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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 SchoolWestminster Public Schools
PMI-NJ Center for Teaching & Learning ~1~ NJCTL.org
Conic Sections and Standard Forms of EquationsA conic section is the intersection of a plane and a double right circular cone. By changing the angle and location of the intersection, we can produce different types of conics. There are four basic types: circles,ellipses, hyperbolas and parabolas. 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
where A, B, C, D, E and F are 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. If B2 – 4AC is less than zero, if a conic exists, it will be either a circle or an ellipse. If B2 – 4AC equals zero, if a conic exists, it will be a parabola. If B2 – 4AC is greater than zero, if a conic exists, it will be a hyperbola.
INSTRUCTION 1: KHAN ACADEMY INSTRUCTION 2: SOPHIA
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CIRCLE
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 rEccentricity 0
VERTICAL HORIZONTAL
PARABOLA
General Form ax2 + bx + dy +e =0 cy2 + dy + bx + e=0
Standard Form y = a(x-h)2 + kx = a(y-k)2
+ hOpens UP if a > 0 RIGHT if a > 0
DOWN if a < 0 LEFT if a < 0Axis of Symmetry x = h y = kVertex (h, k) (h, k)Focus (h, k+p) (h+p, k)Directrix y = k-p x = h-p
a = 1 / 4pp = 1 / 4a
Eccentricity 1
VERTICAL HORIZONTAL
ELIPSE
General Form ax2 + bx + cy2 + dy + e = 0
Standard FormCenter (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 axisMinor Axis: y axis x axisLength of Major Axis 2a 2aLength 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 FormCenter (0, 0) (0, 0)Focci (c, 0), (-c, 0) (0, c), (0, -c)Vertices (a, 0), (-a, 0) (0, a), (0, -a)
Asymptotesc2 = a2 + b2 – b2, a > b > 0
Parabolas
WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION
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x2
a2 + y2
b2 =
1
x2
b2 + y2
a2 = 1
x2
a2 - y2
b2 = 1 y2
a2 - x2
b2 = 1
y=± ba x y=± a
b x
parabola
vertex
axis of symmetry
Standard Form
EXAMPLE: 20. x=−3 ( y+2 )2−6
Vertex:
Axis of symmetry:
Parabola Opens:
Focal Point:
INSTRUCTION 1: KHAN ACADEMY INSTRUCTION 2: SOPHIA
Class WorkWhat is the vertex of the parabola?
1. y=¿2. y=−3¿3. x=5 ( y−7 )2−6
4. x=2 ( y+4 )2+95. y=2¿
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6. y=34¿
7. y=−¿
8. x=53
( y+8 )2−3
Write the following equations in standard form.9. y=x2+4 x10. x= y2−8 y11. y=x2−6 x+812. x= y2+2 y+1013. y=x2+10 x−12
14. x= y2−8 y+1615. y=2 x2+12 x16. x=3 y2−6 y17. y=−4 x2+8 x+618. x=−6 y2−12 y+15
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.
19. y=2 ( x−4 )2−320. x=−3 ( y+2 )2−6
21. y=12
( x+6 )2+5
22. x=34
( y−5 )2+7
23. y=−( x−6 )2−8
24. x=−18
( y+5 )2
HomeworkWhat is the vertex of the parabola?
25. y=¿26. y=−2¿27. x=6 ( y−3 )2−5
28. x=23
( y+8 )2−10
29. y=¿30. y=2¿31. y=−4¿
32. x=23
( y )2
Write the following equations in standard form.33. y=x2+6 x34. x= y2−10 y35. y=x2−4 x+1136. x= y2+8 y+1237. y=x2+16 x+49
38. x=− y2−8 y+839. y=2 x2+8 x40. x=3 y2−9 y41. y=−5x2+10 x+1642. x=−2 y2−12 y−30
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.
43. y=8 (x−2 )2−444. x=−5 ( y+1 )2−7
45. y=−14
( x+9 )2−8
46. x=−312
( y−2 )2+1
47. y= (2x )2−8
x=38
( y+6 )2
Circles
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WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION
center
radius
diameter
tangent
Standard Form
EXAMPLE: ( x−6 )2+ ( y−15 )2=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: SOPHIAClass Work
Name the center and radius of each circle.49. ( x+2 )2+ ( y−4 )2=1 6
50. ( x−3 )2+( y−7 )2=2551. ( x )2+( y+8 )2=152. ( x−7 )2+( y+1 )2=17
PMI-NJ Center for Teaching & Learning ~7~ NJCTL.org
53. ( x+6 )2+( y )2=32
Write the standard form of the equation.54. center (3,2) radius 655. center (-4, -7) radius 856. center (5, -9) radius 1057. center (-8, 0) diameter 1458. 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-axis61. x2+4 x+ y2−8 y=1162. x2−10 x+ y2+2 y=1163. x2+7 x+ y2=11
Homework
Name the center and radius of each circle.
64. ( x−9 )2+ ( y+5 )2=965. ( x+11)2+( y−8 )2=6466. ( x+13 )2+( y−3 )2=14467. ( x−2 )2+( y )2=1968. ( x−6 )2+ ( y−15 )2=40
Write the standard form of the equation.69. center (-2, -4) radius 970. center (-3, 3) radius 1171. center (5, 8) radius 1272. center (0 , 8) diameter 1673. 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-axis76. x2−2 x+ y2+10 y=1177. x2+12 x+ y2+20 y=1178. 4 x2+16 x+4 y2−8 y=12
Ellipses
WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION
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ellipse
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+5 )2
16+
( y−4 )2
9=1
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Write the equation of the ellipse in standard form.x2+10 x+2 y2−12 y=−1
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−2 )2
4+
( y+3 )2
16=1
80. ( x−1 )2
9+ ( y−4 )2
1=1
8 1 . ( x )2
25+
( y+5 )2
36=1
82 . ( x+4 )2
16+ ( y+2 )2
8=1
8 3 . ( x+1 )2
6+
( y−1 )2
20=1
Write the equation of the ellipse in standard form.86. x2+4 x+2 y2−8 y=20
87. 4 x2−8x+3 y2+18 y=584. 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 1086. 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+5 )2
16+
( y−4 )2
9=1
88. ( x−7 )2
4+
( y+1 )2
49=1
89. ( x−2 )2
25+
( y )2
64=1
90. ( x )2
1+
( y )2
4=1
91. ( x+1 )2
36+
( y−1 )2
18=1
Write the equation of the ellipse in standard form.92. x2+10 x+2 y2−12 y=−193. 3 x2−12 x+4 y2+16 y=894. 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 896. Foci (-2, 6) and (-8, 6) with a major
axis of 1
Hyperbolas
WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION
hyperbola
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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.
4 y2−24 y−5 x2+20 x=4
b. Graph the hyperbola
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Class Work
Graph each of the following hyperbolas. Write the equations of the asymptotes.
97. ( y+5 )2
16− ( x−4 )2
9=1
98. ( x−7 )2
4− ( y+1 )2
49=1
99. ( y−2 )2
25− (x )2
64=1
100. ( x )2
1− ( y )2
4=1
101. ( y+1 )2
36− ( x−1 )2
18=1
Write the equation of the hyperbola in standard form.102. x2+4 x−2 y2−8 y=20103. 3 y2+18 y−4 x2−8 x=1
104. Opens horizontally, with center (3,7) and asymptotes with slope m=± 25
105. Opens vertically, with asymptotes y=32
x+8 and y=−32
x−4
Homework
Graph each of the following hyperbolas. Write the equations of the asymptotes.
106. ( x−2 )2
4−
( y+3 )2
16=1
107. ( y−1 )2
9− (x−4 )2
1=1
108. ( x )2
25−
( y+5 )2
36=1
109. ( y+4 )2
16− ( x+2 )2
8=1
110. ( y−6 )2
9− ( x+5 )2
30=1
Write the equation of the hyperbola in standard form.111. 4 y2−24 y−5 x2+20 x=4112. 6 y2+36 y−x2−14 x=1
113. Opens vertically, with center (-4,1) and asymptotes with slope m=± 37
114. Opens horizontally, with asymptotes y= 49
x+10 and y=−49
x−14
Conic Sections Unit Review Multiple Choice
1. What is the vertex of the parabola x=−23
( y−9 )2+2
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a. (9,-2)b. (-2,2)
c. (2,-2)d. (2,9)
2. Write the following equations in standard formx=2 y2+12 y+2a. x=2 ( x+6 )2+2b. x=2 ( x+3 )2−7c. x=2 ( x+3 )2−10d. x=2 ( x+3 )2−16
3. Identify the focus of x=−216
( y−3 )2+2
a. F(0,3)b. F(4,3)
c. F(2,1)d. F(2,5)
4. Write the equation of the parabola with vertex (4,-2) and focus (4,4).
a. y= 116
( x−4 )2−2
b. y=18
( x−4 )2−2
c. y= 124
( x−4 )2−2
d. x= 112
( y+2 )2+4
5. What are the center and the radius of the following circle: ( x−7 )2+( y+6 )2=4a. (-7,6); r=4b. (7,-6); r=16
c. (-7,6); r= 8d. (7,-6); r= 2
6. Write the equation of the circle with a diameter with endpoints (6, 12) and (17, -8).a. ( x−11 )2+ ( y−6 )2=521b. ( x−11 )2+ ( y+6 )2=22.8c. ( x−11 )2+ ( y−2 )2=521d. ( x−11 )2+( y−2 )2=22.8
7. Identify the ellipse’s center and foci: ( x+4 )2
16+
( y−1 )2
36=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+4 )2
16+
( y−1 )2
36=1
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a. Major: 4; Minor: 6b. Major: 6; Minor: 4
c. Major: 36; Minor: 16d. Major: 12; Minor: 8
9. Write the equation in standard form 4 y2−24 y−2 x2+20 x=22
a. ( y−3 )2
2−
( x−5 )2
4=1
b. ( y−3 )2
2−
( x+5 )2
4=1
c. ( y−3 )2
27−
( x−5 )2
54=1
d. ( y−3 )2
27−
( x+5 )2
54=1
10. Write the equation in standard form x2+12 x+3 y2−12 y=−1a. ( x+6 )2+3¿
b. ( x+6 )2
45+¿¿
c. ( x+6 )2+3¿
( x+6 )2
23+3 ¿¿
Extended Response11. A parabola has vertex (3, 4) and focus (4, 4)
a. What direction does the parabola open? CIRCLE ONE: UP DOWNb. 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 A x2+Bx+C y2+ Dy+E=0a. 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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