2. HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.
3. HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.Given two fixed points, called foci, a hyperbola is the setof points whose difference of the distances to the foci isa constant.
4. HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.Given two fixed points, called foci, a hyperbola is the setof points whose difference of the distances to the foci isa constant.If A, B and C are points on a hyperbola as shown C A B
5. HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.Given two fixed points, called foci, a hyperbola is the setof points whose difference of the distances to the foci isa constant.If A, B and C are points on a hyperbola as shown thena1 – a2 C A a1 a2 B
6. HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.Given two fixed points, called foci, a hyperbola is the setof points whose difference of the distances to the foci isa constant.If A, B and C are points on a hyperbola as shown thena1 – a2 = b1 – b2 C A a1 a2 b2 B b1
7. HyperbolasJust as all the other conic sections, hyperbolas are definedby distance relations.Given two fixed points, called foci, a hyperbola is the setof points whose difference of the distances to the foci isa constant.If A, B and C are points on a hyperbola as shown thena1 – a2 = b1 – b2 = c2 – c1 = constant. C c2 A a1 c1 a2 b2 B b1
8. HyperbolasA hyperbola has a “center”,
9. HyperbolasA hyperbola has a “center”, and two straight lines thatcradle the hyperbolas which are called asymptotes.
10. HyperbolasA hyperbola has a “center”, and two straight lines thatcradle the hyperbolas which are called asymptotes.There are two vertices, one for each branch.
11. HyperbolasA hyperbola has a “center”, and two straight lines thatcradle the hyperbolas which are called asymptotes.There are two vertices, one for each branch.The asymptotes are the diagonals of a box with the vertices ofthe hyperbola touching the box.
12. HyperbolasA hyperbola has a “center”, and two straight lines thatcradle the hyperbolas which are called asymptotes.There are two vertices, one for each branch.The asymptotes are the diagonals of a box with the vertices ofthe hyperbola touching the box.
13. HyperbolasThe center-box is defined by the x-radius a, and y-radius bas shown. b a
14. HyperbolasThe center-box is defined by the x-radius a, and y-radius bas shown. Hence, to graph a hyperbola, we find the centerand the center-box first. b a
15. HyperbolasThe center-box is defined by the x-radius a, and y-radius bas shown. Hence, to graph a hyperbola, we find the centerand the center-box first. Draw the diagonals of the boxwhich are the asymptotes. b a
16. HyperbolasThe center-box is defined by the x-radius a, and y-radius bas shown. Hence, to graph a hyperbola, we find the centerand the center-box first. Draw the diagonals of the boxwhich are the asymptotes. Label the vertices and trace thehyperbola along the asympototes. b a
17. HyperbolasThe center-box is defined by the x-radius a, and y-radius bas shown. Hence, to graph a hyperbola, we find the centerand the center-box first. Draw the diagonals of the boxwhich are the asymptotes. Label the vertices and trace thehyperbola along the asympototes. b aThe location of the center, the x-radius a, and y-radius b maybe obtained from the equation.
18. HyperbolasThe equations of hyperbolas have the formAx2 + By2 + Cx + Dy = Ewhere A and B are opposite signs.
19. HyperbolasThe equations of hyperbolas have the formAx2 + By2 + Cx + Dy = Ewhere A and B are opposite signs. By completing the square,they may be transformed to the standard forms below.
20. HyperbolasThe equations of hyperbolas have the formAx2 + By2 + Cx + Dy = Ewhere A and B are opposite signs. By completing the square,they may be transformed to the standard forms below. (x – h)2 (y – k)2 (y – k)2 (x – h)2 a2 – b2 = 1 a2 = 1 – b2
21. HyperbolasThe equations of hyperbolas have the formAx2 + By2 + Cx + Dy = Ewhere A and B are opposite signs. By completing the square,they may be transformed to the standard forms below. (x – h)2 (y – k)2 (y – k)2 (x – h)2 a2 – b2 = 1 a2 = 1 – b2 (h, k) is the center.
22. HyperbolasThe equations of hyperbolas have the formAx2 + By2 + Cx + Dy = Ewhere A and B are opposite signs. By completing the square,they may be transformed to the standard forms below. (x – h)2 (y – k)2 (y – k)2 (x – h)2 a2 – b2 = 1 a2 = 1 – b2 x-rad = a, y-rad = b (h, k) is the center.
23. HyperbolasThe equations of hyperbolas have the formAx2 + By2 + Cx + Dy = Ewhere A and B are opposite signs. By completing the square,they may be transformed to the standard forms below. (x – h)2 (y – k)2 (y – k)2 (x – h)2 a2 – b2 = 1 a2 = 1 – b2 x-rad = a, y-rad = b y-rad = b, x-rad = a (h, k) is the center.
24. HyperbolasThe equations of hyperbolas have the formAx2 + By2 + Cx + Dy = Ewhere A and B are opposite signs. By completing the square,they may be transformed to the standard forms below. (x – h)2 (y – k)2 (y – k)2 (x – h)2 a2 – b2 = 1 a2 = 1 – b2 x-rad = a, y-rad = b y-rad = b, x-rad = a (h, k) is the center. Open in the x direction (h, k)
25. HyperbolasThe equations of hyperbolas have the formAx2 + By2 + Cx + Dy = Ewhere A and B are opposite signs. By completing the square,they may be transformed to the standard forms below. (x – h)2 (y – k)2 (y – k)2 (x – h)2 a2 – b2 = 1 a2 = 1 – b2 x-rad = a, y-rad = b y-rad = b, x-rad = a (h, k) is the center. Open in the x direction Open in the y direction (h, k) (h, k)
26. HyperbolasFollowing are the steps for graphing a hyperbola.
27. HyperbolasFollowing are the steps for graphing a hyperbola.1. Put the equation into the standard form.
28. HyperbolasFollowing are the steps for graphing a hyperbola.1. Put the equation into the standard form.2. Read off the center, the x-radius a, the y-radius b, and draw the center-box.
29. HyperbolasFollowing are the steps for graphing a hyperbola.1. Put the equation into the standard form.2. Read off the center, the x-radius a, the y-radius b, and draw the center-box.3. Draw the diagonals of the box, which are the asymptotes.
30. HyperbolasFollowing are the steps for graphing a hyperbola.1. Put the equation into the standard form.2. Read off the center, the x-radius a, the y-radius b, and draw the center-box.3. Draw the diagonals of the box, which are the asymptotes.4. Determine the direction of the hyperbolas and label the vertices of the hyperbola.
31. HyperbolasFollowing are the steps for graphing a hyperbola.1. Put the equation into the standard form.2. Read off the center, the x-radius a, the y-radius b, and draw the center-box.3. Draw the diagonals of the box, which are the asymptotes.4. Determine the direction of the hyperbolas and label the vertices of the hyperbola. The vertices are the mid-points of the edges of the center-box.
32. HyperbolasFollowing are the steps for graphing a hyperbola.1. Put the equation into the standard form.2. Read off the center, the x-radius a, the y-radius b, and draw the center-box.3. Draw the diagonals of the box, which are the asymptotes.4. Determine the direction of the hyperbolas and label the vertices of the hyperbola. The vertices are the mid-points of the edges of the center-box.5. Trace the hyperbola along the asymptotes.
33. HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 42 22
34. HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 42 22Center: (3, -1)
35. HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 42 22Center: (3, -1)x-rad = 4y-rad = 2
36. HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 42 22Center: (3, -1) 2x-rad = 4 4y-rad = 2 (3, -1)
37. HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 42 22Center: (3, -1) 2x-rad = 4 4y-rad = 2 (3, -1)
38. HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 4 2 22Center: (3, -1) 2x-rad = 4 4y-rad = 2 (3, -1)The hyperbola opensleft-rt
39. HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 4 2 22Center: (3, -1) 2x-rad = 4 4y-rad = 2 (3, -1)The hyperbola opensleft-rt and the verticesare (7, -1), (-1, -1) .
40. HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 4 2 22Center: (3, -1) 2x-rad = 4 (-1, -1) 4 (7, -1)y-rad = 2 (3, -1)The hyperbola opensleft-rt and the verticesare (7, -1), (-1, -1) .
41. HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 4 2 22Center: (3, -1) 2x-rad = 4 (-1, -1) 4 (7, -1)y-rad = 2 (3, -1)The hyperbola opensleft-rt and the verticesare (7, -1), (-1, -1) .
42. HyperbolasExample A. List the center, the x-radius, the y-radius.Draw the box, the asymptotes, and label the vertices.Trace the hyperbola.(x – 3)2 (y + 1)2 – =1 4 2 22Center: (3, -1) 2x-rad = 4 (-1, -1) 4 (7, -1)y-rad = 2 (3, -1)The hyperbola opensleft-rt and the verticesare (7, -1), (-1, -1) .When we use completing the square to get to the standardform of the hyperbolas, because the signs, we add a numberand subtract a number from both sides.
43. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.
44. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:
45. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29
46. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29
47. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square
48. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x ) = 29
49. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29
50. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 16
51. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 16 –9
52. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9
53. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –94(y – 2)2 – 9(x + 1)2 = 36
54. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –94(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1
55. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –94(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 14(y – 2)2 – 9(x + 1)2 = 1 36 36
56. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –94(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 14(y – 2)2 – 9(x + 1)2 = 1 36 9 36 4
57. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –94(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 14(y – 2)2 – 9(x + 1)2 = 1 36 9 36 4 (y – 2)2 – (x + 1)2 = 1 32 22
58. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –94(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 14(y – 2)2 – 9(x + 1)2 = 1 36 9 36 4 (y – 2)2 – (x + 1)2 = 1 32 22Center: (-1, 2),
59. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –94(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 14(y – 2)2 – 9(x + 1)2 = 1 36 9 36 4 (y – 2)2 – (x + 1)2 = 1 32 22Center: (-1, 2), x-rad = 2, y-rad = 3
60. HyperbolasExample B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standardform. Find the center, major and minor axis. Draw and labelthe top, bottom, right, and left most points.Group the x’s and the y’s:4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –94(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 14(y – 2)2 – 9(x + 1)2 = 1 36 9 36 4 (y – 2)2 – (x + 1)2 = 1 32 22Center: (-1, 2), x-rad = 2, y-rad = 3The hyperbola opens up and down.
62. HyperbolasCenter: (-1, 2),x-rad = 2, (-1, 5)y-rad = 3The hyperbola opens up and down.The vertices are (-1, -1) and (-1, 5). (-1, 2) (-1, -1)
63. HyperbolasCenter: (-1, 2),x-rad = 2, (-1, 5)y-rad = 3The hyperbola opens up and down.The vertices are (-1, -1) and (-1, 5). (-1, 2) (-1, -1)
64. HyperbolasExercise A. Write the equation of each hyperbola.1. (4, 2) 2. 3. (2, 4) (–6, –8)4. 5. 6. (5, 3) (2, 4) (–8,–6) (3, 1) (0,0) (2, 4)
65. HyperbolasExercise B. Given the equations of the hyperbolasfind the center and radii. Draw and label the centerand the vertices. 7. 1 = x2 – y2 8. 16 = y2 – 4x2 9. 36 = 4y2 – 9x2 10. 100 = 4x2 – 25y211. 1 = (y – 2)2 – (x + 3)2 12. 16 = (x – 5)2 – 4(y + 7)213. 36 = 4(y – 8)2 – 9(x – 2)214. 100 = 4(x – 5)2 – 25(y + 5)215. 225 = 25(y + 1)2 – 9(x – 4)216. –100 = 4(y – 5)2 – 25(x + 3)2
66. HyperbolasExercise C. Given the equations of the hyperbolasfind the center and radii. Draw and label the centerand the vertices.17. x –4y +8y = 5 2 2 18. x2–4y2+8x = 20 20. y –2x–x +4y = 6 2 219. 4x –y +8y = 52 2 221. x –16y +4y +16x = 16 2 2 22. 4x2–y2+8x–4y = 423. y +54x–9x –4y = 86 2 2 24. 4x2+18y–9y2–8x = 41
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