2. Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

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4. Contents Lesson 8-1Midpoint and Distance Formulas Lesson 8-2Parabolas Lesson 8-3Circles Lesson 8-4Ellipses Lesson 8-5Hyperbolas Lesson 8-6Conic Sections Lesson 8-7Solving Quadratic Systems

5. Lesson 1 Contents Example 1Find a Midpoint Example 2Find the Distance Between Two Points Example 3Find the Farthest Point

6. Example 1-1a Computers A graphing program draws a line segment on a computer screen so that its ends are at (5, 2) and (7, 8). What are the coordinates of its midpoint? or (6, 5) Answer: The coordinates of the midpoint are (6, 5).

7. Example 1-1b Find the midpoint of the segment with endpoints at (3, 6) and (–1, –8). Answer: (1, –1)

8. Example 1-2a What is the distance between P(–1, 4) and Q(2, –3) ? Distance Formula Subtract. Answer: The distance between the two points is units. Let and Simplify.or

9. Example 1-2b What is the distance between P(2, 3) and Q(–3, 1) ? Answer: units

10. Example 1-3a Multiple-Choice Test Item Which point is farthest from (2, –3) ? A (0, 0) B (3, 2) C (–3, 0) D (4, 1) Read the Test Item The word farthest refers to the greatest distance. Solve the Test Item Use the Distance Formula to find the distance from (2, –3) to each point.

11. Example 1-3a Distance to (0, 0) Distance to (3, 2) Distance to (–3, 0) Distance to (4, 1) The greatest distance is units. So, the farthest point from (2, –3) is (–3, 0). Answer: C

12. Example 1-3b Multiple-Choice Test Item Which point is farthest from (2, 3) ? A (0, 0) B (1, 2) C (3, 4) D (–2, 4) Answer: D

13. End of Lesson 1

14. Lesson 2 Contents Example 1Analyze the Equation of a Parabola Example 2Graph Parabolas Example 3Graph an Equation Not in Standard Form Example 4Write and Graph an Equation for a Parabola

15. Example 2-1a Write in standard form. Identify the vertex, axis of symmetry, and direction of opening of the parabola. Original equation Factor –1 from the x -terms. The 1 you added when you completed the square is multiplied by –1. Complete the square on the right side.

16. Example 2-1a Answer: The vertex of this parabola is located at (–1, 4) and the equation of the axis of symmetry is. The parabola opens downward.

17. Example 2-1b Write in standard form. Identify the vertex, axis of symmetry, and direction of opening of the parabola. Answer: axis of symmetry:

18. Example 2-2a Graph. For this equation, and The vertex is at the origin. Since the equation of the axis of symmetry is substitute some small positive integers for x and find the corresponding y -values. xy 12 28 318 Since the graph is symmetric about the y -axis, the points at (–1, 2), (–2, 8) and (–3, 18) are also on the parabola. Use all of these points to draw the graph.

19. Example 2-2a Answer:

20. Example 2-2a Graph. The equation is of the form where The graph of this equation is the graph of in part a translated 1 unit right and 5 units down. The vertex is now at (1, –5). Answer:

21. Example 2-2b Graph each equation. a.b. Answer:

22. Example 2-3a Graph There is a y 2 term, so isolate the y and y 2 terms. Add and subtract 4, since First write the equation in the form Complete the square.

23. Example 2-3a Then use the following information to draw the graph. vertex: (3, 2) axis of symmetry: directrix: direction of opening: left, since length of the latus rectum: or 1 unit focus: or

24. Example 2-3a Answer:

25. Example 2-3b Answer: Graph

26. Example 2-4a Bridges The 52 meter-long Hulme Arch Bridge in Manchester, England, is supported by cables suspended from a parabolic steel arch. The highest point of the arch is 25 meters above the bridge, and the focus of the arch is about 18 meters above the bridge. Let the bridge be the x -axis, and let the y -axis pass through the vertex of the arch. Write an equation that models the arch. The focus is at (0, 12). Use the y -coordinate of the focus to find a. The vertex is at (0, 25), so and

27. Example 2-4a k = 25 ; The y -coordinate of the focus is 18. Subtract 25 from each side. Multiply each side by 4a. Divide each side by –28. Answer: An equation of the parabola is

28. Example 2-4a Graph the equation. The length of the latus rectum is or 28 units, so the graph must pass through (–14, 18) and (14, 18). According to the length of the bridge, the graph must pass through the points (–26, 0) and (26, 0). Use these points and the information from part a to draw the graph.

29. Example 2-4a Answer:

30. Example 2-4b Fountains An outdoor fountain has a jet through which water flows. The water stream follows a parabolic path. The highest point of the water stream is feet above the ground and the water hits the ground 10 feet from the jet. The focus of the fountain is feet above the ground. a. Write an equation that models the path of the water fountain. Answer:

31. Example 2-4b b. Graph the equation. Answer:

32. End of Lesson 2

33. Lesson 3 Contents Example 1Write an Equation Given the Center and Radius Example 2Write an Equation Given a Diameter Example 3Write an Equation Given the Center and a Tangent Example 4Graph an Equation in Standard Form Example 5Graph an Equation Not in Standard Form

34. Example 3-1a Landscaping The plan for a park puts the center of a circular pond, of radius 0.6 miles, 2.5 miles east and 3.8 miles south of the park headquarters. Write an equation to represent the border of the pond, using the headquarters as the origin. Since the headquarters is at (0, 0), the center of the pond is at (2.5, –3.8) with radius 0.6 mile. Answer: The equation is Simplify. Equation of a circle

35. Example 3-1b Landscaping The plan for a park puts the center of a circular pond, of radius 0.5 mile, 3.5 miles west and 2.6 miles north of the park headquarters. Write an equation to represent the border of the pond, using the headquarters as the origin. Answer:

36. Example 3-2a Write an equation for a circle if the endpoints of the diameter are at (2, 8) and (2, –2). ExploreTo write an equation for a circle, you must know the center and the radius. PlanYou can find the center of the circle by finding the midpoint of the diameter. Then you can find the radius of the circle by finding the distance from the center to one of the given points.

37. Example 3-2a SolveFirst, find the center of the circle. Midpoint Formula Add. Simplify.

38. Example 3-2a Now find the radius. Substitute h, k, and r 2 into the standard form of the equation of a circle. Distance Formula Subtract. Simplify. The radius of the circle is 5 units, so

39. Example 3-2a Answer: An equation of the circle is ExamineEach of the given points satisfies the equation, so the equation is reasonable.

40. Example 3-2b Write an equation for a circle if the endpoints of the diameter are at (3, 5) and (3, –7). Answer:

41. Example 3-3a Write an equation for a circle with center at (3, 5) that is tangent to the y -axis. Sketch the circle. Since it is tangent to the y -axis, the radius is 3. Answer: An equation of this circle is.

42. Example 3-3b Write an equation for a circle with center at (2, 3) that is tangent to the x -axis. Answer:

43. Example 3-4a Find the center and radius of the circle with equation Then graph the circle. Answer: The center is at (0, 0) and the radius is 4. The table lists some values for x and y that satisfy the equation. xy 04 13.9 23.5 32.6 40 Since the circle is centered at the origin, it is symmetric about the y -axis. Therefore, the points at (–1, 3.9), (–2, 3.5), (–3, 2.6) and (–4, 0) lie on the graph.

44. Example 3-4a The circle is also symmetric about the x -axis, so the points (–1, –3.9), (–2, –3.5), (–3, –2.6), (1, –3.9), (2, –3.5), (3, – 2.6), and (0, –4) lie on the graph. Graph these points and draw the circle that passes through them. Answer:

45. Example 3-4b Find the center and radius of the circle with equation Then graph the circle. Answer: center (0, 0) ;

46. Example 3-5a Find the center and radius of the circle with equation Then graph the circle. Complete the square. Answer: The center is at (–3, 0) and the radius is 4. (–3, 0)

47. Example 3-5b Find the center and radius of the circle with equation Then graph the circle. Answer: center (–4, 2) ; (–4, 2)

48. End of Lesson 3

49. Lesson 4 Contents Example 1Write an Equation for a Graph Example 2Write an Equation Given the Lengths of the Axes Example 3Graph an Equation in Standard Form Example 4Graph an Equation Not in Standard Form

50. Example 4-1a Write an equation for the ellipse shown. In order to write an equation for the ellipse, we need to find the values of a and b for the ellipse. We know that the length of the major axis of any ellipse is 2 a units. In this ellipse, the length of the major axis is the distance between (0, 5) and (0, –5). This distance is 10 units.

51. Example 4-1a Divide each side by 2. The foci are located at (0, 4) and (0, –4), so c = 4. We can use the relationship between a, b, and c to determine the value of b. Equation relating a, b, and c Solve for b 2. and

52. Example 4-1a Since the major axis is vertical, substitute 25 for a 2 and 9 for b 2 in the form Answer: An equation of the ellipse is

53. Example 4-1b Write an equation for the ellipse shown. Answer:

54. Example 4-2a Sound A listener is standing in an elliptical room 150 feet wide and 320 feet long. When a speaker stands at one focus and whispers, the best place for the listener to stand is at the other focus. Write an equation to model this ellipse, assuming the major axis is horizontal and the center is at the origin. The length of the major axis is 320 feet. Divide each side by 2.

55. Example 4-2a The length of the minor axis is 150 feet. Divide each side by 2. Substitute and into the form Answer: An equation for the ellipse is

56. Example 4-2a How far apart should the speaker and the listener be in this room? The two people should stand at the two foci of the ellipse. The distance between the foci is 2c units. Equation relating a, b, and c Multiply each side by 2. Use a calculator. Substitute and Take the square root of each side.

57. Example 4-2a Answer: The two people should be about 282.7 feet apart.

58. Example 4-2b Sound A listener is standing in an elliptical room 60 feet wide and 120 feet long. When a speaker stands at one focus and whispers, the best place for the listener to stand is at the other focus. a. Write an equation to model this ellipse, assuming the major axis is horizontal and the center is at the origin. b.How far apart should the speaker and the listener be in this room? Answer: 103.9 feet apart Answer:

59. Example 4-3a Find the coordinates of the center and foci and the lengths of the major and minor axes of the ellipse with equation Then graph the equation. The center of this ellipse is at (0, 0). The length of the major axis is 2(6) or 12 units, and the length of the minor axis is 2(3 ) or 6. Since the x 2 term has the greatest denominator, the major axis is horizontal. Since and since

60. Example 4-3a Equation relating a, b, and c Take the square root of each side. The foci are at and

61. Example 4-3a You can use a calculator to find some approximate nonnegative values for x and y that satisfy the equation. xy 03 12.96 22.83 32.60 42.24 51.66 60 Since the ellipse is centered at the origin, it is symmetric about the y -axis. So, the points at (1, 2.96) and (–1, 2.96) lie on the graph. The ellipse is also symmetric about the x -axis, so the points at (1, –2.96) and (–1, –2.96) also lie on the graph.

62. Example 4-3a Graph the intercepts (–6, 0) (6, 0) (0, 3) and (0, –3) and draw the ellipse that passes through them and the other points. Answer: center: (0, 0) ; foci: major axis: 12 ; minor axis: 6

63. Example 4-3b Find the coordinates of the center and foci and the lengths of the major and minor axes of the ellipse with equation Then graph the equation. center: (0, 0) ; foci: major axis: 10 ; minor axis: 4 Answer:

64. Example 4-4a Find the coordinates of the center and foci and the lengths of the major and minor axes of the ellipse with equation Then graph the ellipse. Complete the square to write in standard form. Original equation Complete the squares.

65. Example 4-4a Write the trinomials as perfect squares. Divide each side by 36.

66. Example 4-4a Answer:The center is (3, 2) and the foci are located at and The length of the major axis is 12 units and the length of the minor axis is 6.

67. Example 4-4b Find the coordinates of the center and foci and the lengths of the major and minor axes of the ellipse with equation Then graph the ellipse. center: (–2, 3) ; foci: major axis: 10 ; minor axis: 4 Answer:

68. End of Lesson 4

69. Lesson 5 Contents Example 1Write an Equation for a Graph Example 2Write an Equation Given the Foci and Transverse Axis Example 3Graph an Equation in Standard Form Example 4Graph an Equation Not in Standard Form

70. Example 5-1a Write an equation for the hyperbola. The center is the midpoint of the segment connecting the vertices, or (0, 0). The value of a is the distance from the center to a vertex or 2 units. The value of c is the distance from the center to a focus, or 4 units.

71. Example 5-1a Substitute the values for a 2 and b 2. Equation relating a, b, and c for a hyperbola Evaluate the squares. Solve for b 2. Since the transverse axis is vertical, the equation is of the form Answer: An equation of the hyperbola is

72. Example 5-1b Write an equation for the hyperbola. Answer:

73. Example 5-2a Navigation A ship notes that the difference of its distance from two LORAN stations that are located at (–70, 0) and (70, 0) is 70 nautical miles. Write an equation for the hyperbola on which the ship lies. First draw a figure.

74. Example 5-2a Use the values for a and c to find b for this hyperbola. By studying either of the x -intercepts, you can see that the difference of the distances from any point on the hyperbola to the stations at the foci is the same as the length of the transverse axis, or 2a. Therefore, or According to the coordinates of the foci, Equation relating a, b, and c for a hyperbola Evaluate the squares. Solve for b 2.

75. Example 5-2a Substitute the values for a 2 and b 2. Since the transverse axis is horizontal, the equation is of the form Answer: An equation of the hyperbola is

76. Example 5-2b Navigation A ship notes that the difference of its distance from two LORAN stations that are located at (–60, 0) and (60, 0) is 60 nautical miles. Write an equation for the hyperbola on which the ship lies. Answer:

77. Example 5-3a Find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola with equation Then graph the hyperbola. Answer: The center of the hyperbola is at the origin. According to the equation, andso andThe coordinates of the vertices are (1, 0) and (–1, 0). Equation relating a, b, and c for a hyperbola Simplify. Take the square root of each side.

78. Example 5-3a You can use a calculator to find some approximate nonnegative values for x and y that satisfy the equation. xy 10 21.7 32.8 43.9 54.9 Answer: The foci are at and The equations of the asymptotes are or or

79. Example 5-3a Since the hyperbola is centered at the origin, it is symmetric about the y -axis. Therefore, the points at (–5, 4.9), (–4, 3.9), (–3, 2.8), (–2, 1.7), and (–1, 0) lie on the graph. The hyperbola is also symmetric about the x -axis, so the points at (–5, –4.9), (–4, –3.9), (–3, –2.8), (–2, –1.7), (2, –1.7), (3, –2.8), (4, –3.9), and (5, –4.9 ) also lie on the graph.

80. Example 5-3a Draw a 2 -unit by 2 -unit square. The asymptotes contain the diagonals of the square. Graph the vertices, which, in this case, are the x -intercepts. Use the asymptotes as a guide to draw the hyperbola that passes through the vertices and the other points. The graph does not intersect the asymptotes. Answer:

81. Example 5-3b Answer: Find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola with equation Then graph the hyperbola. vertices: (2, 0), (–2, 0) ; foci: asymptotes:

82. Example 5-4a Complete the square for each variable to write in standard form. Find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola with equation Then graph the hyperbola.

83. Example 5-4a Original equation Write the trinomials as perfect squares. Complete the square.

84. Example 5-4a Answer: The vertices are (–4, 5) and (–2, 5) and the foci areandThe equations of the asymptotes areor and

85. Example 5-4b Answer: Find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola with equation Then graph the hyperbola. vertices: ( 4, 1 ), ( 4, –5 ); foci: ( 4, 3 ), ( 4, –7 ); asymptotes:

86. End of Lesson 5

87. Lesson 6 Contents Example 1Rewrite an Equation of a Conic Section Example 2Analyze an Equation of a Conic Section

88. Example 6-1a Write the equation is standard form. Write the equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. Original equation Divide each side by 18. Isolate terms.

89. Example 6-1a Answer: The graph is an ellipse with center at (0, 0).

90. Write the equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. Example 6-1b Answer: circle

91. Example 6-2a Since A and C have opposite signs, the graph is a hyperbola. Without writing the equation in standard form, state whether the graph of is a parabola, circle, ellipse, or hyperbola. Answer: and

92. Example 6-2a Without writing the equation in standard form, state whether the graph of is a parabola, circle, ellipse, or hyperbola. Answer: and Since the graph is a circle.

93. Example 6-2a Without writing the equation in standard form, state whether the graph of is a parabola, circle, ellipse, or hyperbola. Answer: and Since this graph is a parabola.

94. Without writing the equation in standard form, state whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. a. b. c. Example 6-2b Answer: hyperbola Answer: ellipse Answer: parabola

95. End of Lesson 6

96. Lesson 7 Contents Example 1Linear-Quadratic System Example 2Quadratic-Quadratic System Example 3System of Quadratic Inequalities

97. Example 7-1a You can use a graphing calculator to help visualize the relationships of the graphs of the equations and predict the number of solutions. Solve the system of equations. Solve each equation for y to obtain and

98. Example 7-1a Enter the functions on the Y= screen. The graph indicates that the hyperbola and the line intersect in one point. So, the system has one solution.

99. Example 7-1a Use substitution to solve the system. First, rewrite First equation in the system Substitute 2 – 2y for x. Simplify. Subtract 16 from each side. Divide each side by –32.

100. Example 7-1a Now solve for x. Equation for x in terms of y Substitute the y value. Simplify. Answer: The solution is

101. Example 7-1b Solve the system of equations. Answer: (2, 0) and

102. Example 7-2a Solve the system of equations. A graphing calculator indicates that the circle and ellipse intersect in four points. So, this system has four solutions.

103. Example 7-2a Use the elimination method to solve the system. Rewrite the first original equation. Divide each side by 3. Take the square root of each side. Second original equation Add.

104. Example 7-2a Substitute and in either of the original equations and solve for y. Original equation Substitute for x. Subtract from each side.

105. Example 7-2a Take the square root of each side. Answer: The solutions are and

106. Example 7-2b Answer: (3, 1), (3, –1), (–3, 1), and (–3, –1) Solve the system of equations.

107. Example 7-3a Solve the system of inequalities by graphing. The graph of is the parabola and the region inside and above it. The region is shaded blue. The graph of is the interior of the circle This region is shaded yellow.

108. Example 7-3a Answer: The intersection of these regions, shaded green, represents the solution of the system of inequalities.

109. Example 7-3b Solve the system of inequalities by graphing. Answer:

110. End of Lesson 7

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