Objective #: 3 Name: Date: Period:Practice Assignment: Intercepts and Symmetry

SECTION 1.2 Graphs of Equations in Two Variables; Intercepts; Symmetry 17

Figure 25 infer that if x is a large and positive number, then y : 1 t, a positive number close"t

to 0. we also infer that if x is a positive number close to 0, then y : :is a large

and positive number. Armed with this information, we can graph the equation.

Figure 25 illustrates some of these points and the graph of y : + observe how

the absence of intercepts and the existence of symmetry witir respect to the originwere utilized.

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El cCItulMENT .,Kefer

t'o Example 3 in Appendix b, gecrion b.b, for the graph ot y : ! ueinq al: . .4 qra?ntnq uattf iy. n o

1,4 Assess Ymwr {'$mderstar*d$mq

'Arg You Prepqred?' Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.1. Solvetheequat lpn2(x + 3) - r : -7.(pp.Aa3-Aal z, Solvetheequat ion x2 - g:0.(pp.A43-A47)

Concepts and .Vocabulary3. The points, if any, at which a graph crosses or touches the co-

ordinate axes are called _.

4. The x-intercepts of the graph of an equation are thosex-values for which

5. If for every point (x, y) on the graph of an equation the point(- *, y) is also on the graph, then the graph is symmetriCwithrespect to the _.

6. If the graph of an equation is symmetric with respect to the y-axis and -4 is an x-intercept of this graph, thenis also an x-intercept.

Skil l Building

7. If the graph of an equation is symmetric with respect to theorigin and (3, -4) is a point on the graph, then _is also a point on the graph.

8. True or False To find the y-intercepts of the graph of anequation, let x = 0 and solve for y.

9. True or False The y-coordinate of a point at which the graphcrosses or touches the -r-axis is an x-intercept.

10. True or False' If a graph is symmetric with respect to thex-axis, then it cannot be symmetric with respect to they-axis.

(+,2)

. In Problems 1l-16, determine which of the given points are on the graph of the equation.

\ t l . Equat ion: y: x4 - tG 12. Equat ioni v : x3 -2\ ,G

Poinrs: (0,0); (1, 1) ; ( -1,0)

14.Equat ion: y3:x+IPoints: (1,2); (0,1); ( -1, 0)

Points: (0,0); (1,1); (1, -1)

L5. Equation: x2 + y2 : 4Points: (0,2); (-2,4; (X6, tb)

13. Equation: y2 : x2 + 9Points: (0,3); (3,0); ( -3,0)

16. Equation: xz + 4y2 : 4

Points: (0. 1): (2..f , (, l)

20.y:3x-9

24.Y:-x2+I

28,4x2 + y:4

In Problems 17-28, ftnd the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

I7.Y:1a2

\zr.y=x2- ' ! ,

25.2x+3y:6

18.y:x-6

22.Y=x2-g

26,5x+2y:tO

19.y:2x+8

23.Y:-x2+4

27.9x2 + 4y :36

In Problems 29-38, plot each point. Then plot the point that is symmetric to it with respect to (a) the x-axis; (b) the y-axis; (c) the origin.

\er. 1:,+y 30. (s,3) 3r. (-2,1) 32. (4,-2) 33. (s, _2)

34. (-1, -1) 3s. (-3, -4) 36. (4,0) 37. (0, -3) 38. (-3,0)

SECTION 1.2 Graphs of Equations in Two Variables; Intercepts; Symmetry 17

Figure 25 infer that if x is a large and positive number, then y : 1 t, a positive number close"t

to 0. we also infer that if x is a positive number close to 0, then y : :is a large

and positive number. Armed with this information, we can graph the equation.

Figure 25 illustrates some of these points and the graph of y : + observe how

the absence of intercepts and the existence of symmetry witir respect to the originwere utilized.

- __ff

El cCItulMENT .,Kefer

t'o Example 3 in Appendix b, gecrion b.b, for the graph ot y : ! ueinq al: . .4 qra?ntnq uattf iy. n o

1,4 Assess Ymwr {'$mderstar*d$mq

'Arg You Prepqred?' Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.1. Solvetheequat lpn2(x + 3) - r : -7.(pp.Aa3-Aal z, Solvetheequat ion x2 - g:0.(pp.A43-A47)

Concepts and .Vocabulary3. The points, if any, at which a graph crosses or touches the co-

ordinate axes are called _.

4. The x-intercepts of the graph of an equation are thosex-values for which

5. If for every point (x, y) on the graph of an equation the point(- *, y) is also on the graph, then the graph is symmetriCwithrespect to the _.

6. If the graph of an equation is symmetric with respect to the y-axis and -4 is an x-intercept of this graph, thenis also an x-intercept.

Skil l Building

7. If the graph of an equation is symmetric with respect to theorigin and (3, -4) is a point on the graph, then _is also a point on the graph.

8. True or False To find the y-intercepts of the graph of anequation, let x = 0 and solve for y.

9. True or False The y-coordinate of a point at which the graphcrosses or touches the -r-axis is an x-intercept.

10. True or False' If a graph is symmetric with respect to thex-axis, then it cannot be symmetric with respect to they-axis.

(+,2)

. In Problems 1l-16, determine which of the given points are on the graph of the equation.\ t l . Equat ion: y: x4 - tG 12. Equat ioni v : x3 -2\ ,G

Poinrs: (0,0); (1, 1) ; ( -1,0)

14.Equat ion: y3:x+IPoints: (1,2); (0,1); ( -1, 0)

Points: (0,0); (1,1); (1, -1)

L5. Equation: x2 + y2 : 4Points: (0,2); (-2,4; (X6, tb)

13. Equation: y2 : x2 + 9Points: (0,3); (3,0); ( -3,0)

16. Equation: xz + 4y2 : 4

Points: (0. 1): (2..f , (, l)

20.y:3x-9

24.Y:-x2+I

28,4x2 + y:4

In Problems 17-28, ftnd the intercepts and graph each equation by plotting points. Be sure to label the intercepts.I7.Y:1a2

\zr.y=x2- ' ! ,

25.2x+3y:6

18.y:x-6

22.Y=x2-g

26,5x+2y:tO

19.y:2x+8

23.Y:-x2+4

27.9x2 + 4y :36

In Problems 29-38, plot each point. Then plot the point that is symmetric to it with respect to (a) the x-axis; (b) the y-axis; (c) the origin.

\er. 1:,+y 30. (s,3) 3r. (-2,1) 32. (4,-2) 33. (s, _2)

34. (-1, -1) 3s. (-3, -4) 36. (4,0) 37. (0, -3) 38. (-3,0)

18 CHAPTER 1 Graons

In Problems 39-50, the graph of an equation is given. (a) Find the intercepts. (b) Indicate whether the graph is symmetric with respectto the x-axis, the y-axis, or the origin.

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In Problems 51-54, draw a complete graph so that it has the type of symmetry indicated.51. y-axis 52. x-axis 53. Origin 54. y-axis

In Problems 55-70, list the intercepts and test for symmetry.55.Y2:ra4

\sp. " '+ y - 9: o

63.Y:x3-27

67. Y : ^t*- x"*9

56.y2:*a9

60.x2-y-4:0

64.Y:xa- l

"2-468.v: ' ;

s7. y -- tli

61. 9x2 + 4yz :36

65.Y:x2-3x-4

s8. y : rfG

62.4x2+y2:4

66.Y:x2+4

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174.y:-

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69. v: x2*g

In Problems 7l-74, draw a quick sketch of each equation.7L y: az 72.x:y2 73. y : \/i

75. If (3, b) is a point on the graph of y : 4* + 1, what is b?

77. If (a,4) is a point on the graph of y : *2 * 3-r, what is a?

76. If (-2, b) is a point on the graph of 2x -r 3y : Z, what is b?

78. If (a, -5) is a point on the graph of y = *2 I 6x,whatis a?

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18 CHAPTER 1 Graons

In Problems 39-50, the graph of an equation is given. (a) Find the intercepts. (b) Indicate whether the graph is symmetric with respectto the x-axis, the y-axis, or the origin.

\eo 41..

E]

ltfi 47" 4S" 49. 5lt"

-2 -2 _A

-3

In Problems 51-54, draw a complete graph so that it has the type of symmetry indicated.

51. y-axis 52. x-axis 53. Origin 54. y-axis

In Problems 55-70, list the intercepts and test for symmetry.

55.Y2:ra4

\sp. " '+ y - 9: o

63.Y:x3-27

67. Y : ^t*- x"*9

56.y2:*a9

60.x2-y-4:0

64.Y:xa- l

"2-468.v: ' ;

s7. y -- tli

61. 9x2 + 4yz :36

65.Y:x2-3x-4

s8. y : rfG

62.4x2+y2:4

66.Y:x2+4

-4t ' t70.v:n ' . 'LX-

174.y:-

a-L

69. v: x2*g

In Problems 7l-74, draw a quick sketch of each equation.7L y: az 72.x:y2 73. y : \/i

75. If (3, b) is a point on the graph of y : 4* + 1, what is b?

77. If (a,4) is a point on the graph of y : *2 * 3-r, what is a?

76. If (-2, b) is a point on the graph of 2x -r 3y : Z, what is b?

78. If (a, -5) is a point on the graph of y = *2 I 6x,whatis a?

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18 CHAPTER 1 Graons

In Problems 39-50, the graph of an equation is given. (a) Find the intercepts. (b) Indicate whether the graph is symmetric with respectto the x-axis, the y-axis, or the origin.

\eo 41..

E]

ltfi 47" 4S" 49. 5lt"

-2 -2 _A

-3

In Problems 51-54, draw a complete graph so that it has the type of symmetry indicated.51. y-axis 52. x-axis 53. Origin 54. y-axis

In Problems 55-70, list the intercepts and test for symmetry.55.Y2:ra4

\sp. " '+ y - 9: o

63.Y:x3-27

67. Y : ^t*- x"*9

56.y2:*a9

60.x2-y-4:0

64.Y:xa- l

"2-468.v: ' ;

s7. y -- tli

61. 9x2 + 4yz :36

65.Y:x2-3x-4

s8. y : rfG

62.4x2+y2:4

66.Y:x2+4

-4t ' t70.v:n ' . 'LX-

174.y:-

a-L

69. v: x2*g

In Problems 7l-74, draw a quick sketch of each equation.7L y: az 72.x:y2 73. y : \/i

75. If (3, b) is a point on the graph of y : 4* + 1, what is b?

77. If (a,4) is a point on the graph of y : *2 * 3-r, what is a?

76. If (-2, b) is a point on the graph of 2x -r 3y : Z, what is b?

78. If (a, -5) is a point on the graph of y = *2 I 6x,whatis a?

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18 CHAPTER 1 Graons

In Problems 39-50, the graph of an equation is given. (a) Find the intercepts. (b) Indicate whether the graph is symmetric with respectto the x-axis, the y-axis, or the origin.

\eo 41..

E]

ltfi 47" 4S" 49. 5lt"

-2 -2 _A

-3

In Problems 51-54, draw a complete graph so that it has the type of symmetry indicated.51. y-axis 52. x-axis 53. Origin 54. y-axis

In Problems 55-70, list the intercepts and test for symmetry.55.Y2:ra4

\sp. " '+ y - 9: o

63.Y:x3-27

67. Y : ^t*- x"*9

56.y2:*a9

60.x2-y-4:0

64.Y:xa- l

"2-468.v: ' ;

s7. y -- tli

61. 9x2 + 4yz :36

65.Y:x2-3x-4

s8. y : rfG

62.4x2+y2:4

66.Y:x2+4

-4t ' t70.v:n ' . 'LX-

174.y:-

a-L

69. v: x2*g

In Problems 7l-74, draw a quick sketch of each equation.7L y: az 72.x:y2 73. y : \/i

75. If (3, b) is a point on the graph of y : 4* + 1, what is b?

77. If (a,4) is a point on the graph of y : *2 * 3-r, what is a?

76. If (-2, b) is a point on the graph of 2x -r 3y : Z, what is b?

78. If (a, -5) is a point on the graph of y = *2 I 6x,whatis a?

r

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SECTION 1.3 Lines 19

Applications and Extensions79. Given that the point (1,2) is on the graph of an equation that

is symmetric with respect to the origin, what other point ison the graph?

80. If the graph of an equation is symmetric with respect to they-axis and 6 is an x-intercept of this graph, name anotherx-intercept.

81. If the graph of an equation is symmetric with respect to theorigin and -4 is an x-intercept of this graph, name anotherx-intercept.

82. If the graph of an equation is symmetric with respect to thex-axis and 2 is a y-intercept, name another y-intercept.

83. Microphones In studios and on stages, cardioid micro-phones are often preferred for the richness they add to voicesand for their ability to reduce the level of sound from thesides and rear of the microphone. Suppose one such cardiodpattern is given by the equation (x2 + y2 - *)z : *t + yt.(a) Find the intercepts of the graph of the equation.(b) Test for symmetry with respect to the x-axis, y-axis, and

origin.Source: www.notaviv a. com

Discussion and Writing l

84. Solar Energy The solar electric generating systems atKramer Junction, California, use parabolic troughs to heat aheat-transfer fluid to a high temperature. This fluid is usedto generate steam that drives a power conversion system toproduce electricity. For troughs 7.5 feet wide, an equation forthe cross-section is l6v2 : l20x - 225.

(a) Find the intercepts of the graph of the equation.(b) Test for symmetry with respect to the x-axis, y-axis, and

origin.Source: U.S. Department of Energy

In Problem 85, you may use a graphing utility, but it is not required./

85. (a) Graph y : Yx2,y : x,y : l *1, and y : ( t / i )2,not-ing which graphs are the same.

(b) Explain why the graphs of y : f *z atd, y : l-rl are the

(c) Explain why the graphs of y : x and y : (t/i)2 arenot the same. /

(d) Explain why the graphs of y : y *z and y : r are notthe same.

86. Explain what is meant by a complete graph.87. Draw a graph of an equation that contains two x-intercepts;

at one the graph crosses the x-axis, and at the other the graphtouches the x-axis.

'Are You Prepared?' Answers

88. Make up an equation with the intercepts (2,0), (4,0), and(0, 1). Compare your equation with a friend's equation. Com-ment on any similarities.

89. Draw a graph that contains the points (-2,-l), (0,I),(1,3), and (3,5). Compare your graph with those of otherstudents. Are most of the graphs almost straight lines? Howmany are "curved"? Discuss the various ways that thesepoints might be connected.

90. An equation is beingtested for symmetry with respect to thex-axis, the y-axis, and the origin. Explain why, if two of thesesymmetries are present, the remaining one must also bepresent.

1. { -6i r I -? ?l-. t " ' " '

OBJECTIVES 1z"?

4567I9

10

Calculate and Interpret the Slope of a Line (p.20)Graph Lines Given a Point and the Slope (p.22)Find the Equation of a Vertical Line (p.23)Use the Point-Slope Form of a Line; ldentify Horizontal Lines (p.24)Find the Equation of a Line Given Two Points (p.25)Write the Equation of a Line in Slope-lntercept Form (p.25)ldentify the Slope and y-lntercept of a Line from lts Equation (p. 26)Graph Lines Written in General Form Using Intercepts (p. 27)Find Equations of Parallel Lines (p.28)Find Equations of Perpendicular Lines (p.29)

18 CHAPTER 1 Graons

In Problems 39-50, the graph of an equation is given. (a) Find the intercepts. (b) Indicate whether the graph is symmetric with respectto the x-axis, the y-axis, or the origin.

\eo 41..

E]

ltfi 47" 4S" 49. 5lt"

-2 -2 _A

-3

In Problems 51-54, draw a complete graph so that it has the type of symmetry indicated.51. y-axis 52. x-axis 53. Origin 54. y-axis

In Problems 55-70, list the intercepts and test for symmetry.

55.Y2:ra4

\sp. " '+ y - 9: o

63.Y:x3-27

67. Y : ^t*- x"*9

56.y2:*a9

60.x2-y-4:0

64.Y:xa- l

"2-468.v: ' ;

s7. y -- tli

61. 9x2 + 4yz :36

65.Y:x2-3x-4

s8. y : rfG

62.4x2+y2:4

66.Y:x2+4

-4t ' t70.v:n ' . 'LX-

174.y:-

a-L

69. v: x2*g

In Problems 7l-74, draw a quick sketch of each equation.7L y: az 72.x:y2 73. y : \/i

75. If (3, b) is a point on the graph of y : 4* + 1, what is b?

77. If (a,4) is a point on the graph of y : *2 * 3-r, what is a?

76. If (-2, b) is a point on the graph of 2x -r 3y : Z, what is b?

78. If (a, -5) is a point on the graph of y = *2 I 6x,whatis a?

r

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