answers will vary. Three possible points are: x = −2 + 1 = −1 and y = 3 + 2 = 5 ( −1, 5)

34. Slope = 2 =

x = −1 + 1 = 0 and y = 5 + 2 = 7

( 0, 7 )

3 =

−3

; point: ( 2, −4 )

=−

2 2 If x increases by 2 units, then y decreases by 3 units. Answers will vary. Three possible points are: x = 2 + 2 = 4 and y = −4 − 3 = −7

( 4, −7 )

x = 0 + 1 = 1 and y = −7 − 2 = −9 (1, −9 ) −1

; point: ( 4,1) 1 If x increases by 1 unit, then y decreases by 1 unit. Answers will vary. Three possible points are: x = 4 + 1 = 5 and y = 1 − 1 = 0

( 5, 0 ) x = 5 + 1 = 6 and y = 0 − 1 = −1

( 6, −1) x = 6 + 1 = 7 and y = −1 −1 = −2

x = 4 + 2 = 6 and y = −7 − 3 = −10

( 6, −10 )

x = 6 + 2 = 8 and y = −10 − 3 = −13

( 8, −13)

4 ; point: ( −3, 2 ) 3 If x increases by 3 units, then y increases by 4 units. Answers will vary. Three possible points are: x = −3 + 3 = 0 and y = 2 + 4 = 6 ( 0, 6 )

36. Slope =

x = 0 + 3 = 3 and y = 6 + 4 = 10

( 3,10 )

x = 3 + 3 = 6 and y = 10 + 4 = 14 ( 6,14 ) −2 ; point: ( −2, −3) 1 If x increases by 1 unit, then y decreases by 2 units.

37. Slope = −2 =

x = −1 + 1 = 0 and y = −5 − 2 = −7 ( 0, −7 )

38. Slope = −1 =

x = 0 + 1 = 1 and y = 7 + 2 = 9 (1, 9 ) 35. Slope

Answers will vary. Three possible points are: x = −2 + 1 = −1 and y = −3 − 2 = −5 ( −1, −5)

( 7, −2 )

39. (0, 0) and (2, 1) are points on the line. 1− 0 1 Slope = = 2−0 2 y-intercept is 0; using y = mx + b : 1 y = x+0 2 2y = x 0 = x − 2y x − 2 y = 0 or y =

1 x 2

40. (0, 0) and (–2, 1) are points on the line. 1 1 1− 0 = =− Slope = −2−0 −2 2 y-intercept is 0; using y = mx + b : 1 y = − x+0 2 2 y = −x x + 2y = 0 x + 2 y = 0 or y = −

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1 x 2

Chapter 2: Graphs

Section 2.3: Lines

41. (–1, 3) and (1, 1) are points on the line. 1− 3 −2 Slope = = = −1 1 − (−1) 2 Using y − y1 = m(x − x1 ) y −1 = −1( x −1) y −1 = − x + 1 y = −x + 2 x + y = 2 or y = − x + 2

y − 3 = 3x + 6 y = 3x + 9 3x − y = − 9 or y = 3x + 9

y + 3 = 2x − 8 y = 2x − 11 2x − y = 11 or y = 2x − 11

1 4 x − 3 y = − 4 or y = x + 3 3 y − y1 = m(x − x1 ), m = 2 y − 3 = 2(x − 3)

2 49. Slope = − ; containing (1, –1) 3 y − y1 = m(x − x1 ) 2 ( x − 1) 3 2 2 y +1 = − x + 3 3 2 1 y =− x− 3 3

y − (−1) = −

y − 3 = 2x − 6 y = 2x − 3 2x − y = 3 or y = 2x − 3 44. y − y1 = m(x − x1 ), m = −1 y − 2 = −1( x −1) y − 2 = −x +1

2 x + 3 y = −1 or y = −

1 2

1 y − 2 = − ( x − 1) 2 1 1 y−2= − x+ 2 2 1 5 y = − x+ 2 2

2 1 x− 3 3

1 ; containing the point (3, 1) 2 y − y1 = m(x − x1 ) 1 y − 1 = ( x − 3) 2 1 3 y −1 = x − 2 2 1 1 y = x− 2 2 1 1 x − 2 y = 1 or y = x − 2 2

50. Slope =

y = −x + 3 x + y = 3 or y = − x + 3

x + 2 y = 5 or y = −

y = x+2 x − y = − 2 or y = x + 2

48. Slope = 2; containing the point (4, –3) y − y1 = m(x − x1 ) y − (−3) = 2( x − 4)

1 ( x − (−1) ) 3 1 y −1 = ( x + 1) 3 1 1 y −1 = x + 3 3 1 4 y = x+ 3 3 y −1 =

45. y − y1 = m(x − x1 ), m = −

y − 1 = 1( x − (−1)) y −1 = x +1

47. Slope = 3; containing (–2, 3) y − y1 = m(x − x1 ) y − 3 = 3(x − (− 2))

42. (–1, 1) and (2, 2) are points on the line. 2− 1 1 Slope = = 2 − (−1) 3 Using y − y1 = m(x − x1 )

43.

46. y − y1 = m( x − x1 ), m = 1

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Chapter 2: Graphs

Section 2.3: Lines

56. x-intercept = –4; y-intercept = 4 Points are (–4, 0) and (0, 4) 4− 0 4 m= = =1 0 − (− 4) 4 y = mx + b y = 1x + 4 y = x+4

51. Containing (1, 3) and (–1, 2) 2− 3 −1 1 m= = = −1 −1 − 2 2 y − y1 = m(x − x1 ) 1 ( x − 1) 2 1 1 y−3= x− 2 2 1 5 y = x+ 2 2 y−3=

x − y = − 4 or y = x + 4

x − 2 y = − 5 or y =

57. Slope undefined; containing the point (2, 4) This is a vertical line. x=2 No slope-intercept form.

1 5 x+ 2 2

52. Containing the points (–3, 4) and (2, 5) 5− 4 1 m= = 2 − (−3) 5 y − y1 = m(x − x1 ) 1 y − 5 = ( x − 2) 5 1 2 y −5 = x− 5 5 1 23 y = x+ 5 5

59. Horizontal lines have slope m = 0 and take the form y = b . Therefore, the horizontal line passing through the point ( −3, 2 ) is y = 2 .

60. Vertical lines have an undefined slope and take the form x = a . Therefore, the vertical line passing through the point ( 4, −5 ) is x = 4 .

x − 5 y = − 23 or y =

1 23 x+ 5 5

53. Slope = –3; y-intercept =3 y = mx + b y = −3x + 3 3x + y = 3 or y = −3x + 3 54. Slope = –2; y-intercept = –2 y = mx + b y = − 2 x + (− 2) 2x + y = − 2 or y = − 2 x − 2 55. x-intercept = 2; y-intercept = –1 Points are (2,0) and (0,–1) −1 − 0 −1 1 m= = = 0−2 −2 2 y = mx + b y=

58. Slope undefined; containing the point (3, 8) This is a vertical line. x=3 No slope-intercept form.

61. Parallel to y = 2 x ; Slope = 2 Containing (–1, 2) y − y1 = m( x − x1 ) y − 2 = 2( x − (−1)) y − 2 = 2 x + 2 → y = 2x + 4 2 x − y = − 4 or y = 2 x + 4 62. Parallel to y = −3x ; Slope = –3; Containing the point (–1, 2) y − y1 = m( x − x1 ) y − 2 = −3( x − (−1)) y − 2 = −3x − 3 → y = −3x − 1 3x + y = −1 or y = −3x − 1

1 x −1 2

x − 2 y = 2 or y =

1 x −1 2

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Chapter 2: Graphs

Section 2.3: Lines

63. Parallel to 2 x − y = − 2 ; Slope = 2 Containing the point (0, 0) y − y1 = m( x − x1 ) y − 0 = 2( x − 0) y = 2x 2 x − y = 0 or y = 2 x

1 1 3 ( x − (−3)) → y = x + 2 2 2 1 3 x − 2 y = − 3 or y = x + 2 2 y−0 =

64. Parallel to x − 2 y = − 5 ; 1 Slope = ; Containing the point ( 0, 0 ) 2 y − y1 = m( x − x1 ) 1 1 ( x − 0) → y = x 2 2 1 x − 2 y = 0 or y = x 2 y−0 =

65. Parallel to x = 5 ; Containing (4,2) This is a vertical line. x = 4 No slope-intercept form. 66. Parallel to y = 5 ; Containing the point (4, 2) This is a horizontal line. Slope = 0 y=2 67. Perpendicular to y =

69. Perpendicular to 2x + y = 2 ; Containing the point (–3, 0) 1 Slope of perpendicular = 2 y − y1 = m( x − x1 )

1

x + 4; Containing (1, –2) 2 Slope of perpendicular = –2 y − y1 = m( x − x1 )

70. Perpendicular to x − 2 y = −5 ; Containing the point (0, 4) Slope of perpendicular = –2 y = mx + b y = −2x + 4 2 x + y = 4 or y = −2 x + 4 71. Perpendicular to x = 8 ; Containing (3, 4) Slope of perpendicular = 0 (horizontal line) y=4 72. Perpendicular to y = 8 ; Containing the point (3, 4) Slope of perpendicular is undefined (vertical line). x = 3 No slope-intercept form. 73. y = 2 x + 3 ; Slope = 2; y-intercept = 3

y − (− 2) = − 2( x − 1) y + 2 = − 2 x + 2 → y = − 2x 2 x + y = 0 or y = − 2x 68. Perpendicular to y = 2 x − 3 ; Containing the point (1, –2) 1 Slope of perpendicular = − 2 y − y1 = m( x − x1 )

74. y = −3x + 4 ; Slope = –3; y-intercept = 4

1 y − (− 2) = − ( x − 1) 2 1 1 1 3 y+2 = − x+ → y = − x− 2 2 2 2 1 3 x + 2 y = −3 or y = − x − 2 2

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Chapter 2: Graphs

75.

76.

Section 2.3: Lines

1 y = x −1 ; y = 2x − 2 2 Slope = 2; y-intercept = –2

79. x + 2 y = 4 ; 2 y = − x + 4 → y = − 1 Slope = − ; y-intercept = 2 2

1 1 x+ y = 2; y = − x+2 3 3 1 Slope = − ; y-intercept = 2 3

77. y =

1 x+2 2

80. − x + 3 y = 6 ; 3 y = x + 6 → y = Slope =

1 x+2 3

1 ; y-intercept = 2 3

1 1 x + 2 ; Slope = ; y-intercept = 2 2 2 81. 2 x − 3 y = 6 ; −3 y = − 2 x + 6 → y = Slope =

78. y = 2x +

2 ; y-intercept = –2 3

1 1 ; Slope = 2; y-intercept = 2 2

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2 x−2 3

Chapter 2: Graphs

Section 2.3: Lines

3 82. 3x + 2 y = 6 ; 2 y = − 3x + 6 → y = − x + 3 2 3 Slope = − ; y-intercept = 3 2

86. y = −1 ; Slope = 0; y-intercept = –1

87. y = 5 ; Slope = 0; y-intercept = 5 83. x + y = 1 ; y = − x + 1 Slope = –1; y-intercept = 1

88. x = 2 ; Slope is undefined y-intercept - none 84. x − y = 2 ; y = x − 2 Slope = 1; y-intercept = –2

89. y − x = 0 ; y = x Slope = 1; y-intercept = 0

85. x = − 4 ; Slope is undefined y-intercept - none

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Chapter 2: Graphs

Section 2.3: Lines

90. x + y = 0 ; y = − x Slope = –1; y-intercept = 0

93. a.

x-intercept: 2 x + 3 ( 0 ) = 6 2x = 6 x=3 The point ( 3, 0 ) is on the graph. y-intercept: 2 ( 0 ) + 3 y = 6 3y = 6 y=2 The point ( 0, 2 ) is on the graph. y

b. 91. 2 y − 3x = 0 ; 2 y = 3x → y =

5

3 x 2

(0, 2) (3, 0)

3 Slope = ; y-intercept = 0 2

−5

x

5 −5

94. a.

x-intercept: 3x − 2 ( 0 ) = 6 3x = 6 x=2 The point ( 2, 0 ) is on the graph.

92. 3x + 2 y = 0 ; 2 y = −3x → y = −

y-intercept: 3 ( 0 ) − 2 y = 6

3 x 2

−2 y = 6 y = −3

3 Slope = − ; y-intercept = 0 2

The point ( 0, −3) is on the graph. y

b.

5 (2, 0)

−5

5 (0, −3)

−5

186 186 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

x

Chapter 2: Graphs

95. a.

Section 2.3: Lines

x-intercept: −4 x + 5 ( 0 ) = 40

97. a.

x-intercept: 7 x + 2 ( 0 ) = 21

−4 x = 40

7 x = 21

x = −10 The point ( −10, 0 ) is on the graph.

x=3 The point ( 3, 0 ) is on the graph.

y-intercept: −4 ( 0 ) + 5 y = 40

y-intercept: 7 ( 0 ) + 2 y = 21

5 y = 40

2 y = 21 21 y= 2

y =8

The point ( 0, 8) is on the graph. The point 0,

b.

21  is on the graph. 2 

b.

96. a.

x-intercept: 6 x − 4 ( 0 ) = 24 6 x = 24 x=4 The point ( 4, 0 ) is on the graph.

98. a.

x-intercept: 5x + 3 ( 0 ) = 18 5x = 18 18 x= 5

y-intercept: 6 ( 0 ) − 4 y = 24 −4 y = 24 y = −6

The point

The point ( 0, −6 ) is on the graph.

18  , 0 is on the graph. 5 

y-intercept: 5 ( 0 ) + 3 y = 18 3 y = 18

b.

y=6 The point ( 0, 6 ) is on the graph. b.

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Chapter 2: Graphs

99. a.

Section 2.3: Lines

1 1 x + (0) = 1 2 3 1 x =1 2 x=2 The point ( 2, 0 ) is on the graph.

x-intercept:

y-intercept:

101. a.

x-intercept: 0.2 x − 0.5 ( 0 ) = 1 0.2x = 1 x=5 The point ( 5, 0 ) is on the graph. y-intercept: 0.2 ( 0 ) − 0.5 y = 1 −0.5 y = 1

1 1 (0) + y = 1 2 3 1 y =1 3 y=3

y = −2 The point ( 0, −2 ) is on the graph. b.

The point ( 0, 3) is on the graph. b.

102. a.

x-intercept: −0.3x + 0.4 ( 0 ) = 1.2 −0.3x = 1.2

100. a.

x = −4 The point ( −4, 0 ) is on the graph.

2 x-intercept: x − ( 0 ) = 4 3 x=4 The point ( 4, 0 ) is on the graph. 2 y=4 3 2 − y=4 3 y = −6 The point ( 0, −6 ) is on the graph.

y-intercept: −0.3 ( 0 ) + 0.4 y = 1.2 0.4 y = 1.2 y=3

y-intercept: ( 0 ) −

The point ( 0, 3) is on the graph. b.

b.

103. The equation of the x-axis is y = 0 . (The slope is 0 and the y-intercept is 0.) 104. The equation of the y-axis is x = 0 . (The slope is undefined.) 105. The slopes are the same but the y-intercepts are different. Therefore, the two lines are parallel.

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Chapter 2: Graphs

Section 2.3: Lines

106. The slopes are opposite-reciprocals. That is, their product is −1 . Therefore, the lines are perpendicular. 107. The slopes are different and their product does not equal −1 . Therefore, the lines are neither parallel nor perpendicular. 108. The slopes are different and their product does not equal −1 (in fact, the signs are the same so the product is positive). Therefore, the lines are neither parallel nor perpendicular. 109. Intercepts: ( 0, 2 ) and ( −2, 0 ) . Thus, slope = 1. y = x + 2 or x − y = − 2 110. Intercepts: ( 0,1) and (1, 0 ) . Thus, slope = –1. y = − x + 1 or x + y = 1

115. P1 = ( −1, 0 ) , P2 = ( 2, 3) , P3 = (1, −2 ) , P4 = ( 4,1) m12 =

3− 0 3 = = 1; m 2 − ( −1) 3

24

=

1− 3 = −1 ; 4−2

1 − ( −2 ) 3 −2− 0 = = 1 ; m13 = = −1 4 −1 3 1 − ( −1) Opposite sides are parallel (same slope) and adjacent sides are perpendicular (product of slopes is −1 ). Therefore, the vertices are for a rectangle. m34 =

116. P1 = ( 0, 0 ) , P2 = (1, 3) , P3 = ( 4, 2 ) , P4 = ( 3, −1) 3− 0 2− 3 1 m12 = = 3 ; m23 = =− ; 1− 0 4 −1 3 −1 − 2 −1 − 0 1 = 3 ; m14 = =− m34 = 3− 4 3−0 3 d12 =

(1 − 0 )2 + ( 3 − 0 )

1 111. Intercepts: ( 3, 0 ) and ( 0,1) . Thus, slope = − . 3 1 y = − x + 1 or x + 3 y = 3 3

d 23 =

( 4 − 1)2 + ( 2 − 3)2

d 34 =

( 3 − 4 )2 + ( −1 − 2 )2

= 1 + 9 = 10

d14 =

( 3 − 0 )2 + ( −1 − 0 )2

= 9 + 1 = 10

112. Intercepts: ( 0, −1) and ( −2, 0 ) . Thus,

Opposite sides are parallel (same slope) and adjacent sides are perpendicular (product of slopes is −1 ). In addition, the length of all four sides is the same. Therefore, the vertices are for a square.

slope = −

1 . 2

1 y = − x −1 or x + 2 y = − 2 2 5−3 2 2 = =− −2 −1 −3 3 3− 0 3 = P2 = (1, 3) , P3 = ( −1, 0 ) : m2 = 1 − ( −1) 2

113. P1 = ( −2, 5) , P2 = (1, 3) : m1 =

Since m1 ⋅ m2 = −1 , the line segments P1 P2 and P2 P3 are perpendicular. Thus, the points P1 , P2 , and P3 are vertices of a right triangle. 114. P1 = (1, −1) , P2 = ( 4,1) , P3 = ( 2, 2 ) , P4 = ( 5, 4 ) 1 − ( −1) 2 4− 1 = 3; = ; m24 = 5−4 4 −1 3 2− ( −1) 4− 2 2 =3 m34 = = ; m13 = 2 −1 5 −2 3 Each pair of opposite sides are parallel (same slope) and adjacent sides are not perpendicular. Therefore, the vertices are for a parallelogram. m12 =

2

= 1 + 9 = 10 = 9 + 1 = 10

117. Let x = number of miles driven, and let C = cost in dollars. Total cost = (cost per mile)(number of miles) + fixed cost C = 0.60 x + 39 When x = 110, C = ( 0.60 )(110 ) + 39 = $105.00 .

When x = 230, C = ( 0.60 )( 230 ) + 39 = $177.00 .

118. Let x = number of pairs of jeans manufactured, and let C = cost in dollars. Total cost = (cost per pair)(number of pairs) + fixed cost C = 8x + 500 When x = 400, C = ( 8 ) ( 400 ) + 500 = $3700 . When x = 740, C = ( 8 ) ( 740 ) + 500 = $6420 .

119. Let x = number of miles driven annually, and let C = cost in dollars. Total cost = (approx cost per mile)(number of miles) + fixed cost C = 0.17 x + 4462

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Chapter 2: Graphs

Section 2.3: Lines

120. Let x = profit in dollars, and let S = salary in dollars. Weekly salary = (% share of profit)(profit) + weekly pay S = 0.05x + 375 121. a.

C = 0.0821x + 15.37 ; 0 ≤ x ≤ 800

b.

c.

For 200 kWh, C = 0.0821(200) + 15.37 = $31.79

d.

For 500 kWh, C = 0.0821(500) + 15.37 = $56.42

e.

For each usage increase of 1 kWh, the monthly charge increases by $0.0821 (that is, 8.21 cents).

122. a.

123. (°C, °F ) = (0, 32); (°C, °F ) = (100, 212) 212 − 32 180 9 = = 100 − 0 100 5 9 °F − 32 = (°C − 0) 5 9 °F − 32 = (°C ) 5 5 °C = (°F − 32) 9 If °F = 70 , then 5 5 °C = (70 − 32) = (38) 9 9 °C ≈ 21.1° slope =

124. a. b.

C = 0.0907 x + 7.24 ; 0 ≤ x ≤ 1000

b. 125. a.

b. c. d. e.

For 200 kWh, C = 0.0907 ( 200 ) + 7.24 = $25.38 For 500 kWh,

C = 0.0907 (500 ) + 7.24 = $52.59 For each usage increase of 1 kWh, the monthly charge increases by $0.0907 (that is, 9.07 cents).

K =º C + 273 5 º C = (º F − 32) 9 5 K = (°F − 32) + 273 9 5 160 K = ºF − + 273 9 9 5 2297 K = ºF+ 9 9 The y-intercept is (0, 30), so b = 30. Since the ramp drops 2 inches for every 25 inches −2 2 of run, the slope is m = = − . Thus, 25 25 2 the equation is y = − x + 30 . 25 Let y = 0. 0=−

2 x + 30 25

2 x = 30 25 25 2 25 x = ( 30 ) 2 25  2 x = 375 The x-intercept is (375, 0). This means that the ramp meets the floor 375 inches (or 31.25 feet) from the base of the platform.

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Chapter 2: Graphs

Section 2.3: Lines

c.

No. From part (b), the run is 31.25 feet which exceeds the required maximum of 30 feet.

d.

First, design requirements state that the maximum slope is a drop of 1 inch for each 1 . 12 inches of run. This means m ≤ 12 Second, the run is restricted to be no more than 30 feet = 360 inches. For a rise of 30 inches, this means the minimum slope is 30 1 1 = . That is, m ≥ . Thus, the 360 12 12 1 only possible slope is m = . The 12 diagram indicates that the slope is negative. Therefore, the only slope that can be used to obtain the 30-inch rise and still meet design 1 requirements is m = − . In words, for 12 every 12 inches of run, the ramp must drop exactly 1 inch.

126. a.

b.

c.

d.

127. a.

The year 2000 corresponds to x = 0, and the year 2012 corresponds to x = 12. Therefore, the points (0, 20.6) and (12, 9.3) are on the line. Thus, 9.3− 20.6 11.3 m= =− = −0.942 . The y12 − 0 12 intercept is 20.6, so b = 20.6 and the equation is y = −0.942x + 20.6 0 = −0.942x + 20.6 0.942 x = 20.6 x = 21.9 y-intercept: y = −0.942 ( 0 ) + 20.6 = 20.6 The intercepts are (21.9, 0) and (0, 20.6). The y-intercept represents the percentage of twelfth graders in 2000 who had reported daily use of cigarettes. The x-intercept represents the number of years after 2000 when 0% of twelfth graders will have reported daily use of cigarettes. x-intercept:

The year 2025 corresponds to x = 25. y = −0.942 ( 25) + 20.6 = −2.95 This prediction is not reasonable.

( x2 , A2 ) = (200, 000, 60, 000) 60, 000 − 40, 000 200, 000 −100, 000 20, 000 1 = = 100, 000 5 1 A − 40, 000 = ( x − 100, 000 ) 5 1 A − 40, 000 = x − 20, 000 5 1 A = x + 20, 000 5 slope =

b.

If x = 300,000, then 1 A = ( 300, 000 ) + 20, 000 = $80, 000 5

c.

Each additional box sold requires an additional $0.20 in advertising.

128. Find the slope of the line containing ( a, b ) and

( b, a ) : a−b = −1 b−a The slope of the line y = x is 1. slope =

Since −1⋅1 = −1 , the line containing the points (a, b) and (b, a) is perpendicular to the line y= x. The midpoint of (a, b) and (b, a) is M =

a+b b+a , . 2 2 

Since the coordinates are the same, the midpoint lies on the line y = x . Note:

a+b b+a = 2 2

129. 2 x − y = C Graph the lines: 2x − y = − 4 2x − y = 0 2x − y = 2 All the lines have the same slope, 2. The lines

Let x = number of boxes to be sold, and A = money, in dollars, spent on advertising. We have the points ( x1 , A1 ) = (100, 000, 40, 000);

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Chapter 2: Graphs

Section 2.3: Lines

slope 1 and y-intercept (0, −1). Thus, the lines are parallel with positive slopes. One line has a positive y-intercept and the other with a negative y-intercept. 134. (d) The equation y − 2 x = 2 has slope 2 and yintercept (0, 2). The equation x + 2 y = −1 has 1 1 slope − and y-intercept 0, − . The lines 2 2  1 are perpendicular since 2 − = −1 . One line 2  has a positive y-intercept and the other with a negative y-intercept.

are parallel.

130. Refer to Figure 47. length of OA = d ( O, A ) =

1 + m12

length of OB = d ( O, B ) =

1 + m2 2

135 – 137. Answers will vary.

length of AB = d ( A, B ) = m1 − m2

138. No, the equation of a vertical line cannot be written in slope-intercept form because the slope is undefined.

Now consider the equation

(

1+ m 21

) ( 2

+

1 + m2 2

)

2

= ( m1 − m2 ) 2

If this equation is valid, then ΔAOB is a right triangle with right angle at vertex O.

( 1+ m ) + ( 2 1

2

1 + m2

2

)

2

= ( m1 − m2 )

2

1 + m12 + 1 + m2 2 = m12 − 2m1m2 + m2 2 2 + m12 + m2 2 = m12 − 2m1m2 + m2 2 But we are assuming that m1m2 = −1 , so we have

139. No, a line does not need to have both an xintercept and a y-intercept. Vertical and horizontal lines have only one intercept (unless they are a coordinate axis). Every line must have at least one intercept. 140. Two lines with equal slopes and equal y-intercepts are coinciding lines (i.e. the same).

2 + m12 + m22 = m12 − 2( −1) + m 22

141. Two lines that have the same x-intercept and yintercept (assuming the x-intercept is not 0) are the same line since a line is uniquely defined by two distinct points.

2 + m12 + m2 2 = m12 + 2 + m2 2 0=0 Therefore, by the converse of the Pythagorean Theorem, ΔAOB is a right triangle with right angle at vertex O. Thus Line 1 is perpendicular to Line 2.

142. No. Two lines with the same slope and different xintercepts are distinct parallel lines and have no points in common. Assume Line 1 has equation y = mx + b1 and Line 2 has equation y = mx + b2 ,

131. (b), (c), (e) and (g) The line has positive slope and positive y-intercept. 132. (a), (c), and (g) The line has negative slope and positive y-intercept.

b1 and y-intercept b1 . m b Line 2 has x-intercept − 2 and y-intercept b . 2 m Assume also that Line 1 and Line 2 have unequal x-intercepts. If the lines have the same y-intercept, then b1 = b2 . Line 1 has x-intercept −

133. (c) The equation x − y = −2 has slope 1 and yintercept (0, 2). The equation x − y = 1 has

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Chapter 2: Graphs

b1 = b2

b1

Section 2.4: Circles

= m

b2

− m

b1

=− m

b2

146.

m b1 b2 But − = − Line 1 and Line 2 have the m m same x-intercept, which contradicts the original assumption that the lines have unequal x-intercepts. Therefore, Line 1 and Line 2 cannot have the same y-intercept. 143. Yes. Two distinct lines with the same y-intercept, but different slopes, can have the same x-intercept if the x-intercept is x = 0 . Assume Line 1 has equation y = m1x + b and Line 2 has equation y = m2 x + b , b Line 1 has x-intercept − and y-intercept b . m1

−2

=

x2 4 5 3 x y y

=

1 x 2 y8

x 2 y8 = 1

−2

−2

2

= x 4 y16

2 2 2 147. h = a + b

= 82 + 152 = 16 + 225 = 289 h = 289 = 17 148.

b and y-intercept b . m2 Assume also that Line 1 and Line 2 have unequal slopes, that is m1 ≠ m2 . If the lines have the same x-intercept, then b b − =− . m1 m2 Line 2 has x-intercept −

b b − =− m1 m2

x 2 y −3 x4 y 5

( x − 3)2 + 25 = 49 ( x − 3)2 = 24 x − 3 = ± 24 x − 3 = ±2 6 x = 3± 2 6

{

149.

2 x − 5 + 7 < 10 2x − 5 < 3

−m2 b = −m1b −m2 b + m1b = 0 But − m2 b + m1b = 0 b ( m1 − m2 ) = 0 b=0 or m1 − m2 = 0 m1 = m2 Since we are assuming that m1 ≠ m2 , the only way that the two lines can have the same x-intercept is if b = 0.

−3 < 2x − 5 < 3 2 < 2x < 8 1< x < 4 The solution set is: { x | 1 < x < 4} . Interval notation: (1, 4 )

144. Answers will vary. 145. m =

y2 − y1 x2 − x1

=

−4− 2

1 − ( −3)

=

−6 4

=−

3 2

It appears that the student incorrectly found the slope by switching the direction of one of the subtractions.

}

The solution set is: 3 − 2 6 , 3 + 2 6 .

Section 2.4 1. add; 2.

( 12 ⋅10 )

( x − 2)2

2

= 25

=9

x−2= ± 9 x − 2 = ±3 x = 2±3 x = 5 or x = −1 The solution set is {−1, 5}.

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Chapter 2: Graphs

Section 2.4: Circles

3. False. For example, x 2 + y 2 + 2x + 2 y + 8 = 0 is not a circle. It has no real solutions.

General form: x 2 + y 2 − 4 = 0

4. radius 5. True; r 2 = 9 → r = 3 6. False; the center of the circle

( x + 3 ) 2 + ( y − 2 )2

= 13 is ( −3, 2 ) .

7. d 8. a

14.

9. Center = (2, 1) Radius = distance from (0,1) to (2,1) = (2 − 0) 2 + (1−1) 2 =

( x − h) 2 + ( y − k ) 2 = r 2 ( x − 0) 2 + ( y − 0) 2 = 32 x 2 + y2 = 9

4=2

General form: x 2 + y 2 − 9 = 0

Equation: ( x − 2) 2 + ( y −1) 2 = 4 10. Center = (1, 2) Radius = distance from (1,0) to (1,2) = (1− 1) 2 + (2 − 0) 2 =

4=2

Equation: ( x − 1) + ( y − 2) = 4 2

2

11. Center = midpoint of (1, 2) and (4, 2) 1+ 4 2+ 2 5 =

(

) ( )

2

,

=

2

2

r 15. ( x h) ( y k ) − 2+ − 2= 2

,2

( )

( x − 0) 2 + ( y − 2) 2 = 22

Radius = distance from 5 , 2 to (4,2) 2 5 = 4− 2

2

5 Equation: x − 2

2

+ (2 − 2) =

9

2

+ ( y − 2) 2 = 4 

4

= 2

x 2 + ( y − 2) 2 = 4 3

General form: x 2 + y 2 − 4 y + 4 = 4 x2 + y 2 − 4y = 0

9

12. Center = midpoint of (0, 1) and (2, 3) 0+ 2 1 + 3 = (1, 2 ) = , 2 2  Radius = distance from (1, 2 ) to (2,3) = Equation:

( 2 −1)2 + (3 − 2) 2

= 2

( x −1)2 + ( y − 2) 2 = 2

2 2 2 16. ( x − h) + ( y − k) = r

13. ( x − h) 2 + ( y − k ) 2 = r 2

( x − 1) 2 + ( y − 0)2 = 32

( x − 0) + ( y − 0) = 2

( x − 1) 2 + y 2 = 9

2

2

2

x 2 + y2 = 4

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Chapter 2: Graphs

Section 2.4: Circles

General form: x 2 − 2x + 1 + y 2 = 9 x 2 + y 2 − 2x − 8 = 0

19. ( x − h) 2 + ( y − k ) 2 = r 2 ( x − ( −2 )) 2 + ( y − 1) 2 = 42 ( x + 2) 2 + ( y − 1) 2 = 16 General form: x 2 + 4x + 4 + y 2 − 2 y + 1 = 16 x 2 + y 2 + 4x − 2 y −11 = 0

17. ( x − h) 2 + ( y − k ) 2 = r 2 ( x − 4) 2 + ( y − (−3)) 2 = 52 ( x − 4) 2 + ( y + 3) 2 = 25 General form: x 2 − 8x + 16 + y 2 + 6 y + 9 = 25 x + y − 8x + 6 y = 0 2

2

20. ( x − h) 2 + ( y − k) 2 = r 2 ( x − ( −5 )) 2 + ( y − (−2)) 2 = 7 2 ( x + 5) 2 + ( y + 2) 2 = 49 General form: x 2 + 10 x + 25 + y 2 + 4 y + 4 = 49 x 2 + y2 + 10 x + 4 y − 20 = 0

18. ( x − h) 2 + ( y − k ) 2 = r 2 ( x − 2) 2 + ( y − (−3)) 2 = 4 2 ( x − 2) 2 + ( y + 3) 2 = 16 General form: x 2 − 4x + 4 + y 2 + 6 y + 9 = 16 x 2 + y 2 − 4x + 6 y − 3 = 0

21. ( x − h) 2 + ( y − k) 2 = r 2 x− 2

1 2

1 + ( y − 0) 2 =  2 2 1 1 x − + y2 = 2 4

2

1 1 + y2 = 4 4 x 2 + y2 − x = 0

General form: x 2 − x +

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

Chapter 2: Graphs

Section 2.4: Circles

c.

x-intercepts: x 2 + ( 0 ) = 4 2

x2 = 4 x = ± 4 = ±2 y-intercepts: ( 0 ) + y 2 = 4 2

y2 = 4 y = ± 4 = ±2 The intercepts are ( −2, 0 ) , ( 2, 0 ) , ( 0, −2 ) , and ( 0, 2 ) . 22. ( x − h) 2 + ( y − k ) 2 = r 2

( x − 0 )2 +

y− −

1  2

2

24. x 2 + ( y − 1) 2 = 1 1 =  2

2

x 2 + ( y − 1) 2 = 12 a.

2

1 1 x + y+ = 2 4 2

Center:(0, 1); Radius = 1

b.

1 1 = 4 4 x2 + y2 + y = 0

General form: x 2 + y 2 + y +

c.

x-intercepts: x 2 + (0 − 1) 2 = 1 x2 + 1 = 1 x2 = 0 x=± 0 =0 y-intercepts: ( 0 ) + ( y − 1) 2 = 1 2

23. x + y = 4 2

2

( y − 1) 2 = 1

x 2 + y 2 = 22 a.

y −1 = ± 1

Center: (0, 0); Radius = 2

y −1 = ±1 y = 1±1 y = 2 or y = 0

b.

The intercepts are ( 0, 0 ) and ( 0, 2 ) . 25. 2 ( x − 3) + 2 y 2 = 8 2

( x − 3 )2 a.

+ y2 = 4

Center: (3, 0); Radius = 2

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Chapter 2: Graphs

Section 2.4: Circles

b.

c.

x-intercepts: ( x + 1) + ( 0 − 1) = 2 2

2

( x + 1) + ( −1) = 2 ( x + 1)2 + 1 = 2 2 ( x + 1) = 1 2

2

x +1 = ± 1

c.

x + 1 = ±1 x = −1 ± 1 x = 0 or x = −2

x-intercepts: ( x − 3) + ( 0 ) = 4 2

2

( x − 3)

2

y-intercepts: ( 0 + 1) + ( y −1) = 2 2

=4

2

(1)2 + ( y −1)2

x−3 = ± 4 x − 3 = ±2

1 + ( y −1) = 2

( y −1)2

x = 3± 2 x = 5 or x = 1 2

( −3)

y − 1 = ±1 y = 1±1 y = 2 or y = 0 The intercepts are ( −2, 0 ) , ( 0, 0 ) , and ( 0, 2 ) .

+y =4 2

9 + y2 = 4 y 2 = −5 No real solution. The intercepts are (1, 0 ) and ( 5, 0 ) .

27. x 2 + y 2 − 2x − 4 y − 4 = 0 x 2 − 2x + y 2 − 4 y = 4

26. 3 ( x + 1) + 3 ( y −1) = 6 2

a. b.

( x 2 − 2x + 1) + ( y 2 − 4 y + 4) = 4 + 1 + 4 (

2

( x + 1)2 + ( y −1)2

=1

y −1 = ± 1

y-intercepts: ( 0 − 3) + y 2 = 4 2

=2

2

x −1) 2 + ( y − 2) 2 = 32

=2

Center: (–1,1); Radius =

2

a.

Center: (1, 2); Radius = 3

b.

c.

x-intercepts: ( x − 1) 2 + (0 − 2) 2 = 32 ( x −1) 2 + (−2) 2 = 32

( x − 1)2 + 4 = 9 ( x − 1)2 = 5

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Chapter 2: Graphs

Section 2.4: Circles x −1 = ± 5 x = 1± 5

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Chapter 2: Graphs

Section 2.4: Circles

y-intercepts: (0 −1) 2 + ( y − 2) 2 = 32

29.

(−1) 2 + ( y − 2) 2 = 32 1 + ( y − 2) = 9 2

( y − 2 )2

=8

x 2 + y 2 + 4x − 4 y −1 = 0 x 2 + 4x + y 2 − 4 y = 1 ( x 2 + 4x + 4) + ( y 2 − 4 y + 4) = 1 + 4 + 4 ( x + 2) 2 + ( y − 2) 2 = 32 a.

y−2 = ± 8

b.

y − 2 = ±2 2

Center: (–2, 2); Radius = 3 y 5

y = 2±2 2

( )( 2 ) , and ( 0, 2 + 2 2 ) .

(−2, 2)

)

The intercepts are 1 − 5, 0 , 1 + 5, 0 ,

( 0, 2 − 2

−5

5 x

x 2 + y 2 + 4 x + 2 y − 20 = 0

28.

−5

x 2 + 4x + y 2 + 2 y = 20 c.

( x 2 + 4x + 4) + ( y2 + 2 y + 1) = 20 + 4 + 1 (

x-intercepts: ( x + 2) 2 + (0 − 2) 2 = 32 ( x + 2) 2 + 4 = 9

x + 2) 2 + ( y + 1) 2 = 52 a. b.

( x + 2) 2 = 5

Center: (–2,–1); Radius = 5

x+2= ± 5 x = −2 ± 5 y-intercepts: (0 + 2) + ( y − 2) 2 = 32 2

4 + ( y − 2)2 = 9 ( y − 2)2 = 5 y−2= ± 5

c.

( −2 +

( x + 2) 2 + 1 = 25 x + 2 = ± 24 x + 2 = ±2 6 x = −2 ± 2 6 y-intercepts: (0 + 2) + ( y + 1) = 52 2

2

4 + ( y + 1) = 25 2

30.

a.

Center: (3, –1); Radius = 1

b.

y + 1 = ± 21 y = −1 ± 21

)

The intercepts are −2 − 2 6, 0 ,

) (

)

)

(

x2 + y2 − 6 x + 2y + 9 = 0 x 2 − 6x + y 2 + 2 y = −9 2 ( x − 6x + 9) + ( y 2 + 2 y + 1) = −9 + 9 + 1 ( x − 3) 2 + ( y + 1) 2 = 12

( y + 1) = 21

( −2 + 2

)(

)

5, 0 , 0, 2 − 5 , and 0, 2 + 5 .

2

(

)

The intercepts are −2 − 5, 0 ,

x-intercepts: ( x + 2) 2 + (0 + 1) 2 = 52 ( x + 2) 2 = 24

(

y = 2± 5

6 , 0 , 0, −1 − 21 , and

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Chapter 2: Graphs

( 0, −1 +

Section 2.4: Circles

)

21 .

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Chapter 2: Graphs

c.

Section 2.4: Circles

x-intercepts: ( x − 3) 2 + (0 + 1) 2 = 12 ( x − 3) + 1 = 1 2

( x − 3)

=0 x−3= 0 x=3 2

y-intercepts: (0 − 3) 2 + ( y + 1) 2 = 12 9 + ( y + 1) 2 = 1

( y + 1)

= −8 No real solution. The intercept only intercept is ( 3, 0 ) . 2

32.

1 =0 2 1 x 2 + x + y2 + y = 2 1 1 1 1 1 2 2 x + x+ + y + y+ = + + 4 2 4 4 4 2 2 1 1 x+ + y+ = 12 2 2 x2 + y2 + x + y −

a. b.

1 1 Center:  − , − ; Radius = 1 2 2

x2 + y2 − x + 2 y + 1 = 0

31.

x 2 − x + y2 + 2 y = −1 1 1 x 2 − x + + ( y2 + 2 y + 1) = −1 + + 1 4 4 2 2  1 1 + ( y + 1) 2 = x− 2 2 a.

Center:

1 1 , −1 ; Radius =  2 2 c.

b.

x-intercepts: x +

1

2 1  + 0 +  = 12 2 2 2  x + 1  + 1 =1  2 4 2  1 3 x+ = 4  2

2

x+

c.

x-intercepts: x −

1 2

2

2 1  + (0 + 1) 2 = 2

y-intercepts: 0 +

2

1 1 +1 = x− 4 2   12 3 x− =− 4 2 No real solutions y-intercepts: 0 −

1 2

2 1  + ( y + 1) 2 = 2 1 1 2 + ( y + 1) = 4 4 2 ( y + 1) = 0 y +1 = 0 y = −1 2

The only intercept is ( 0, −1) .

1 2

2

1 + y+ 2

1 3 =± 2 2 −1 ± 3 x= 2 2

=2 1 2

1 1 =1 + y + 4  2 2  1 3 y+ = 2 4 1 3 y+ =± 2 2 −1 ± 3 y= 2  −1+ 3  −1− 3 The intercepts are  ,0,  ,0, 2 2   −1 + 3 −1 − 3 0, , and 0, .

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2

2

Chapter 2: Graphs

Section 2.4: Circles

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Chapter 2: Graphs

Section 2.4: Circles

2x 2 + 2 y 2 −12 x + 8 y − 24 = 0

33.

x 2 + y 2 − 6 x + 4 y = 12 x 2 − 6 x + y 2 + 4 y = 12 ( x 2 − 6x + 9) + ( y 2 + 4 y + 4) = 12 + 9 + 4 ( x − 3)2 + ( y + 2) 2 = 52 a.

Center: (3,–2); Radius = 5

b.

b. c.

1 2 1 = 2

x-intercepts: ( x + 2) 2 + ( 0 ) = 2

( x + 2 )2

1 2 2 x+2=± 2

x+2=±

c.

x-intercepts: ( x − 3) 2 + (0 + 2) 2 = 52

x = −2 ±

( x − 3) + 4 = 25 ( x − 3)2 = 21 2

1 2 1 2 4+ y = 2

y-intercepts: (0 + 2) 2 + y2 =

x − 3 = ± 21 x = 3 ± 21

7 y2 = − 2 No real solutions. 2  The intercepts are −2 − , 0 and 2 

y-intercepts: (0 − 3) 2 + ( y + 2) 2 = 52 2 9 + ( y + 2 ) = 25

( y + 2 )2 = 16

(

y + 2 = ±4 y = −2 ± 4 y = 2 or y = −6

) (3 +

The intercepts are 3 − 21, 0 ,

( 0, −6 ) , 34. a.

and ( 0, 2 ) .

2x 2 + 2 y 2 + 8x + 7 = 0 2 x 2 + 8x + 2 y 2 = −7 7 x 2 + 4x + y 2 = − 2 7 2 2 ( x + 4x + 4) + y = − + 4 2 1 2 2 ( x + 2) + y = 2 2 2  ( x + 2) 2 + y 2 = 2  Center: (–2, 0); Radius =

2 2

−2 +

)

21, 0 ,

35.

2  , 0. 2

2 x 2 + 8x + 2 y 2 = 0 x 2 + 4x + y 2 = 0 x 2 + 4x + 4 + y 2 = 0 + 4 a.

( x + 2 )2 + y 2 = 22 Center: ( −2, 0 ) ;

b.

2

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Radius: r = 2

Chapter 2: Graphs

Section 2.4: Circles 2

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Chapter 2: Graphs

Section 2.4: Circles

x-intercepts: ( x + 2 ) + ( 0 ) = 2 2 2

c.

2

37. Center at (0, 0); containing point (–2, 3).

( −2 − 0 )

+ ( 3 − 0 ) = 4 + 9 = 13

( x + 2)2 = 4

r=

( x + 2)

Equation: ( x − 0)2 + ( y − 0) 2 =

2

=± 4

x + 2 = ±2 x = −2 ± 2 x = 0 or x = −4 y-intercepts: ( 0 + 2 ) + y 2 = 22 2

4+ y = 4

2

2

y =0 y=0 The intercepts are ( −4, 0 ) and ( 0, 0 ) . 36. 3x 2 + 3 y 2 −12 y = 0

2

x 2 + y2 = 13 38. Center at (1, 0); containing point (–3, 2). r=

( −3 − 1)2 + ( 2 − 0 )2

= 16 + 4 = 20 = 2 5

2 2

( 13)

Equation: ( x −1) 2 + ( y − 0) 2 =

(

20

)

2

( x −1) 2 + y 2 = 20 39. Center at (2, 3); tangent to the x-axis. r =3 Equation: ( x − 2) 2 + ( y − 3) 2 = 32

x 2 + y2 − 4 y = 0

( x − 2) 2 + ( y − 3) 2 = 9

x + y − 4y + 4 = 0 + 4 2

2

x 2 + ( y − 2) = 4 2

a.

Center:

( 0, 2 ) ; Radius:

r=2

40. Center at (–3, 1); tangent to the y-axis. r =3 Equation: ( x + 3) 2 + ( y − 1) 2 = 32 ( x + 3) 2 + ( y − 1) 2 = 9

b.

41. Endpoints of a diameter are (1, 4) and (–3, 2). The center is at the midpoint of that diameter: 1 + (−3) 4+ 2 = ( −1, 3 ) Center: , 2 2 Radius: r = (1 − (−1)) 2 + (4 − 3) 2 = 4 + 1 = 5 Equation: ( x − (−1)) 2 + ( y − 3) 2 = c.

x-intercepts: x 2 + ( 0 − 2 ) = 4

( 5)

2

( x + 1) 2 + ( y − 3) 2 = 5

2

x2 + 4 = 4 x2 = 0 x=0

y-intercepts: 0 2 + ( y − 2 ) = 4 2

( y − 2)

2

=4

42. Endpoints of a diameter are (4, 3) and (0, 1). The center is at the midpoint of that diameter: 4+ 0 3+ 1 , = ( 2, 2 ) Center: 2 2 Radius: r = (4 − 2) 2 + (3 − 2) 2 = Equation: ( x − 2) 2 + ( y − 2) 2 =

y−2= ± 4 y − 2 = ±2 y = 2±2 y = 4 or y = 0 The intercepts are ( 0, 0 ) and ( 0, 4 ) .

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( 5)

( x − 2) 2 + ( y − 2) 2 = 5

4 +1 = 5 2

Chapter 2: Graphs

Section 2.4: Circles

43. Center at (–1, 3); tangent to the line y = 2. This means that the circle contains the point (–1, 2), so the radius is r = 1. Equation: ( x + 1) 2 + ( y − 3) 2 = (1) 2

44. Center at (4, –2); tangent to the line x = 1. This means that the circle contains the point (1, –2), so the radius is r = 3. Equation: ( x − 4) 2 + ( y + 2) 2 = (3) 2 ( x − 4) 2 + ( y + 2) 2 = 9 45. (c); Center: (1, −2 ) ; Radius = 2

( −3, 3) ; Radius = 3

47. (b) ; Center:

( −1, 2 ) ; Radius = 2

48. (a) ; Center:

( −3, 3) ; Radius = 3

49. Let the upper-right corner of the square be the point ( x, y ) . The circle and the square are both centered about the origin. Because of symmetry, we have that x = y at the upper-right corner of the square. Therefore, we get x2 + y 2 = 9 x2 + x2 = 9 2x 2 = 9 9 x2 = 2 9 32 x= = 2 2 The length of one side of the square is 2x . Thus, the area is 2 2 3 2 2 A = s = 2⋅ = 3 2 = 18 square units. 2

(

x 2 + x 2 = 36 2 x 2 = 36 x 2 = 18

( x + 1) 2 + ( y − 3) 2 = 1

46. (d) ; Center:

x 2 + y2 = 36

)

x=3 2 The length of one side of the square is 2x . Thus,

(

the area of the square is 2 ⋅ 3 2

)

2

units. From the equation of the circle, we have r = 6 . The area of the circle is 2 πr 2 = π( 6 ) = 36π square units. Therefore, the area of the shaded region is A = 36π− 72 square units. 51. The diameter of the Ferris wheel was 250 feet, so the radius was 125 feet. The maximum height was 264 feet, so the center was at a height of 264 −125 = 139 feet above the ground. Since the center of the wheel is on the y-axis, it is the point (0, 139). Thus, an equation for the wheel is:

( x − 0 )2 + ( y − 139 )2 = 1252 2 x 2 + ( y − 139 ) = 15, 625

52. The diameter of the wheel is 520 feet, so the radius is 260 feet. The maximum height is 550 feet, so the center of the wheel is at a height of 550 − 260 = 290 feet above the ground. Since the center of the wheel is on the y-axis, it is the point (0, 290). Thus, an equation for the wheel is:

( x − 0)2 + ( y − 290)2 2 x 2 + ( y − 290 )

= 260 2 = 67, 600

2 2 53. x + y + 2 x + 4 y − 4091 = 0

x 2 + 2x + y2 + 4 y − 4091 = 0 x 2 + 2x + 1 + y 2 + 4 y + 4 = 4091 + 5



= 72 square

Therefore, we get

50. The area of the shaded region is the area of the circle, less the area of the square. Let the upperright corner of the square be the point ( x, y ) . The circle and the square are both centered about the origin. Because of symmetry, we have that x = y at the upper-right corner of the square.

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Chapter 2: Graphs

( x + 1)

2

Section 2.4: Circles

+ ( y + 2 ) = 4096 2

The circle representing Earth has center ( −1, −2 ) and radius = 4096 = 64 . So the radius of the satellite’s orbit is 64 + 0.6 = 64.6 units. The equation of the orbit is

( x + 1)

2

+ ( y + 2 ) = ( 64.6 ) 2

2

x 2 + y 2 + 2x + 4 y − 4168.16 = 0

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Chapter 2: Graphs

Section 2.4: Circles

x 2 + (mx + b) 2 = r2

54. a.

Equation of the tangent line is: 2 y − 2 2 = − ( x − 1) 4 2 2 y−2 2 = − x+ 4 4 4y −8 2 = − 2 x + 2

x 2 + m 2 x 2 + 2bmx + b 2 = r2 (1 + m 2 ) x 2 + 2bmx + b 2 − r2 = 0 There is one solution if and only if the discriminant is zero. (2bm) 2 − 4(1+ m 2 )(b 2 − r2 ) = 0

2 x + 4y = 9 2

4b 2 m 2 − 4b 2 + 4r 2 − 4b 2 m 2 + 4m 2 r 2 = 0

2 x + 4y − 9 2 = 0

− 4b 2 + 4r 2 + 4m 2 r 2 = 0 − b 2 + r2 + m 2 r 2 = 0 r (1 + m ) = b 2

b.

2

56. x 2 + y 2 − 4x + 6 y + 4 = 0 ( x2 − 4x + 4) + ( y 2 + 6 y + 9) = −4 + 4 + 9

Using the quadratic formula, the result from part (a), and knowing that the discriminant is zero, we get: (1 + m 2 ) x 2 + 2bmx + b 2 − r2 = 0

( x − 2) 2 + ( y + 3) 2 = 9 Center: (2, –3)

− 2bm −bm −bmr 2 −mr 2 = 2 = 2 = 2 b b 2(1 + m ) b 2  r 

2 2 − 3 − (−3) 2 2 = =2 2 3− 2 1

x=

y=m = c.

2

−mr  +b b 2

r2 −m 2 r2 −m 2 r 2 + b 2 +b = = b b b

The slope of the tangent line is m . The slope of the line joining the point of tangency and the center is: 2  r − 0  b b 1 r2 = ⋅ =− 2 2  b −mr m −mr − 0 b  Therefore, the tangent line is perpendicular to the line containing the center of the circle and the point of tangency.

55. x 2 + y2 = 9 Center: (0, 0) Slope from center to 1, 2 2

(

Slope of the tangent line is:

)

)

−1 =2 =− 4 2 2

Equation of the tangent line: 2 y − 2 2 − 3 = − ( x − 3) 4 2 32 y−2 2 +3= − x+ 4 4 4 y − 8 2 + 12 = − 2 x + 3 2

(

)

2 x + 4 y −11 2 + 12 = 0 57. Let (h, k ) be the center of the circle. x − 2y + 4 = 0 2y = x + 4 y=

1 x+2 2

The slope of the tangent line is

1 . The slope 2

from (h, k ) to (0, 2) is –2. is

2 2− 0 2 2 = =2 2. 1− 0 1 Slope of the tangent line is

(

Slope from center to 3, 2 2 − 3 is

2 −1 =− . 4 2 2

2− k = −2 0−h 2 − k = 2h The other tangent line is y = 2x − 7 , and it has slope 2.

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Chapter 2: Graphs

Section 2.4: Circles

The slope from (h, k ) to (3, –1) is − −1 − k 1 =− 3−h 2 2 + 2k = 3 − h

Therefore, the path of the center of the circle has the equation y = 2 .

1 . 2 60.

2k = 1 − h h = 1 − 2k Solve the two equations in h and k : 2 − k = 2(1− 2k ) 2 − k = 2 − 4k 3k = 0 k =0 h = 1 − 2(0) = 1 The center of the circle is (1, 0).

61. (b), (c), (e) and (g) We need h, k > 0 and ( 0, 0 ) on the graph. 62. (b), (e) and (g) We need h < 0 , k = 0 , and h > r .

58. Find the centers of the two circles: x 2 + y 2 − 4x + 6 y + 4 = 0 ( x 2 − 4x + 4) + ( y2 + 6 y + 9) = − 4 + 4 + 9 ( x − 2) 2 + ( y + 3) 2 = 9

63. Answers will vary. 64. The student has the correct radius, but the signs of the coordinates of the center are incorrect. The student needs to write the equation in the 2

2

( x 2 + 6x + 9) + ( y2 + 4 y + 4) = − 9 + 9 + 4 ( x + 3) 2 + ( y + 2)2 = 4 Center: ( −3, −2 ) Find the slope of the line containing the centers: − 2− (−3) 1 m= =− −3 − 2 5 Find the equation of the line containing the centers: 1 y + 3 = − ( x − 2) 5 5 y + 15 = − x + 2

2

65. A = πr 2 = π(13)

2

2 = 169π cm

C = 2πr = 2π(13) = 26π cm 2 3 2 2 66. (3x − 2)(x − 2x + 3) = 3x − 6x + 9x − 2x + 4x − 6

= 3x3 − 8x 2 + 13x − 6

x + 5 y + 13 = 0 67.

2 x 2 + 3x − 1 = x + 1 2 x 2 + 3x − 1 = ( x + 1)

2

2 x 2 + 3x − 1 = x 2 + 2x + 1

(2,2)

2

( x + 3) + ( y − 2) = 16 2 ( x − ( −3)) + ( y − 2) 2 = 42 Thus, ( h, k ) = ( −3, 2 ) and r = 4 .

x 2 + y 2 + 6x + 4 y + 9 = 0

59. Consider the following diagram:

2

standard form ( x − h ) + ( y − k ) = r .

Center: ( 2, −3)

x + 5 y = −13

C = 2πr 6π = 2πr 6π 2πr = 2π 2π 3=r The radius is 3 units long.

x2 + x − 2 = 0 ( x + 2)(x − 1) = 0 x = −2 or x = 1

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Chapter 2: Graphs

Section 2.4: Circles We need to check each possible solution:

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Chapter 2: Graphs

Section 2.5: Variation

Check x = −2

5. y = kx 2 = 10k

2(−2) + 3(−2) − 1 = (−2) + 1 2

2 1 = 10 5 1 y= x 5

k=

2(4) − 6 − 1 = −1 no Check x = 1

6. v = kt 16 = 2k 8=k

2(1) + 3(1) − 1 = (1) + 1 2

2 + 3 −1 = 2 4=2

v = 8t

yes 7. A = kx 2 4π = k (2) 2

The solution is {1} 68. Let t represent the time it takes to do the job together. Time to do job Part of job done in one minute 1 Aaron 22 22 1 Elizabeth 28 28 Together

t

1 t

1 1 1 + = 22 28 t 14t + 11t = 308

4π = 4k

π= k A = πx 2 8. V = kx3 36π = k (3) 3 36π = 27k 36π 4 k= = π 27 3 4 3 V = πx 3

25t = 308 t = 12.32 Working together, the job can be done in 12.32 minutes.

9. F = 10 =

52 k 10 = 25 k = 250 250 F= 2 d

Section 2.5 1. y = kx 2. False. If y varies directly with x, then y = kx, where k is a constant. 3. b 4. c

k d2 k

10.

k x k 4= 9 k 4= 3 k = 12 12 y= y=

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Chapter 2: Graphs

Section 2.5: Variation

x

212 212 Copyright © 2016 Pearson Education, Inc. Inc. Copyright © 2016 Pearson Education,

Chapter 2: Graphs

( 5 = k (3

11. z = k x 2 + y 2 2

+ 42

) )

5 = k (25) 5 1 k= = 25 5 1 2 z = x + y2 5

(

Section 2.5: Variation

15. T 2 = 22 =

)

2

3

18 = k (18 )

13. M = 24 =

( x )(d )

23

2

3

kd

d2

( = k (9

2

+4

2

) )

8 97 8 2 3 z = x + y2 97 k=

( )

(

k 42

9 16k 24 = 3 3 9 k = 24 = 16 2 9d 2 M = 2 x

17. V =

)

4π 3 r 3

18. c 2 = a 2 + b 2 19. A =

) +3 )

1 bh 2

20. p = 2( l + w )

14. z = k x3 + y2 3

8a 3

8 = k ( 97 )

2

x

( 1 = k (2

( )

k 23

2 2 16. z 3 = k x + y

1= k T=

T2 =

2

3

d2

42 k (8) 4= 16 k 4= 2 k =8

( x )(d ) 18 = k ( 8 ) ( 3 )

12. T = k

ka 3

2

(

1 = k (17 )

21. F = 6.67 ×10−11

1 17 1 3 2 z= x +y 17

22. T =

k=

(

)

23.

) mM  d 2

2π l 32 p = kB

6.49 = k (1000 ) 0.00649 = k Therefore we have the linear equation p = 0.00649B . If B = 145000 , then p = 0.00649 (145000 ) = $941.05 .

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Chapter 2: Graphs

Section 2.5: Variation

If R = 576, then 12,288 576 = l 576l = 12, 288 12, 288 64 l= = 576 3 inches

p = kB

24.

8.99 = k (1000 ) 0.00899 = k Therefore we have the linear equation p = 0.00899B . If B = 175000 , then p = 0.00899 (175000 ) = $1573.25 . 25.

29.

s = kt 2 16 = k (1)

47.40 = k (12 ) 3.95 = k Therefore, we have the linear equation R = 3.95g .

2

k = 16 Therefore, we have equation s = 16t 2 .

If g = 10.5 , then R = ( 3.95 ) (10.5 ) ≈ $41.48 .

If t = 3 seconds, then s = 16 ( 3) = 144 feet. 2

30.

If s = 64 feet, then 64 = 16t 2 t =4 t = ±2 Time must be positive, so we disregard t = −2. It takes 2 seconds to fall 64 feet. 2

26.

R = kg

v = kt

C = kA 23.75 = k ( 5 ) 4.75 = k Therefore, we have the linear equation C = 4.75A. If A = 3.5 , then C = ( 4.75 ) ( 3.5 ) = $16.63 .

31. D = a.

64 = k ( 2 )

k=

k 2.75 k = 429 429 So, D = . p b.

3 20

Therefore, we have the linear equation E = If W = 15, then E =

28.

R=

3 (15) = 2.25 . 20

k

3 W. 20

D = 156 , p = 2.75 ; 156 =

k = 32 Therefore, we have the linear equation v = 32t. If t = 3 seconds, then v = 32 ( 3) = 96 ft/sec. 27. E = kW 3 = k ( 20 )

k p

D=

429 = 143 bags of candy 3

k s a. t = 40 , s = 30 ; k 40 = 30 k = 1200

32. t =

So, we have the equation t =

l k 256 = 48 k = 12, 288

b.

Therefore, we have the equation R =

12,288 . l

t=

1200 40

= 30 minutes

k P V = 600, P = 150 ;

33. V =

201 201 Copyright © 2016 Pearson Education, Copyright © 2016 Pearson Education, Inc. Inc.

1200 . s

Chapter 2: Graphs

Section 2.5: Variation

k 150 k = 90, 000

39. I =

600 =

k

d2 If I = 0.075, d = 2 , then

So, we have the equation V = If P = 200 , then V =

90, 000 P

0.075 =

90, 000 = 450 cm3 . 200

k and k = 0.3 . 22

So, we have the equation I = If d = 5, then I =

k 34. i = R If i = 30, R = 8 , then 30 =

k 8 and k = 240 .

0.3 . d2

0.3 = 0.012 foot-candles. 52

2 40. F = kAv

11 = k (20)(22) 2 11 = 9860k 11 1 k= = 9680 880

240 So, we have the equation i = . R 240 If R = 10, then i = = 24 amperes . 10

1 Av 2 . 880 If A = 47.125 and v = 36.5 , then 1 F= ( 47.125 ) ( 36.5)2 ≈ 71.34 pounds. 880 So, we have the equation F =

k

35. W =

d2 If W = 125, d = 3960 then k and k = 1, 960, 200, 000 3960 2 1, 960,200, 000 So, we have the equation W = . d At the top of Mt. McKinley, we have d = 3960 + 3.8 = 3963.8 , so 1, 960,200, 000 W= ≈ 124.76 pounds. ( 3963.8 )2 125 =

41.

h = ksd 3 36 = k (75)(2)3 36 = 600k 0.06 = k So, we have the equation h = 0.06sd 3 . If h = 45 and s = 125, then 45 = (0.06)(125)d 3

36. W =

k

45 = 7.5d 3

d2

6 = d3

k

55 =

d = 3 6 ≈ 1.82 inches

3960 2 k = 862, 488, 000

862,488, 000 So, we have the equation W = . d2 If d =3965, then W=

862,488, 000 39652

37. V = πr 2 h 38. V =

≈ 54.86 pounds.

42.

kT P k (300) 100 = 15 100 = 20k V =

5=k So, we have the equation V =

π 3

r2 h

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5T . P

Chapter 2: Graphs

Section 2.5: Variation

If V = 80 and T = 310, then 5(310) 80 = P 80P = 1550 1550 P= = 19.375 atmospheres 80 43.

K = kmv

46.

kwt 2 l k (4)(2)2 750 = 8 750 = 2k S=

375 = k 375wt 2 . l If l = 10, w = 6, and t = 2, then

2

1250 = k (25)(10) 2 1250 = 2500k k = 0.5 So, we have the equation K = 0.5mv 2 . If m = 25 and v = 15, then K = 0.5( 25 ) (15 ) = 2812.5 Joules

So, we have the equation S =

S=

375(6)(2) 2 = 900 pounds. 10

47 – 50. Answers will vary.

2

51. 44.

R= 1.24 =

3x3 + 25x 2 − 12 x − 100 = (3x3 + 25x 2 ) − (12x + 100)

kl 2

= x 2 (3x + 25) − 4(3x + 25) = (x 2 − 4)(3x + 25) = (x − 2)(x + 2)(3x + 25)

d k ( 432 )

(4) 2 1.24 = 27k 1.24 k= 27

52. 5

So, we have the equation R =

1.24l 27d 2

x+3

.

+

x−2 x 2 + 7x + 12

If R = 1.44 and d = 3, then 1.24l 1.44 = 27(3) 2 1.24l 1.44 = 243 349.92 = 1.24l l=

45.

5 =

5( x + 4)+ ( x − 2) ( x + 3)(x + 4) 5x + 20+ x − 2 = ( x + 3)(x + 4) 6 x + 18 = ( x + 3)(x + 4) 6( x + 3) 6 = = ( x + 3)(x + 4) ( x + 4)

349.92 ≈ 282.2 feet 1.24

S=

53.

0.6 = k 0.6pd . t If p = 40, d = 8, and t = 0.50, then 0.6(40)(8) S= = 384 psi. 0.50 So, we have the equation S =

x−2

x + 3 ( x + 3)(x + 4) 5( x + 4) x−2 = + ( x + 3)(x + 4) ( x + 3)(x + 4) =

kpd t k (25)(5) 100 = 0.75 75 = 125k

+

4 25 2 5

3 2

3

=

=

203 203 Copyright © 2016 Pearson Education, Copyright © 2016 Pearson Education, Inc. Inc.

4 25 8 125

1 2

3

Chapter 2: Graphs

Section 2.5: Variation

54. The term needed to rationalize the denominator is 7+2.

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Chapter 2: Graphs

Section 2.5: Variation

4. y = x 2 + 4

Chapter 2 Review Exercises

y

1. P1 = ( 0, 0 ) and P2 = ( 4, 2 ) a.

d ( P1 , P2 ) =

( 4 − 0)

2

+ ( 2 − 0)

(−1, 5)

= 16 + 4 = 20 = 2 5 b. The coordinates of the midpoint are: x+x y+y ( x, y) = 1 2 , 1 2  2 2  0+ 4 0+ 2 4 2 = , = , = ( 2,1) 2 2 2 2 Δy 2− 0 2 1 = = = Δx 4 − 0 4 2

c.

slope =

d.

For each run of 2, there is a rise of 1.

2. P1 = (1, −1) and P2 = ( −2, 3) a.

d ( P1 , P2 ) =

( −2 −1)2 + (3 − (−1 ) )

2

= 9 + 16 = 25 = 5

4 Δy 3− ( −1) 4 = = =− Δx −2 −1 −3 3

c.

slope =

d.

For each run of 3, there is a rise of −4.

3. P1 = ( 4, −4 ) and P2 = ( 4, 8 ) a.

d ( P1 , P2 ) =

( 4 − 4)

+ (8 − ( −4 ) )

b. The coordinates of the midpoint are: x+x y+y  ( x, y) = 1 2 , 1 2  2 2  4+ 4 −4+ 8 8 4 = , = , = ( 4, 2 ) 2 2 2 2 slope =

(1, 5) (0, 4)

−1

x5

5. x-intercepts: −4, 0, 2 ; y-intercepts: −2, 0, 2 Intercepts: (−4, 0), (0, 0), (2, 0), (0, −2), (0, 2) 6. 2 x = 3 y 2 x-intercepts: y-intercepts: 2 2 x = 3(0) 2(0) = 3y 2 2x = 0 0 = y2 y=0 x=0 The only intercept is (0, 0). 2x = 3(− y) 2 2x = 3y 2 same Testy-axissymmetry: Let x = − x 2(− x) = 3y2 −2x = 3 y 2 different Testoriginsymmetry: Let x = − x and y = − y . 2(− x) = 3(− y) 2 −2x = 3 y 2 different Therefore, the graph will have x-axis symmetry. 7. x 2 +4 y 2 =16 x-intercepts:

2

= 0 + 144 = 144 = 12

c.

−5

(2, 8)

Testx-axissymmetry: Let y = − y

b. The coordinates of the midpoint are: x+x y+y  ( x, y) = 1 2 , 1 2  2 2 1 + ( −2 ) −1 + 3   = , 2 2  −1 2 1  = , = − ,1 2 2 2 

2

9

(−2, 8)

2

Δy 8− ( −4 ) 12 = = , undefined Δx 4−4 0

x 2 +4 ( 0 ) =16 2

y-intercepts:

( 0 )2 +4 y 2 =16

x 2 = 16 x = ±4

4 y 2 = 16 y2 = 4

y = ±2 The intercepts are (−4, 0), (4, 0), (0, −2), and (0, 2). Testx-axissymmetry: Let y = − y x 2 + 4( − y ) =16 x 2 + 4 y 2 =16 same 2

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Chapter 2: Graphs d.

Section 2.5: Variation

An undefined slope means the points lie on a vertical line. There is no change in x.

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Chapter 2 Review Exercises

Chapter 2: Graphs

Testy-axissymmetry: Let x = − x

(−x)

2

Testy-axissymmetry: Let x = − x y = (− x)3 − (− x)

+ 4 y =16 2

y = − x3 + x

x + 4 y2 =16 same 2

Testoriginsymmetry: Let x = − x and y = − y .

Testoriginsymmetry: Let x = − x and y = − y .

(−x)

2

− y = (− x)3 − (− x) − y = − x3 + x y = x3 − x same

+ 4 ( − y ) =16 2

x 2 +4 y 2 =16

same

Therefore, the graph will have x-axis, y-axis, and origin symmetry.

Therefore, the graph will have origin symmetry. 10. x 2 + x + y2 + 2 y = 0

8. y = x 4 +2 x 2 +1 x-intercepts: 0 = x 4 +2 x 2 +1

(

)(

x-intercepts: x 2 + x + (0) 2 + 2(0) = 0 x2 + x = 0 x( x + 1) = 0 x = 0, x = −1

y-intercepts: y = (0) 4 +2(0) 2 +1

)

=1

0 = x2 + 1 x 2 + 1 x +1 = 0 2

y-intercepts: (0) 2 + 0 + y 2 + 2 y = 0 y2 + 2y = 0 y( y + 2) = 0 y = 0, y = −2

x 2 = −1 no real solutions The only intercept is (0, 1). Testx-axissymmetry: Let y = − y

The intercepts are (−1, 0), (0, 0), and (0, −2).

− y = x 4 + 2x 2 + 1 y = − x 4 − 2x 2 − 1 different

Testx-axissymmetry: Let y = − y x 2 + x + (− y) 2 + 2(− y) = 0

Testy-axissymmetry: Let x = − x

x 2 + x + y 2 − 2 y = 0 different Testy-axissymmetry: Let x = − x 2 2 (− x) + (− x) + y + 2 y = 0 2 2 x − x + y + 2 y = 0 different Testoriginsymmetry: Let x = − x and y = − y .

y = ( − x ) + 2( − x ) + 1 4

2

y = x 4 + 2x 2 + 1

same

Testoriginsymmetry: Let x = − x and y = − y . − y = ( − x ) + 2( − x ) + 1 4

2

− y = x 4 + 2x 2 + 1 y = − x 4 − 2x 2 − 1

(− x) 2 + (− x) + (− y) 2 + 2(− y) = 0 x 2 − x + y 2 − 2 y = 0 different The graph has none of the indicated symmetries.

different

Therefore, the graph will have y-axis symmetry. 9. y = x3 − x

11.

x-intercepts: 0 = x3 − x

(

)

0 = x x −1 2

different

2 ( x − ( −2 ) ) + ( y − 3)2

y-intercepts: y = (0)3 − 0 =0

0 = x ( x + 1) ( x − 1)

( x − h) 2 + ( y − k) 2 = r 2

( x + 2) 12.

2

= 42

+ ( y − 3 ) = 16 2

( x − h) 2 + ( y − k ) 2 = r 2

( x − ( −1) ) + ( y − ( −2 ) )

x = 0, x = −1, x = 1

2

The intercepts are (−1, 0), (0, 0), and (1, 0). Testx-axissymmetry: Let y = − y

2

= 12

( x + 1)2 + ( y + 2 )2 = 1

− y = x3 − x y = − x 3 + x different

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Chapter 2 Review Exercises

Chapter 2: Graphs

13. x 2 + ( y − 1) = 4

x-intercepts: ( x − 1) + ( 0 + 2 ) = 32

2

2

2

( x −1) + 4 = 9 2 ( x −1) = 5

x 2 + ( y − 1) = 2 2 Center: (0,1); Radius = 2

2

2

x −1 = ± 5 x = 1± 5 y-intercepts: ( 0 − 1) + ( y + 2 ) = 32 2

2

1 + ( y + 2) = 9 2

( y + 2)

=8

2

y+2= ± 8 x-intercepts: x 2 + ( 0 −1) = 4

y + 2 = ±2 2

2

x2 + 1 = 4

y = −2 ± 2 2

(

) (

)

The intercepts are 1 − 5, 0 , 1 + 5, 0 ,

x2 = 3

( 0, −2 − 2 2 ) , and ( 0, −2 + 2 2 ) .

x=± 3 y-intercepts: 0 + ( y − 1) = 4 2

2

y −1 = ±2 y = 1± 2 y = 3 or y = −1

(

)(

The intercepts are − 3, 0 , and ( 0, 3) .

3x 2 + 3 y 2 − 6x + 12 y = 0

15.

( y − 1) 2 = 4

x 2 + y 2 − 2x + 4 y = 0

)

3, 0 , ( 0, −1) ,

(x

x 2 − 2x + y 2 + 4 y = 0 2

) (

)

− 2x + 1 + y 2 + 4 y + 4 = 1 + 4

( x −1)

2

+ ( y + 2) = 2

Center: (1, –2) Radius =

( 5)

2

5

x + y − 2x + 4 y − 4 = 0 2

14.

(x

2

x 2 − 2 x + y2 + 4 y = 4 2

) (

)

− 2x + 1 + y 2 + 4 y + 4 = 4 + 1 + 4

( x −1)

+ ( y + 2 ) = 32 Center: (1, –2) Radius = 3 2

2

x-intercepts: ( x − 1) + ( 0 + 2 ) = 2

2

( 5)

2

( x − 1)2 + 4 = 5 ( x −1)2 = 1 x −1 = ±1 x = 1±1 x = 2 or x = 0

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Chapter 2 Review Exercises

Chapter 2: Graphs

y-intercepts: ( 0 − 1) + ( y + 2 ) = 2

2

( 5)

2

1 + ( y + 2) = 5 2

( y + 2)

2

=4

y + 2 = ±2 y = −2 ± 2 y = 0 or y = −4 The intercepts are ( 0, 0 ) , ( 2, 0 ) , and ( 0, −4 ) . 16. Slope = –2; containing (3,–1) y − y1 = m ( x − x1 ) y − (−1) = −2 ( x − 3 ) y + 1 = −2 x + 6 y = −2 x + 5 or 2 x + y = 5 17. vertical; containing (–3,4) Vertical lines have equations of the form x = a, where a is the x-intercept. Now, a vertical line containing the point (–3, 4) must have an x-intercept of –3, so the equation of the line is x = −3. The equation does not have a slopeintercept form. 18. y-intercept = –2; containing (5,–3) Points are (5,–3) and (0,–2) − 2− (−3) 1 1 m= = =− 0−5 −5 5 y = mx + b 1 y = − x − 2 or x + 5 y = −10 5

20. Parallel to 2 x − 3 y = −4 2x − 3y = − 4 −3 y = −2 x − 4 −3 y −2x − 4 = −3 −3 2 4 y = x+ 3 3 2 Slope = ; containing (–5,3) 3 y − y1 = m ( x − x1 ) 2 ( x − (−5) ) 3 2 y − 3 = ( x + 5) 3 2 10 y−3= x+ 3 3 2 19 y = x+ or 2 x − 3 y = −19 3 3

y−3=

21. Perpendicular to x + y = 2 x+ y =2 y = −x + 2 The slope of this line is −1 , so the slope of a line perpendicular to it is 1. Slope = 1; containing (4,–3) y − y1 = m(x − x1 ) y − (−3) = 1( x − 4) y+3= x−4 y = x − 7 or x − y = 7

19. Containing the points (3,–4) and (2, 1) 1− (−4) 5 = −5 m= = 2−3 −1 y − y1 = m ( x − x1 )

y − (− 4) = −5 ( x − 3 ) y + 4 = −5x + 15 y = −5x + 11 or 5x + y = 11

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Chapter 2 Review Exercises

Chapter 2: Graphs

22. 4 x − 5 y = − 20 −5 y = −4 x − 20 y= slope =

4 x+4 5

24. 2x − 3 y = 12 x-intercept: 2x − 3(0) = 12 2 x = 12

y-intercept: 2(0) − 3 y = 12 −3 y = 12

x=6 y = −4 The intercepts are ( 6, 0 ) and ( 0, −4 ) .

4 ; y-intercept = 4 5

x-intercept: Let y = 0. 4 x − 5(0) = − 20 4 x = − 20 x = −5

25.

23.

1 1 1 x− y = − 2 3 6 1 1 1 − y = − x− 3 2 6 3 1 y = x+ 2 2 3 1 slope = ; y-intercept = 2 2 x-intercept: Let y = 0. 1 1 1 x − (0) = − 2 3 6 1 1 x=− 2 6 1 x=− 3

1 1 x+ y = 2 2 3 x-intercept: y-intercept: 1 1 1 1 x + (0) = 2 (0) + y = 2 2 3 2 3 1 1 x=2 y=2 2 3 x=4 y=6 The intercepts are ( 4, 0 ) and ( 0, 6 ) .

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Chapter 2 Review Exercises

Chapter 2: Graphs

30. Given the points A = (− 2, 0), B = (− 4, 4), and

26. y = x3

C = (8, 5). a.

Find the distance between each pair of points. d ( A, B ) = (− 4 − (− 2)) 2 + (4 − 0) 2 = 4 + 16 = 20 = 2 5 2 2 d ( B, C ) = (8 − (− 4)) + (5 − 4)

= 144 + 1 = 145 2 2 d ( A, C ) = (8 − (− 2)) + (5 − 0)

= 100 + 25

27. y = x

= 125 = 5 5 d ( A, B )

2

+ d ( A, C )

( 20) + ( 2

125

2

= d ( B, C )

) = ( 145) 2

2

2

20 + 125 = 145 145 = 145 The Pythagorean Theorem is satisfied, so this is a right triangle. 28. slope =

2 , containing the point (1,2) 3

b.

Find the slopes: m AB mBC

4− 0 4 = −2 = = − 4 − (− 2) − 2 5−4 1 = = 8 − ( − 4 ) 12

m AC =

5− 0 5 1 = = 8 − ( − 2 ) 10 2

1 = −1 , the sides AB 2 and AC are perpendicular and the triangle is a right triangle. Since mAB ⋅ m AC = − 2 ⋅

29. Find the distance between each pair of points. 2 2 d A, B = (1 − 3) + (1− 4) =

4 + 9 = 13

d B,C = (− 2 −1) 2 + (3 − 1) 2 = d A,C = (− 2 − 3) 2 + (3 − 4) 2 =

9 + 4 = 13 25 + 1 = 26

Since AB = BC, triangle ABC is isosceles.

31. Endpoints of the diameter are (–3, 2) and (5,–6). The center is at the midpoint of the diameter: −3+ 5 2+ ( − 6 )  , Center:  = (1, − 2 ) 2 2 Radius: r = =

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=

Chapter 2 Review Exercises

Chapter 2: Graphs (1 − (−3)) 2 + (− 2 − 2) 2 16 + 16 32 = 4 2

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Chapter 2 Review Exercises

Chapter 2: Graphs

Chapter 2 Test Equation:

( x − 1)2 + ( y + 2 ) 2 = ( 4 ( x − 1)

2

2

)

2

+ ( y + 2 ) = 32

= 6 2 + ( −4 )

2

= 52 = 2 13 2. The coordinates of the midpoint are: x+x y+y  ( x, y) = 1 2 , 1 2 2 2

p = kB

−1 + 5 3+ (−1) , 2 2  4 2 = ,  2 2  = ( 2, 1)

=

854 = k (130, 000 ) k=

854 427 = 130, 000 65, 000

Therefore, we have the equation p =

34. w =

3. a.

427 (165, 000 ) = $1083.92 . 65, 000 k

y −y 2

1

2

1

x −x

=

−1 − 3 5 − (−1)

=

−4

=−

6

k 3960 2

(

)

k = ( 200 ) 3960 2 = 3,136, 320, 000 Therefore, we have the equation 3,136, 320, 000 w= . d2 If d = 3960 + 1 = 3961miles, then 3,136, 320, 000 w= ≈ 199.9 pounds. 39612 35.

H = ksd 135 = k (7.5)(40) 135 = 300k k = 0.45 So, we have the equation H = 0.45sd. If s = 12 and d = 35, then H = 0.45 (12)( 35) = 189 BTU

2 3

b. If x increases by 3 units, y will decrease by 2 units. 4. y = x 2 − 9

d2

200 =

427 B. 65, 000

m=

If B = 165, 000 , then p=

2

= 36 + 16

1− 5 = −1 6−2 −1 − 5 slope of AC = = −1 8−2 Therefore, the points lie on a line.

32. slope of AB =

33.

( 5 − (−1) )2 + ( −1 − 3)2

1. d (P1 , P2 ) =

5. y 2 = x y 5 (1, 1)

(4, 2)

(9, 3) y2 = x 10 x

(0, 0) (1,−1)

(4,−2)

−5

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(9,−3)

Chapter 2 Test

Chapter 2: Graphs

6. x 2 + y = 9 x-intercepts: x2 + 0 = 9

9. y-intercept: (0) 2 + y = 9 y=9

x2 = 9

x = ±3 The intercepts are ( −3, 0 ) , ( 3, 0 ) , and ( 0, 9 ) .

x 2 + y 2 + 4x − 2 y − 4 = 0 x 2 + 4x + y 2 − 2 y = 4 2 ( x + 4x + 4) + ( y 2 − 2 y + 1) = 4 + 4 + 1 ( x + 2) 2 + ( y −1) 2 = 32 Center: (–2, 1); Radius = 3 y 5

Testx-axissymmetry: Let y = − y x + (− y) = 9 x 2 − y = 9 different 2

(−2, 1) −5

Testy-axissymmetry: Let x = − x

5 x

( − x )2 + y = 9 x 2 + y = 9 same

−5

Testoriginsymmetry: Let x = − x and y = − y

10. 2 x + 3 y = 6

( − x )2 + ( − y ) = 9

3 y = −2 x + 6

x − y = 9 different 2

2 y = − x+2 3

Therefore, the graph will have y-axis symmetry. 7. Slope = −2 ; containing (3, −4)

Parallelline Any line parallel to 2 x + 3 y = 6 has slope 2 m = − . The line contains (1, −1) : 3 y − y1 = m(x − x1 )

y − y1 = m(x − x1 ) y − (−4) = −2( x − 3) y + 4 = −2 x + 6 y = −2 x + 2

2 y − (−1) = − ( x − 1) 3 2 2 y +1 = − x + 3 3 2 1 y = − x− 3 3 Perpendicularline Any line perpendicular to 2 x + 3 y = 6 has slope 8. ( x − h) 2 + ( y − k ) 2 = r 2

( x − 4 )2 + ( y − (−3) )2 ( x − 4 ) 2 + ( y + 3)2 General form:

3 . The line contains (0, 3) : 2 y − y1 = m(x − x1 ) 3 y − 3 = ( x − 0) 2 3 y−3= x 2 3 y = x+3 2

m=

= 52 = 25

( x − 4)

2

+ ( y + 3) = 25 2

x 2 − 8x + 16 + y 2 + 6 y + 9 = 25 x 2 + y2 − 8x + 6 y = 0

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Chapter 2 Test

Chapter 2: Graphs

11. Let R = the resistance, l = length, and r = radius. l Then R = k ⋅ 2 . Now, R = 10 ohms, when r l = 50 feet and r = 6 ×10−3 inch, so 50 10 = k ⋅ 2 6 ×10−3

(

( 6 ×10 k = 10 ⋅

4. x 2 − 2x − 2 = 0 x=

2± 4+ 8 2 2± 12 = 2 2± 2 3 = 2 = 1± 3

) )

−3 2

)

)

(

{

5. x 2 + 2x + 5 = 0

)

x=

−2 ± 22 − 4()(1 ) 5 2 (1)

−2± 4− 20 2 −2± −16 = 2 No real solutions =

Chapter 2 Cumulative Review 1. 3x − 5 = 0 3x = 5 5 x= 3

(

The solution set is . 3 

2x + 1

x − x − 12 = 0

( x − 4 ) ( x + 3) = 0

Check:

The solution set is {−3, 4} .

= 32

2(4) + 1 = 3?

9 = 3? 3 = 3 True The solution set is {4} .

2x 2 − 5x − 3 = 0 ( 2x + 1) ( x − 3) = 0 7.

1 or x = 3 2

1  The solution set is − ,3 . 2

2

2x = 8 x=4

2

x=−

)

2x +1 = 9

x = 4 or x = −3

3.

2x + 1 = 3

6. 5

2.



}

The solution set is 1 − 3, 1 + 3 .

If l = 100 feet and r = 7 ×10−3 inch, then 100 R = 7.2 ×10−6 ≈ 14.69 ohms. 2 7 ×10−3

(

( −2 )2 − 4 (1) ( −2 ) 2 (1)

=

= 7.2 ×10 −6 50 Therefore, we have the equation l R = 7.2 ×10−6 2 . r

(

− ( −2 ) ±

x−2 =1 x − 2 = 1 or x − 2 = −1 x=3 x =1 The solution set is {1, 3} .

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Chapter 2 Cumulative Review

Chapter 2: Graphs

8.

(

12. −1 < x + 4 < 5 −5 < x < 1 { x − 5 < x < 1} or ( −5,1)

x 2 + 4x = 2

x2 + 4x ) = 2 2 x 2 + 4x = 4 x 2 + 4x − 4 = 0 2

−4 ± 4 2 − 4(1)(−4) −4 ± 16 + 16 = 2(1) 2 −4± 32 −4± 4 2 = = = −2 ± 2 2 2 2

x=

x−2 ≤1

13.

−1 ≤ x − 2 ≤ 1 1≤ x ≤ 3 { x 1 ≤ x ≤ 3} or [1, 3]

Check x = −2 + 2 2 :

( −2 + 2 2 )2 + 4 ( −2 + 2 2 ) = 2? 4 − 8 2 + 8 − 8 + 8 2 = 2? 4 = 2 True Check x = −2 − 2 2 :

( −2 − 2 2 )2 + 4( −2 − 2 2 ) = 2?

14.

2+ x >3 2 + x < −3 or 2 + x > 3 x < − 5 or x >1

{x

x < −5 or x > 1} or ( −∞, − 5 ) ∪ (1, ∞ )

4 + 8 2 + 8 − 8 − 8 2 = 2?

{

4 = 2 True

}

The solution set is −2 − 2 2 , −2 + 2 2 .

15. d ( P, Q ) =

9. x 2 = −9

=

x = ± −9

Midpoint =

( −2 )2 − 4 (1) ( 5 ) 2 (1)

=

2 ± 4 − 20 2

2± −16 2± 4i = = = 1 ± 2i 2 2 The solution set is {1 − 2i, 1 + 2i} . 11. 2x − 3 ≤ 7

a.

b.

x≤5

x ≤ 5} or ( −∞, 5]

3 1 −1 + 4 3+ ( −2 )  , = , 2 2 2 2

16. y = x3 − 3x + 1

2 x ≤ 10

{x

( −5)2 + ( 5 )2

= 50 = 5 2

10. x 2 − 2x + 5 = 0 x=

2

= 25 + 25

x = ±3i The solution set is {−3i, 3i} .

− ( −2 ) ±

( −1 − 4 ) 2 + ( 3 − ( −2 ) )

c.

( −2, −1) : ( −2 )3 − ( 3) ( −2 ) + 1 = −8 + 6 + 1 = −1 ( −2, −1) is on the graph. ( 2, 3) : ( 2 )3 − ( 3 ) ( 2 ) + 1 = 8 − 6 + 1 = 3 ( 2, 3) is on the graph. ( 3,1) : ( 3)3 − ( 3) ( 3) + 1 = 27 − 9 + 1 = 19 ≠ 1 ( 3,1) is not on the graph.

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Chapter 2 Cumulative Review

Chapter 2: Graphs

17. y = x3

20.

x 2 + y 2 − 4x + 8 y − 5 = 0 x 2 − 4x + y 2 + 8 y = 5 ( x 2 − 4x + 4) + ( y 2 + 8 y + 16) = 5 + 4 + 16 ( x − 2) 2 + ( y + 4) 2 = 25 ( x − 2) 2 + ( y + 4) 2 = 52 Center: (2,–4); Radius = 5

18. The points (–1,4) and (2,–2) are on the line. −6 −2− 4 = = −2 Slope = 2 − (−1) 3 y − y1 = m(x − x1 ) y − 4 = −2 ( x − ( −1) ) y − 4 = −2 ( x + 1)

y = −2x − 2 + 4 y = −2x + 2 19. Perpendicular to y = 2 x + 1 ; Contains ( 3, 5 ) 1 Slope of perpendicular = − 2 y − y1 = m(x − x1 )

Chapter 2 Project Internet Based Project

1 y − 5 = − ( x − 3) 2 1 3 y −5 = − x+ 2 2 1 13 y = − x+ 2 2

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Chapter 2 Cumulative Review

Chapter 2: Graphs

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