Pre Calc. Conics.

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2 Pre Calc Conics

3 Table of Contents click on the topic to go to that section Review of Midpoint and Distance Formulas Intro to Conic Sections Parabolas Circles Ellipses Hyperbolas Recognizing Conic Sections from the General Form 3

4 Midpoint and Distance Formula Return to Table of Contents 4

5 Midpoint and Distance Formula The Midpoint Formula Give points A(x 1,y 1 ) and B (x 2,y 2 ), the point midway between A and B is Examples: Find the midpoint of the segment with the given endpoints. 5

6 Midpoint and Distance Formula Example: If M is the midpoint of line segment WZ, and given M(2, 4) and W(5, 2), find the coordinates of Z. 6

7 Midpoint and Distance Formula 1 Find the midpoint of K(1,8) & L(5,2). A (2,3) B (3,5) C ( 2, 3) D ( 3, 5) 7

8 Midpoint and Distance Formula 2 Find the midpoint of H( 4, 8) & L(6, 10). A (1,9) B (2,18) C ( 2, 18) D ( 1, 9) 8

9 Midpoint and Distance Formula 3 Given the midpoint of a segment is (4, 9) and one endpoint is ( 3, 10), find the other midpoint. A ( 10, 8) B (11, 8) C ( 10, 11) D (.5, 9.5) 9

10 Midpoint and Distance Formula The Distance Formula The distance between points A(x 1,y 1 ) and B (x 2,y 2 ) is Find the distance between the following points: 10

11 Midpoint and Distance Formula Example: If the distance between (3, 2) and (8, y) is 6, find the possible values of y. 11

12 Midpoint and Distance Formula 4 What is the distance between (2, 4) and ( 1, 8)? 12

13 Midpoint and Distance Formula 5 What is the distance between (0, 7) and (5, 5)? 13

14 Midpoint and Distance Formula Note: The distance between points A and B can be notated as AB 14

15 Midpoint and Distance Formula 6 Given A( 4, 5) and B(x, 1) and AB=5, find all of the possible values of x. A 7 B 5 C 3 D 1 E 0 F 1 G 3 H 5 I 7 J 9 15

16 Intro to Conic Sections Return to Table of Contents 16

17 Intro to Conic Sections Conic Sections come from cutting through 2 cones, which is called taking cross sections. Conic Sections are often times not functions because they do not pass the Vertical Line Test. 17

18 Intro to Conic Sections A Circle comes from cutting parallel to the "base". The term base is mis leading because cones continue on, like lines. 18

19 Intro to Conic Sections An Ellipse comes from cutting skew (diagonal) to the "base". 19

20 Intro to Conic Sections A Parabola comes from cutting the cone an intersecting the "base" and parallel to a side. 20

21 Intro to Conic Sections A Hyperbola comes from cutting the cones perpendicular to the "bases". This is the only cross section that intersects both cones. 21

22 Parabolas Return to Table of Contents 22

23 Parabolas As we've studied earlier, Parabolas come from a quadratic equation of the form y=ax 2 +bx+c and have a "U" shaped graph. Another helpful form of the equation is called Standard Form. Standard Form is (x h) 2 = 4p(y k), where (h,k) is the vertex. This is also called Vertex Form. Example: What is the vertex of: (x 4) 2 = 3(y 5) (x + 7) 2 = 2(y 2) (x 3) 2 = y 23

24 Parabolas 7 What is the vertex of A (3, 2) B ( 3, 2) C (2, 3) D ( 2, 3) 24

25 Parabolas 8 What is the vertex of A (3, 2) B ( 3, 2) C (2, 3) D ( 2, 3) 25

26 Parabolas 9 What is the vertex of A (3, 2) B ( 3, 2) C (2, 3) D ( 2, 3) 26

27 Parabolas In this section on Conic Sections parabolas that open left and right are also included. (Conics don't have to be functions.) These relations have the general equation of and the standard form of Where (h,k) is the vertex 27

28 Parabolas Examples What is the vertex of Where (h,k) is the vertex What is the vertex of What is the vertex of What is the vertex of What is the vertex of 28

29 Parabolas 10 What is the vertex of A (3, 2) B ( 3, 2) C (2, 3) D ( 2, 3) 29

30 Parabolas 11 What is the vertex of A (3, 2) B ( 3, 2) C (2, 3) D ( 2, 3) 30

31 Parabolas 12 What is the vertex of A (3, 2) B ( 3, 2) C (2, 3) D (2, 3) 31

32 Parabolas Converting from General Form to Standard Form Note: To convert into Standard Form, we use a process called Completing the Square. Steps: 1) Group the quadratic and its linear term on one side, and move the other linear and constant terms to the other side. 2) If there is a number in front of the quadratic, factor it out of the group. 3) Take the number in front of the linear term, divide it in half and square it. 4) Add this number inside the parenthesis; multiply it by the number you factored out in step two, and add it to the other side of the equation as well. 5) Factor the quadratic function inside the parenthesis 32

33 Parabolas Example: Find the vertex of the parabola 33

34 Parabolas Example: Find the vertex of the parabola 34

35 Parabolas 13 What value completes the square of 35

36 Parabolas 14 What value goes on the blank line after 2? 36

37 Parabolas 15 What is y coordinate of the vertex? 37

38 Parabolas 16 What is x coordinate of the vertex? 38

39 Parabolas 17 What is is the vertex of of y 2 x=y 10y 2 10y+29 x + 29 = 0? A (4, 5) B ( 4, 5) C ( 5, 4) D (5, 4) 39

40 Parabolas 18 What is the vertex of A (4, 5) B ( 4, 5) C ( 5, 4) D (5, 4) 40

41 Parabolas Converting from General Form to Standard Form } +18 } 12 41

42 Parabolas 19 What should be factored out of (4y 2 8y + ) = x 9 +? 42

43 Parabolas 20 What value completes the square of 4(y 2 2y + ) = x 9 +? 43

44 Parabolas 21 What value should follow " 9" in 4(y 2 2y + ) = x 9 +? 44

45 Parabolas 22 Which is the correct standard form of 4(y 2 2y + ) = x 9 + A B C D 45

46 Parabolas 23 What should be factored out of ( 5x 2 20x + ) = y 7 +? 46

47 Parabolas 24 What value completes the square of 5(x 2 + 4x + ) = y 7 +? 47

48 Parabolas 25 What value should follow " 7" in 5(x 2 + 4x + ) = y 7 +? 48

49 Parabolas 26 Which is the correct standard form of ( 5x 2 20x + ) = y 7 + A B C D 49

50 Parabolas Focus and Directrix of a Parabola Every point on the parabola is the same distance from the directrix and the focus. L 1 =L 2 L 1 L 2 Focus Axis of Symmetry Directrix The focal distance is the distance from the vertex to the focus, which is the same as the distance from the vertex to the directrix. 50

51 Parabolas Eccentricity of a Parabola L 1 =L 2 L 1 L 2 Focus Directrix 51

52 Parabolas Parts of a Parabola Whether a quadratic has the x 2 or y 2, they have the same parts. ax 2 +bx+dy+e=0 cy 2 +dy+bx+e=0 Focus Vertex Axis of Symmetry Vertex Focus Directrix Directrix Axis of Symmetry 52

53 Parabolas General Form Standard Form Vertex Axis of Symmetry Focal Distance Opens Directrix Focus p>0 opens up p<0 opens down p>0 opens up p<0 opens down Eccentricity

54 Parabolas Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity? 54

55 Parabolas Graph the equation from the last example. 55

56 Parabolas Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity? 56

57 Parabolas Graph 57

58 Parabolas Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity? Convert the equation from general to standard form. 58

59 Parabolas Graph 59

60 Parabolas 27 Given the following equation, which direction does it open? A B C D UP DOWN LEFT RIGHT 60

61 Parabolas 28 Where is the vertex for the following equation? A ( 3, 4) B (3, 4) C (4, 3) D (4, 3) 61

62 Parabolas 29 What is the equation of the axis of symmetry for the following equation? A y = 3 B y = 3 C x = 4 D x = 4 62

63 Parabolas 30 What is the focal distance in the following equation? 63

64 Parabolas 31 What is the equation of the directrix for the following equation? A y = 2 B y = 4 C x = 3 D x = 5 64

65 Parabolas 32 Where is the focus for the following equation? A ( 3, 5) B (3, 5) C (5, 3) D (5, 3) 65

66 Parabolas 33 What is the eccentricity of the following conic section? 66

67 Parabolas 34 Given the following equation, which direction does it open? A B C D UP DOWN LEFT RIGHT 67

68 Parabolas 35 Where is the vertex for the following equation? A ( 3, 2) B ( 3, 2) C (2, 3) D ( 2, 3) 68

69 Parabolas 36 What is the equation of the axis of symmetry for the following equation? A y = 2 B y = 2 C x = 3 D x = 3 69

70 Parabolas 37 What is the focal distance in the following equation? 70

71 Parabolas 38 What is the equation of the directrix for the following equation? A y = 2.5 B y = 1.5 C x = 3.5 D x =

72 Parabolas 39 Where is the focus for the following equation? A ( 2.5, 2) B ( 3.5, 2) C ( 3, 2.5) D ( 3, 1.5) 72

73 Parabolas 40 What is the eccentricity of the following conic section? 73

74 Parabolas 41 Given the following equation, which direction does it open? A B C D UP DOWN LEFT RIGHT 74

75 Parabolas 42 Where is the vertex for the following equation? A (0, 4) B (0, 4) C (4, 0) D ( 4, 0) 75

76 Parabolas 43 What is the equation of the axis of symmetry for the following equation? A y = 0 B y = 0 C x = 4 D x = 4 76

77 Parabolas 44 What is the focal distance in the following equation? 77

78 Parabolas 45 What is the equation of the directrix for the following equation? A y = 0 B y = 4 C x = 8 D x = 0 78

79 Parabolas 46 Where is the focus for the following equation? A (4, 8) B ( 4, 4) C (4, 4) D (4, 4) 79

80 Parabolas 47 What is the eccentricity of the following conic section? 80

81 Circles Return to Table of Contents 81

82 Circles Standard Form of a Circle Where (h, k) is the center, r is the radius and (x, y) represent every point on the circle: Note: Circles have an eccentricity of 0. 82

83 Circles Write the equations given the following centers and radii. 83

84 Circles State the center and radius, given the following equation. 84

85 Circles 48 Write the equation of the circle with center (5, 2) and radius 6 A B C D 85

86 Circles 49 Write the equation of the circle with center ( 5, 0) and radius 7 A B C D 86

87 Circles 50 Write the equation of the circle with center ( 2, 1) and radius A B C D 87

88 Circles 51 What is the center and radius of the following equation? A B C D 88

89 Circles 52 What is the center and radius of the following equation? A B C D 89

90 Circles 53 What is the center and radius of the following equation? A B C D 90

91 Circles 54 What is eccentricity of a circle? 91

Since the midpoint of the diameter is the center use the midpoint

92 Circles Ex: Write the equation of the circle that meets the following criteria: Diameter with endpoints (4, 7) and ( 2, 1). Since the midpoint of the diameter is the center use the midpoint formula. The radius is distance from the center to either of the given points. 92

Since we know the center we only need to find the radius.

93 Circles Ex: Write the equation of the circle that meets the following criteria: Center (1, 2) and passes through (4, 6) Since we know the center we only need to find the radius. The radius is the distance from the center to the point. 93

94 Circles Ex: Write the equation of the circle that meets the following criteria: Center at ( 5, 6) and tangent to the y axis. "Tangent to the y axis" means the circle only touches the y axis at one point. Look at the graph. ( 5, 6) y x

95 Circles Write the equation of the circle in standard form that meets the following criteria: Complete the square for both the x's and the y's 95

96 Circles Write the equation of the circle in standard form that meets the following criteria: Complete the square for the x's 96

97 Circles 55 What is the equation of the circle that has a diameter with endpoints (0, 0) and (16, 12)? A B C D 97

98 Circles 56 What is the equation of the circle with center ( 3, 5) and contains point (1, 3)? A B C D 98

99 Circles 57 What is the equation of the circle with center (7, 3) and tangent to the x axis? A B C D 99

100 Circles 58 What is the equation of the circle, in standard form, for A B C D 100

101 Circles 59 What is the equation of the circle, in standard form, for A B C D 101

102 Ellipses Return to Table of Contents 102

103 Ellipses An ellipse is the set of points the same total distance from 2 points. In this example, P As point moves along the ellipse, L 1 and L 2 will change but their sum will stay ten. 103

104 Ellipses In this graph F 1 and F 2 are foci. (Plural of focus) They lie on the major axis. (The longest distance) The shortest distance is the minor axis. Where the axes intersect is the ellipse's center. P The more elongated the ellipse the closer the eccentricity is to 1. The closer an ellipse is to being a circle, the closer the eccentricity is to 0. (0 < e < 1) 104

105 Ellipses 60 What letter or letters corresponds with ellipse's center? A B C D B A C E D E 105

106 Ellipses 61 What letter or letters corresponds with ellipse's foci? A B C D B A C E D E 106

107 Ellipses 62 What letter or letters corresponds with ellipse's major axis? A B C D B A C E D E 107

108 Ellipses 63 Which choice best describes an ellipse's eccentricity? A e = 0 B 0< e < 1 C e = 1 D e > 1 108

109 Ellipses Standard Form of an Ellipse Where the center is (h,k),the horizontal distance from the center to the ellipse is a and the vertical distance is b

Ellipses Standard Form of an Ellipse 8 3 8 The Foci are equidistant from the center and are on the major axis.

110 Ellipses Standard Form of an Ellipse The Foci are equidistant from the center and are on the major axis. 3 If a > b, then the ellipse is horizontal. The distance from the center to the foci is If b>a, then the ellipse is vertical. The distance from the center to the foci is 110

111 Ellipses Find the Foci. Standard Form of an Ellipse

112 Ellipses 64 What is the center of A (9, 4) B (5, 6) C ( 5, 6) D (3, 2) 112

113 Ellipses 65 How long is the major axis of A 9 B 4 C 3 D 2 113

114 Ellipses 66 How long is the minor axis of A 9 B 4 C 3 D 2 114

115 Ellipses 67 Name one foci of A B C D 115

116 Ellipses 68 Name one foci of A B C D 116

117 Ellipses Graphing an Ellipse Find and graph the center Find the length and direction of the major and minor axes From the center go half the length the axis from the center for each Graph the ellipse The center is (4, 2) The major axis is 6 units and horizontal The minor axis is 4 units and vertical 117

118 Ellipses Graph: 118

119 Ellipses Graph: 119

120 Ellipses What is equation of an ellipse with foci (3, 2) and (3, 6) and minor axis of length 8? 120

121 Ellipses 69 Given that an ellipse has foci (4, 1) and ( 4, 1) and major axis of length 10, what is the center of the ellipse? A (8, 2) B (0, 2) C (0, 1) D ( 8, 1) 121

122 Ellipses 70 Given that an ellipse has foci (4, 1) and ( 4, 1) and major axis of length 10, in which direction is the ellipse elongated? A B C D horizontally vertically obliquely it is not elongated 122

123 Ellipses 71 Given that an ellipse has foci (4, 1) and ( 4, 1) and major axis of length 10, how far is it from the center to an endpoint of the major axis? A 10 B 100 C 5 D

124 Ellipses 72 Given that an ellipse has foci (4, 1) and ( 4, 1) and major axis of length 10, which equation would be used to find the distance from the center to an endpoint of the minor axis? A B C D 124

125 Ellipses 73 Given that an ellipse has foci (4, 1) and ( 4, 1) and major axis of length 10, find a. A B C D 125

126 Ellipses 74 Given that an ellipse has foci (4, 1) and ( 4, 1) and major axis of length 10, which is the equation of the ellipse? A C B D 126

127 Ellipses Converting to Standard Form complete the square for x and/or y factor the x's and y's divide by the constant Ex: Ex: 127

128 Ellipses Convert the ellipse from general form to standard form. 128

129 Ellipses 75 Convert the following ellipse to standard form. A B C D 129

130 Ellipses 76 Convert the following ellipse to standard form. A B C D 130

131 Ellipses 77 Convert the following ellipses to standard form. A B C D 131

132 Hyperbolas Return to Table of Contents 132

133 Hyperbolas The standard form of a horizontal hyperbola is 133

134 Hyperbolas The standard form of a vertical hyperbola is 134

135 Hyperbolas To graph a hyperbola in standard form: graph (h,k) as center of graph go a right and left of the center, and b up and down make a rectangle through the four points from previous step draw asymptotes that contain the diagonals of the rectangle decide if hyperbola goes left & right or up & down left & right: the "x term" is first up & down: the "y term" is first graph hyperbola 135

136 Hyperbolas Example: Graph The center of the rectangle is? From the center move left/right? From the center move up/down? The hyperbola opens? What are the slopes of the asymptotes? How does this relate to a and b? Why? 136

137 Hyperbolas Example: Graph The center of the rectangle is? From the center move left/right? From the center move up/down? The hyperbola opens? 137

138 Hyperbolas 78 What is the center of the following hyperbola? A ( 3, 2 ) B ( 3, 2 ) C ( 2, 3 ) D ( 2, 3) 138

139 Hyperbolas 79 How far left of the center is the rectangle? A 5 B 10 C 4 D 8 139

140 Hyperbolas 80 How wide is the rectangle? A 5 B 10 C 4 D 8 140

141 Hyperbolas 81 How far below the center is the rectangle? A 5 B 10 C 4 D 8 141

142 Hyperbolas 82 What is the height of the rectangle? A 5 B 10 C 4 D 8 142

143 Hyperbolas 83 What is the slope of the asymptote that has a positive slope? 143

144 Hyperbolas 84 The hyperbola opens up and down? True False 144

145 Hyperbolas Graph 145

146 Hyperbolas Standard Form of an Hyperbola The Foci are equidistant from the center in the horizontal direction if the x term comes first, or in the vertical direction if the y term comes first. The distance from the center to the foci is In this example, the focal distance is And their location is at and 146

147 Hyperbolas 85 What is the focal distance for the following equation? A 12 B 13 C 5 D 8 147

148 Hyperbolas 86 What is the location of one of the foci for this hyperbola? A ( 13, 6) B (10, 6) C ( 10, 6) D (13, 6) 148

149 Hyperbolas 87 What is the location of one of the foci for this hyperbola? A (3,7) B (3,19) C (3, 7) D (3,13) 149

150 Hyperbolas Convert to standard form: 150

151 Hyperbolas 88 Which is the standard form of A B C D 151

152 Recognizing Conic Sections from the General Form Return to Table of Contents 152

153 Recognizing Conic Sections General Form: In a parabola either a=0 or c=0 ax2 + bx + dy +e =0 In a circle a=c cy 2 + dy + bx + e=0 In an ellipse a>0 and c>0, but a = c ax 2 + bx + cy 2 + dy + e=0 In a hyperbola either a<0 or c<0 ax2 + bx cy 2 + dy + e=0 cy 2 + dy ax 2 + bx + e=0 153

154 Recognizing Conic Sections 89 Identify the Conic Section A B C D Parabola Circle Ellipse Hyperbola 154

155 Hyperbolas 90 Identify the Conic Section A B C D Parabola Circle Ellipse Hyperbola 155

156 Hyperbolas 91 Identify the Conic Section A B C D Parabola Circle Ellipse Hyperbola 156

157 Hyperbolas 92 Identify the Conic Section A B C D Parabola Circle Ellipse Hyperbola 157

158 Hyperbolas 93 Identify the Conic Section A B C D Parabola Circle Ellipse Hyperbola 158

159 Hyperbolas 94 Identify the Conic Section A B C D Parabola Circle Ellipse Hyperbola 159

160 Hyperbolas 95 Identify the Conic Section A B C D Parabola Circle Ellipse Hyperbola 160

Pre-Calc. Slide 1 / 160. Slide 2 / 160. Slide 3 / 160. Conics Table of Contents. Review of Midpoint and Distance Formulas

Pre-Calc. Slide 1 / 160. Slide 2 / 160. Slide 3 / 160. Conics Table of Contents. Review of Midpoint and Distance Formulas Slide 1 / 160 Pre-Calc Slide 2 / 160 Conics 2015-03-24 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 160 Review of Midpoint and Distance Formulas Intro to Conic Sections