92
CHAPTER 2
C O O R D I N A T E S AND G R A P H S
EXAMPLE 10 • A Circle That Has All Three Types of Symmetry
Test the equation of the circle x2 + y2 = 4 for symmetry.
SOLUTION
The equation x2 + y2 = 4 remains unchanged when x is replaced by — x and y is
replaced by — y, since (— jc)2 = x2 and (— y}2 = y2, so the circle exhibits all three
types of symmetry. It is symmetric with respect to the jc-axis, the y-axis, and the
origin, as shown in Figure 14.
FIGURE 14
2.2
EXERCISES
1-6 • Determine whether the given points are on the graph of
the equation.
1. y = 2x + 3;
(0, 0), (|, 4), (I, 4)
15-40 • Make a table of values and isketch the graph of the
equation. Find x- and j-intercepts
^-intercepts and test for symmetry.
15. y = x
2. y = V* + 1; (1,0), (0, 1), (3,2)
17. y = x- 1
3. 2 y - ; t + 1 = 0 ;
19. 3jc - y = 5
4. y(x2 + ! ) = !;
(0, 0), (1, 0), (-1, -1)
(1, 1), (l, £), (-1, £)
5. x2 + xy + y2 = 4; (0, -2), (1, -2), (2, -2)
16. v
21. y=l-x*
23. 4>> = x2
7-14 • Find the x- and y-intercepts of the graph of the
equation.
7. y = x - 3
8. y = x2 - 5x + 6
9. y = x2 - 9
10. y - 2xy + 2x =
11. x2+y2 = 4
U. y = Jx+ ]
13. xy = 5
U. x2 - xy + y= I
40. .y = 16 - x4
SECTION 2.2
GRAPHS OF EQUATIONS
93
41-46 • Test the equation for symmetry.
59-60 • Find the equation of the circle shown in the figure.
41. y = jc4 + x2
42. x = y4 - y2
59.
43. x2y2 + xy = I
44. jc4/ + x2y2 = 1
45. y =
yi
60.
46. y = x2
\0x
47-50 • Complete the graph using the given symmetry
property.
47. Symmetric with respect
2
x
48. Symmetric with respect
61-68 • Show that the equation represents a circle, and find
the center and radius of the circje.
61. x2 + y2 - 2x + 4y + I = 0
62. x2 + y2 - 2x - 2y = 2
63. x2 + y2 - 4x + \0y + 1 3 = 0
64. x2 + y2 + 6y + 2 = 0
65. x2 + y2 + x = 0
49. Symmetric with respect
to the origin
50. Symmetric with respect
to the origin
66. x2 + y2 + 2x + y + 1 = 0
67. x2 + y2 - {x + {y = |
68. x2 + y2 + {x + 2y + ^ = 0
69-72 • Sketch the graph of the equation.
69. x2 + y2 + 4x - 10y = 21
70. 4x2 + 4y2 + 2x = 0
71. x2 + y2 + bx - \2y + 45 = 0
72. x2 + y2 - 16* + I2y + 200 = 0
51-58 • Find an equation of the circle that satisfies the given
conditions.
73-76 • Sketch the region given by the set.
51. Center (2, -1);
73. { ( x , y ) \ x 2 + y2^ 1)
52. Center (-1, -4);
radius 3
radius 8
53. Center at the origin;
54. Center (-1, 5);
passes through (4, 7)
75. { ( x , y ) \ ^x2 + y2<9\. {(x, y) \x < x2 + y2 ^ 4)
passes through (—4, —6)
55. Endpoints of a diameter are P(—l, 1) and Q(5, 5)
56. Endpoints of a diameter are P(—\, 3) and Q(l, -5)
57. Center (7, —3);
74. {(x,y) \2 + r >4)
tangent to the jc-axis
58. Circle lies in the first quadrant, tangent to both x- and
y-axes; radius 5
77. Find the area of the region that lies outside the circle
x2 + y2 = 4 but inside the circle
X2 + y2 - 4y - 12 = 0
78. Sketch the region in the coordinate plane that satisfies both
the inequalities x2 + y2 = 9 and y ~s* \ . What is the area
of this region?
,