Algebra / Geometry III: Unit 7- Conic Sections
SUCCESS CRITERIA:
1. Be able to identify x & y-intercepts and average rate of change using graphs, tables, & equations.
2. Be able to identify and describe key features of graphs, tables and equations.
3. Be able to analyze the transformations of functions given graphs or equations.
INSTRUCTOR: Craig Sherman
Hidden Lake High School
Westminster Public Schools
Conic Sections and Standard Forms of Equations
Aconic sectionis the intersection of a plane and a double right circularcone. By changing the angle and location of the intersection, we can produce different types of conics. There are four basic types:circles,ellipses,hyperbolasandparabolas. None of the intersections will pass through the vertices of the cone.
If the right circular cone is cut by a plane perpendicular to the axis of the cone, the intersection is a circle. If the plane intersects one of the pieces of the cone and its axis but is not perpendicular to the axis, the intersection will be an ellipse. To generate a hyperbola the plane intersects both pieces of the cone without intersecting the axis. And finally, to generate a parabola, the intersecting plane must intersect one piece of the double cone and its base.
The general equation for any conic section is
whereA, B, C, D, EandFare constants.
As we change the values of some of the constants, the shape of the corresponding conic will also change. It is important to know the differences in the equations to help quickly identify the type of conic that is represented by a given equation.
IfB2– 4ACis less than zero, if a conic exists, it will be either a circle or an ellipse.
IfB2– 4ACequals zero, if a conic exists, it will be a parabola.
IfB2– 4ACis greater than zero, if a conic exists, it will be a hyperbola.
INSTRUCTION 1: KHAN ACADEMY INSTRUCTION 2: SOPHIA
CONIC SECTION FORMULASCIRCLE / General Form / ax2 + bx + cy2 + dy + e = 0 + (y – k)2 = r2
Standard Form / (x – h)2 + (y – k)2 = r2
Center / (h, k)
Radius / r
Eccentricity / 0
VERTICAL / HORIZONTAL
PARABOLA / General Form / ax2 + bx + dy +e =0 / cy2 + dy + bx + e=0
Standard Form / y = a(x-h)2 + k / x = a(y-k)2
+ h
Opens / UP if a > 0 / RIGHT if a > 0
DOWN if a < 0 / LEFT if a < 0
Axis of Symmetry / x = h / y = k
Vertex / (h, k) / (h, k)
Focus / (h, k+p) / (h+p, k)
Directrix / y = k-p / x = h-p
a = 1 / 4p
p = 1 / 4a
Eccentricity / 1
VERTICAL / HORIZONTAL
ELIPSE / General Form / ax2 + bx + cy2 + dy + e = 0
Standard Form / /
Center / (0, 0) / (0, 0)
Focci / (c, 0), (-c, 0) / (0, c), (0, -c)
Vertices / (a, 0), (-a, 0) / (0, a), (0, -a)
y Intercepts / (0, b), (0, -b) / (b, 0), (-b, 0)
Major Axis / x axis / y axis
Minor Axis: / y axis / x axis
Length of Major Axis / 2a / 2a
Length of Minor Axis / 2b / 2b
c2 = a2 – b2, a > b > 0
Transverse Axis is VERTICAL / Transverse Axis is HORIZONTAL
HYPERBOLA / General Form / ax2 + bx - cy2 + dy + e = 0 / cy2 + dy - ax2 + bx + e = 0
Standard Form / /
Center / (0, 0) / (0, 0)
Focci / (c, 0), (-c, 0) / (0, c), (0, -c)
Vertices / (a, 0), (-a, 0) / (0, a), (0, -a)
Asymptotes / /
c2 = a2 + b2 – b2, a > b > 0
Parabolas
WORD or CONCEPT / DEFINITION or NOTES / EXAMPLE or GRAPHIC REPRESENTATIONparabola
vertex
axis of symmetry
Standard Form
EXAMPLE: 20. x=-3y+22-6
Vertex:
Axis of symmetry:
Parabola Opens:
Focal Point:
INSTRUCTION 1: KHAN ACADEMY INSTRUCTION 2: SOPHIA
Class Work
What is the vertex of the parabola?
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1. y=(x-2)2+4
2. y=-3(x+5)2+5
3. x=5y-72-6
4. x=2y+42+9
5. y=2(x-7)2-9
6. y=34(x)2+4
7. y=-(x-7)2
8. x=53y+82-3
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Write the following equations in standard form.
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9. y=x2+4x
10. x=y2-8y
11. y=x2-6x+8
12. x=y2+2y+10
13. y=x2+10x-12
14. x=y2-8y+16
15. y=2x2+12x
16. x=3y2-6y
17. y=-4x2+8x+6
18. x=-6y2-12y+15
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Graph each of the following. State the direction of the opening. Identify vertex and the focus and give the equation of the axis of symmetry.
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19. y=2x-42-3
20. x=-3y+22-6
21. y=12x+62+5
22. x=34y-52+7
23. y=-x-62-8
24. x=-18y+52
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Homework
What is the vertex of the parabola?
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25. y=(x+3)2+7
26. y=-2(x+4)2+8
27. x=6y-32-5
28. x=23y+82-10
29. y=(x-12)2-11
30. y=2(x-3)2
31. y=-4(x)2+5
32. x=23y2
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Write the following equations in standard form.
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33. y=x2+6x
34. x=y2-10y
35. y=x2-4x+11
36. x=y2+8y+12
37. y=x2+16x+49
38. x=-y2-8y+8
39. y=2x2+8x
40. x=3y2-9y
41. y=-5x2+10x+16
42. x=-2y2-12y-30
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Graph each of the following. State the direction of the opening. Identify vertex and the focus and give the equation of the axis of symmetry.
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43. y=8x-22-4
44. x=-5y+12-7
45. y=-14x+92-8
46. x=-312y-22+1
47. y=2x2-8
48. x=38y+62
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Circles
WORD or CONCEPT / DEFINITION or NOTES / EXAMPLE or GRAPHIC REPRESENTATIONcenter
radius
diameter
tangent
Standard Form
EXAMPLE: x-62+y-152=40
Center:
Radius:
Focal Point(s):
Write the standard form of the equation.
center (-2, -4) radius 9
INSTRUCTION: KHAN ACADEMY EQUATION of a CIRCLE INSTRUCTION 2: SOPHIA
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Class Work
Name the center and radius of each circle.
49. x+22+y-42=16
50. x-32+y-72=25
51. x2+y+82=1
52. x-72+y+12=17
53. x+62+y2=32
Write the standard form of the equation.
54. center (3,2) radius 6
55. center (-4, -7) radius 8
56. center (5, -9) radius 10
57. center (-8, 0) diameter 14
58. center (4,5) and point on the circle (3, -7)
59. diameter with endpoints (6, 4) and (10, -8)
60. center (4, 9) and tangent to the x-axis
61. x2+4x+y2-8y=11
62. x2-10x+y2+2y=11
63. x2+7x+y2=11
Homework
Name the center and radius of each circle.
64. x-92+y+52=9
65. x+112+y-82=64
66. x+132+y-32=144
67. x-22+y2=19
68. x-62+y-152=40
Write the standard form of the equation.
69. center (-2, -4) radius 9
70. center (-3, 3) radius 11
71. center (5, 8) radius 12
72. center (0 , 8) diameter 16
73. center (-4,6) and point on the circle (-2, -8)
74. diameter with endpoints (5, 14) and (11, -8)
75. center (4, 9) and tangent to the y-axis
76. x2-2x+y2+10y=11
77. x2+12x+y2+20y=11
78. 4x2+16x+4y2-8y=12
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Ellipses
WORD or CONCEPT / DEFINITION or NOTES / EXAMPLE or GRAPHIC REPRESENTATIONellipse /
center
vertices
focci
major axis
minor axis
Standard Form
INSTRUCTION1: KHAN ACADEMY
INSTRUCTION 2: SOPHIA
a. Identify the ellipse’s center and foci.
b. State the length of the major and minor axes.
c. Graph the ellipse.
92. x+5216+y-429=1
Write the equation of the ellipse in standard form.
x2+10x+2y2-12y=-1
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Class Work
a. Identify the ellipse’s center and foci.
b. State the length of the major and minor axes.
c. Graph the ellipse.
79.. x-224+y+3216=1
80. x-129+y-421=1
81.x225+y+5236=1
82.x+4216+y+228=1
83.x+126+y-1220=1
Write the equation of the ellipse in standard form.
86. x2+4x+2y2-8y=20
87. 4x2-8x+3y2+18y=5
84. Center (1,4), a horizontal major axis of 10 and a minor axis of 6.
85. Foci (2,5) and (2,11) with a minor axis of 10
86. Foci (-2,4) and (-6,4) with a major axis of 18
Homework
d. Identify the ellipse’s center and foci.
e. State the length of the major and minor axes.
f. Graph the ellipse.
87. x+5216+y-429=1
88. x-724+y+1249=1
89. x-2225+y264=1
90. x21+y24=1
91. x+1236+y-1218=1
Write the equation of the ellipse in standard form.
92. x2+10x+2y2-12y=-1
93. 3x2-12x+4y2+16y=8
94. Center (-1,2), a vertical major axis of 8 and a minor axis of 4.
95. Foci (3, 5) and (3,11) with a minor axis of 8
96. Foci (-2, 6) and (-8, 6) with a major axis of 1
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Hyperbolas
WORD or CONCEPT / DEFINITION or NOTES / EXAMPLE or GRAPHIC REPRESENTATIONhyperbola /
center
vertices
focci
major axis
minor axis
asymptotes
Standard Form
INSTRUCTION 1: KHAN ACADEMY INSRTUCTION 2: SOPHIA
a. Write the equation of the hyperbola in standard form.
4y2-24y-5x2+20x=4
b. Graph the hyperbola
Class Work
Graph each of the following hyperbolas. Write the equations of the asymptotes.
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97. y+5216-x-429=1
98. x-724-y+1249=1
99. y-2225-x264=1
100. x21-y24=1
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101. y+1236-x-1218=1
Write the equation of the hyperbola in standard form.
102. x2+4x-2y2-8y=20
103. 3y2+18y-4x2-8x=1
104. Opens horizontally, with center (3,7) and asymptotes with slope m=±25
105. Opens vertically, with asymptotes y=32x+8 and y=-32x-4
Homework
Graph each of the following hyperbolas. Write the equations of the asymptotes.
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106. x-224-y+3216=1
107. y-129-x-421=1
108. x225-y+5236=1
109. y+4216-x+228=1
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110. y-629-x+5230=1
Write the equation of the hyperbola in standard form.
111. 4y2-24y-5x2+20x=4
112. 6y2+36y-x2-14x=1
113. Opens vertically, with center (-4,1) and asymptotes with slope m=±37
114. Opens horizontally, with asymptotes y=49x+10 and y=-49x-14
Conic Sections Unit Review
Multiple Choice
1. What is the vertex of the parabola x=-23y-92+2
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a. (9,-2)
b. (-2,2)
c. (2,-2)
d. (2,9)
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2. Write the following equations in standard form x=2y2+12y+2
a. x=2x+62+2
b. x=2x+32-7
c. x=2x+32-10
d. x=2x+32-16
3. Identify the focus of x=-216y-32+2
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a. F(0,3)
b. F(4,3)
c. F(2,1)
d. F(2,5)
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4. Write the equation of the parabola with vertex (4,-2) and focus (4,4).
a. y=116x-42-2
b. y=18x-42-2
c. y=124x-42-2
d. x=112y+22+4
5. What are the center and the radius of the following circle: x-72+y+62=4
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a. (-7,6); r=4
b. (7,-6); r=16
c. (-7,6); r= 8
d. (7,-6); r= 2
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6. Write the equation of the circle with a diameter with endpoints (6, 12) and (17, -8).
a. x-112+y-62=521
b. x-112+y+62=22.8
c. x-112+y-22=521
d. x-112+y-22=22.8
7. Identify the ellipse’s center and foci: x+4216+y-1236=1
a. C(-4,1); Foci: -4±20,1
b. C(4,-1); Foci: 4±20,-1
c. C(-4,1); Foci: -4,1±20
d. C(4,-1); Foci: 4,1±20
8. State the length of the major and minor axes of x+4216+y-1236=1
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a. Major: 4; Minor: 6
b. Major: 6; Minor: 4
c. Major: 36; Minor: 16
d. Major: 12; Minor: 8
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9. Write the equation in standard form 4y2-24y-2x2+20x=22
a. y-322-x-524=1
b. y-322-x+524=1
c. y-3227-x-5254=1
d. y-3227-x+5254=1
10. Write the equation in standard form x2+12x+3y2-12y=-1
a. x+62+3(y-2)2=47
b. x+6245+(y-2)215=45
c. x+62+3(y-2)2=23
d. x+6223+3(y-2)223=4
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Extended Response
11. A parabola has vertex (3, 4) and focus (4, 4)
a. What direction does the parabola open? CIRCLE ONE: UP DOWN
b. What is the equation of the axis of symmetry?
c. Write the equation of the parabola.
12. Given the general form of a conic section as Ax2+Bx+Cy2+Dy+E=0
a. What do A & C tell us about the conic?
b. What is center of the conic if A≠0 & C≠0?
13. Consider a circle and a parabola.
a. At how many points can they intersect? ______
b. If the circle has equation x2+y2=4 and the parabola has equation y=x2, what are the point(s) of intersection?
c. If the parabola were reflected over the x-axis, what would be the point(s) of intersection
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