Graphs Of Equations; Circles

Chapter 1

7

Graphs

1.2 Graphs of Equations; Circles

1. add, 4 3. intercepts

5. (3,–4) 7. True

9. 11.

13. 15.

17.

19. (a) (–1, 0), (1, 0) (b) symmetric with respect to the x-axis,

y-axis, and origin

Chapter 1 Graphs

8

21. (a) -

p

2

,0

æ

è

ç

ö

ø

÷ ,

p

2

, 0

æ

è

ç

ö

ø

÷ , (0,1) (b) symmetric with respect to the y-axis

23. (a) (0, 0) (b) symmetric with respect to the x-axis

25. (a) (1, 0) (b) not symmetric with respect to x-axis,

y-axis, or origin

27. (a) (–1, 0), (1, 0), (0, –1) (b) symmetric with respect to the y-axis

29. (a) none (b) symmetric with respect to the origin

31. y = x 4 - x

0 = 04 - 0

0 = 0

1=14 - 1

1¹ 0

0 = (-1)4 - -1

0 ¹1- -1

(0, 0) is on the graph of the equation.

33. y 2 = x2 + 9

32 = 02 + 9

9 = 9

02 = 32 + 9

0 ¹18

02 = (-3)2 + 9

0 ¹18

(0, 3) is on the graph of the equation.

35. x 2 + y 2 = 4

02 + 22 = 4

4 = 4

(-2)2 + 22 = 4

8 ¹ 4

( 2)2

+ ( 2)2

= 4

4 = 4

(0, 2) and ( 2, 2) are on the graph of the equation.

37. x2 = y

y - intercept : Let x = 0, then 0 2 = yÞ y = 0 (0,0)

x - intercept : Let y = 0, then x2 = 0Þ x = 0 (0,0)

Test for symmetry:

x - axis : Replace y by - y : x2 = -y, which is not equivalent to x 2 = y.

y - axis : Replace x by - x : (-x)2 = y or x2 = y, which is equivalent to x 2 = y.

Origin : Replace x by - x and y by - y : (-x)2 = -y or x2 = -y,

which is not equivalent to x 2 = y.

Therefore, the graph is symmetric with respect to the y - axis .

39. y = 3x

y - intercept : Let x = 0, then y = 3× 0 = 0 (0,0)

x - intercept : Let y = 0, then 3 x = 0Þ x = 0 (0,0)

Section 1.2 Graphs of Equations; Circles

9

Test for symmetry:

x - axis : Replace y by - y : - y = 3x, which is not equivalent to y = 3x.

y - axis : Replace x by - x : y = 3(-x ) or y = -3x,

which is not equivalent to y = 3x.

Origin : Replace x by - x and y by - y : - y = 3(-x) or y = 3x,

which is equivalent to y = 3x.

Therefore, the graph is symmetric with respect to the origin.

41. x 2 + y - 9 = 0

y - intercept : Let x = 0, then 0 + y -9 = 0 Þ y = 9 (0,9)

x - intercept : Let y = 0, then x2 - 9 = 0Þ x = ±3 (-3,0),(3,0)

Test for symmetry:

x - axis : Replace y by - y : x2 + (-y)- 9 = 0 or x2 - y - 9 = 0,

which is not equivalent to x2 + y -9 = 0.

y - axis : Replace x by - x : (-x)2 + y - 9 = 0 or x2 + y -9 = 0,

which is equivalent to x 2 + y -9 = 0.

Origin : Replace x by - x and y by - y : (-x)2 + (-y)-9 = 0 or x 2 - y -9 = 0,

which is not equivalent to x 2 + y - 9 = 0.

Therefore, the graph is symmetric with respect to the y-axis.

43. 9 x2 + 4 y2 = 36

y - intercept : Let x = 0, then 4y2 = 36Þ y 2 = 9Þ y = ±3 (0,-3),(0,3)

x - intercept : Let y = 0, then 9x 2 = 36Þ x 2 = 4Þ x = ±2 (-2,0),(2,0)

Test for symmetry:

x - axis : Replace y by - y : 9x2 + 4(-y)2 = 36 or 9x 2 + 4y 2 = 36,

which is equivalent to 9x 2 + 4y 2 = 36.

y - axis : Replace x by - x : 9(-x )2 + 4y2 = 36 or 9x 2 + 4y 2 = 36,

which is equivalent to 9x 2 + 4y 2 = 36.

Origin : Replace x by - x and y by - y : 9(-x)2 + 4(-y)2 = 36 or 9x2 + 4y2 = 36,

which is equivalent to 9x 2 + 4y 2 = 36.

Therefore, the graph is symmetric with respect to the x-axis, the y-axis, and the origin.

45. y = x3 - 27

y - intercept : Let x = 0, then y = 03 - 27Þ y = -27 (0,-27)

x - intercept : Let y = 0, then 0 = x3 -27 Þ x 3 = 27 Þ x = 3 (3,0)

Chapter 1 Graphs

10

Test for symmetry:

x - axis : Replace y by - y : - y = x 3 -27, which is not equivalent to y = x 3 - 27.

y - axis : Replace x by - x : y = (-x)3 -27 or y = -x3 -27,

which is not equivalent to y = x 3 -27.

Origin : Replace x by - x and y by - y : - y = (-x)3 -27 or

y = x 3 + 27, which is not equivalent to y = x 3 -27.

Therefore, the graph is not symmetric with respect to the x-axis, the y-axis, or the origin.

47. y = x2 - 3x - 4

y - intercept : Let x = 0, then y = 02 - 3(0) - 4 Þ y = -4 (0,-4)

x - intercept : Let y = 0, then 0 = x 2 - 3x - 4 Þ(x - 4)(x +1) = 0Þ x = 4, x = -1 (4,0), (-1,0)

Test for symmetry:

x - axis : Replace y by - y : - y = x 2 - 3x - 4, which is not

equivalent to y = x 2 - 3x - 4.

y - axis : Replace x by - x : y = (-x)2 - 3(-x) - 4 or y = x 2 + 3x - 4,

which is not equivalent to y = x 2 - 3x - 4.

Origin : Replace x by - x and y by - y : - y = (-x)2 - 3(-x) - 4 or

y = -x 2 - 3x + 4, which is not equivalent to y = x 2 - 3x - 4.

Therefore, the graph is not symmetric with respect to the x-axis, the y-axis, or the origin.

49. y =

3x

x 2 + 9

y - intercept : Let x = 0, then y =

0

0 + 9

= 0 (0, 0)

x - intercept : Let y = 0, then 0 =

3x

x 2 + 9

Þ 3x = 0Þ x = 0 (0, 0)

Test for symmetry:

x - axis : Replace y by - y : - y =

3x

x2 + 9

, which is not

equivalent to y = 3x

x2 + 9

.

y - axis : Replace x by - x : y =

3(-x)

(-x)2 + 9

or y =

-3x

x2 + 9

,

which is not equivalent to y =

3x

x2 + 9

.

Origin : Replace x by - x and y by - y : - y =

-3x

(-x)2 + 9

or

y =

3x

x2 + 9

, which is equivalent to y =

3x

x 2 + 9

.

Therefore, the graph is symmetric with respect to the origin.

Section 1.2 Graphs of Equations; Circles

11

51. y = -x3

x 2 - 9

y - intercept : Let x = 0, then y =

0

-9

= 0 (0,0)

x - intercept : Let y = 0, then 0 =

-x 3

x2 -9

Þ-x 3 = 0 Þ x = 0 (0,0)

Test for symmetry:

x - axis : Replace y by - y : - y = -x 3

x 2 - 9

, which is not equivalent to y = -x3

x 2 - 9

.

y - axis : Replace x by - x : y =

-(-x)3

(-x)2 - 9

or y =

x 3

x 2 -9

,

which is not equivalent to y =

-x 3

x2 -9

.

Origin : Replace x by - x and y by - y : - y =

-(-x)3

(-x)2 -9

or

- y =

x 3

x2 - 9

, which is equivalent to y =

-x 3

x 2 -9

.

Therefore, the graph is symmetric with respect to the origin.

53. y = x3 55. y = x

57. y = 3x + 5

2 = 3a + 5Þ 3a = - 3Þ a = -1

59. 2x + 3y = 6

2a+ 3b = 6Þb = 2-

2

3

a

Chapter 1 Graphs

12

61. Center = (2, 1)

Radius = distance from (0, 1) to (2,1)

= (2 - 0)2 + (1-1)2

= 4 = 2

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

63. Center = midpoint of (1,2) and (4,2)

=

1+ 4

2

,

2 + 2

2

æ

è

ç

ö

ø

÷ =

5

2

,2

æ

è

ç

ö

ø

÷

Radius = distance from

5

2

,2

æ

è

ç

ö

ø

÷ to (4, 2)

= 4 -

5

2

æ

è

ç

ö

ø

÷

2

+ (2 -2)2

=

9

4

=

3

2

x -

5

2

æ

è

ç

ö

ø

÷

2

+ (y - 2)2 =

9

4

65. ( x - h)2 + (y - k)2 = r2

(x - 0)2 + (y - 0)2 = 22

x 2 + y2 = 4

General form: x 2 + y 2 - 4 = 0

67. ( x - h)2 + (y - k)2 = r2

(x -1)2 + (y - (-1))2 = 12

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

General form:

x 2 -2x +1+ y 2 + 2y +1=1

x 2 + y 2 - 2x + 2y +1= 0

69. ( x - h)2 + (y - k)2 = r2

(x - 0)2 + (y - 2)2 = 22

x2 + (y - 2)2 = 4

General form:

x2 + y2 - 4y + 4 = 4

x2 + y2 - 4y = 0

Section 1.2 Graphs of Equations; Circles

13

71. ( x - h)2 + (y - k)2 = r2

(x - 4)2 + (y - (-3))2 = 52

(x - 4)2 + (y + 3)2 = 25

General form:

x2 - 8x +16 + y2 + 6y + 9 = 25

x2 + y2 - 8x + 6y = 0

73. ( x - h)2 + ( y - k)2 = r 2

(x -(-3))2 + (y -(-6))2 = 62

(x + 3)2 + (y + 6)2 = 36

General form:

x 2 + 6x + 9+ y2 +12y + 36 = 36

x2 + y 2 + 6x +12 y + 9 = 0

75. ( x - h)2 + ( y - k)2 = r 2

(x - 0)2 + (y -(-3))2 = 32

x2 + (y + 3)2 = 9

General form:

x 2 + y 2 + 6 y + 9 = 9

x 2 + y2 + 6y = 0

77. x2 + y2 = 4

x 2 + y 2 = 22

(a) Center : (0,0); Radius = 2

(b)

(c) x-intercepts: y = 0

x 2 + 0 = 4

x = ±2

(-2,0),(2,0)

y-intercepts: x = 0

0 + y 2 = 4

y = ±2

(0,-2),(0,2)

Chapter 1 Graphs

14

79. 2(x - 3)2 + 2y 2 = 8

(x - 3)2 + y2 = 4

(x - 3)2 + y2 = 22

(a) Center: (3,0); Radius = 2

(b)

(c) x-intercepts: y = 0

(x - 3)2 + 0 = 4

(x - 3)2 = 4

x - 3 = ±2

x = 5, x =1

(1,0),(5,0)

y-intercepts: x = 0

9 + y 2 = 4

y 2 = -5

no solution Þno y - intercepts

81. x2 + y2 + 4x - 4y -1 = 0

x 2 + 4x + y 2 - 4y =1

(x2 + 4x + 4) + (y 2 - 4y + 4) =1+ 4 + 4

(x + 2)2 + (y -2)2 = 32

(a) Center: (–2,2); Radius = 3

(b)

(c) x-intercepts: y = 0

(x + 2)2 + 4 = 9

(x + 2)2 = 5

x + 2 = ± 5

x = 5 - 2, x = - 5 - 2

(- 5 - 2,0),( 5 - 2,0)

y-intercepts: x = 0

4 + (y -2)2 = 9

(y - 2)2 = 5

y -2 = ± 5

y = 5 + 2, y = - 5 + 2

(0,- 5 + 2),(0, 5 + 2)

Section 1.2 Graphs of Equations; Circles

15

83. x 2 + y 2 - 2x + 4 y - 4 = 0

x 2 -2x + y 2 + 4y = 4

(x2 -2x +1) + (y 2 + 4y + 4) = 4 +1+ 4

(x -1)2 + (y + 2)2 = 32

(a) Center: (1,–2); Radius = 3

(b)

(c) x-intercepts: y = 0

(x -1)2 + 4 = 9

(x -1)2 = 5

x +1= ± 5

x = 5 -1, x = - 5 -1

(- 5 -1,0),( 5 -1,0)

y-intercepts: x = 0

1+ (y + 2)2 = 9

(y + 2)2 = 8

y + 2 = ± 8

y = 8 - 2, y = - 8 - 2

(0,- 8 - 2),(0, 8 - 2)

85. x 2 + y 2 - x + 2y +1= 0

x 2 - x + y 2 + 2y = -1

x 2 - x +

1

4

æ

è

ç

ö

ø

÷ + (y 2 + 2y +1) = -1+

1

4

+1

x -

1

2

æ

è

ç

ö

ø

÷

2

+ (y +1)2 =

1

2

æ

è

ç

ö

ø

÷

2

(a) Center: 12

æ ,-1

è

ö

ø ; Radius = 12

(b)

(c) x-intercepts: y = 0

x -

1

2

æ

è

ç

ö

ø

÷

2

+1=

1

4

x -

1

2

æ

è

ç

ö

ø

÷

2

= -

3

4

no solution Þno x - intercepts

y-intercepts: x = 0

1

4

+ (y +1)2 =

1

4

(y +1)2 = 0

y +1= 0

y = -1

(0,-1)

Chapter 1 Graphs

16

87. 2 x2 + 2 y2 - 12x + 8y - 24 = 0

x 2 + y 2 - 6x + 4y =12

x 2 - 6x + y 2 + 4y =12

(x2 -6x + 9) + (y 2 + 4y + 4) =12 + 9 + 4

(x - 3)2 + (y + 2)2 = 52

(a) Center: (3,–2); Radius = 5

(b)

(c) x-intercepts: y = 0

(x - 3)2 + 4 = 25

(x - 3)2 = 21

x - 3 = ± 21

x = 21 + 3, x = - 21+ 3

(- 21 + 3,0),( 21 + 3,0)

y-intercepts: x = 0

9 + (y + 2)2 = 25

(y + 2)2 =16

y + 2 = ±4

y = 2, y = -6

(0,-6),(0,2)

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

r = (-3- 0)2 + (2- 0)2 = 9 + 4 = 13

Equation:

(x - 0)2 + (y - 0)2 = ( 13)2

x 2 + y 2 =13

91. Center at (2,3); tangent to the x-axis.

r = 3

Equation:

(x -2)2 + (y - 3)2 = 32

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

x 2 + y2 - 4x - 6y + 4 = 0

93. Endpoints of a diameter are (1,4) and (–3,2).

The center is at the midpoint of that diameter:

Center:

1+ (-3)

2

,

4 + 2

2

æ

è

ç

ö

ø

÷ = (-1,3)

Radius: r = (1- (-1))2 + (4 - 3)2 = 4 +1 = 5

Equation: (x - (-1))2 + (y - 3)2 = ( 5)2

x 2 + 2x +1+ y 2 - 6y + 9 = 5

x 2 + y 2 + 2x - 6y + 5 = 0

95. (c) 97. (b)

99. (b), (c), (e) and (g)

101. x 2 + y 2 + 2 x + 4 y - 4091= 0

x 2 + 2x + y2 + 4y - 4091= 0

x2 + 2x +1+ y 2 + 4y + 4 = 4091+ 5

(x +1)2 + (y + 2)2 = 4096

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.

Section 1.2 Graphs of Equations; Circles

17

The equation of the orbit is

(x +1)2 + (y + 2)2 = (64.6)2

x 2 + y 2 + 2x + 4y - 4168.16 = 0

103–105. Answers will vary

...