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

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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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CONIC SECTION FORMULAS

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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

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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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