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4.1 stem hyperbolas
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- 2. Hyperbolas Just as allthe other conic sections, hyperbolas are defined by distance relations.
- 3. Hyperbolas Just as allthe other conic sections, hyperbolas are defined by distance relations. Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant.
- 4. Hyperbolas Just as allthe other conic sections, hyperbolas are defined by distance relations. Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant. If A, B and C are points on a hyperbola as shown C A B
- 5. Hyperbolas Just as allthe other conic sections, hyperbolas are defined by distance relations. Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant. If A, B and C are points on a hyperbola as shown then a1 – a2 C A a1 a2 B
- 6. Hyperbolas Just as allthe other conic sections, hyperbolas are defined by distance relations. Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant. If A, B and C are points on a hyperbola as shown then a1 – a2 = b1 – b2 C A a1 a2 b2 B b1
- 7. Hyperbolas Just as allthe other conic sections, hyperbolas are defined by distance relations. Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant. If A, B and C are points on a hyperbola as shown then a1 – a2 = b1 – b2 = c2 – c1 = constant. C c2 A a1 c1 a2 b2 B b1
- 8. Hyperbolas A hyperbola hasa “center”,
- 9. Hyperbolas A hyperbola hasa “center”, and two straight lines that cradle the hyperbolas which are called asymptotes.
- 10. Hyperbolas A hyperbola hasa “center”, and two straight lines that cradle the hyperbolas which are called asymptotes. There are two vertices, one for each branch.
- 11. Hyperbolas A hyperbola hasa “center”, and two straight lines that cradle 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 of the hyperbola touching the box.
- 12. Hyperbolas A hyperbola hasa “center”, and two straight lines that cradle 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 of the hyperbola touching the box.
- 13. Hyperbolas The center-box isdefined by the x-radius a, and y-radius b as shown. b a
- 14. Hyperbolas The center-box isdefined by the x-radius a, and y-radius b as shown. Hence, to graph a hyperbola, we find the center and the center-box first. b a
- 15. Hyperbolas The center-box isdefined by the x-radius a, and y-radius b as shown. Hence, to graph a hyperbola, we find the center and the center-box first. Draw the diagonals of the box which are the asymptotes. b a
- 16. Hyperbolas The center-box isdefined by the x-radius a, and y-radius b as shown. Hence, to graph a hyperbola, we find the center and the center-box first. Draw the diagonals of the box which are the asymptotes. Label the vertices and trace the hyperbola along the asympototes. b a
- 17. Hyperbolas The center-box isdefined by the x-radius a, and y-radius b as shown. Hence, to graph a hyperbola, we find the center and the center-box first. Draw the diagonals of the box which are the asymptotes. Label the vertices and trace the hyperbola along the asympototes. b a The location of the center, the x-radius a, and y-radius b may be obtained from the equation.
- 18. Hyperbolas The equations ofhyperbolas have the form Ax2 + By2 + Cx + Dy = E where A and B are opposite signs.
- 19. Hyperbolas The equations ofhyperbolas have the form Ax2 + By2 + Cx + Dy = E where A and B are opposite signs. By completing the square, they may be transformed to the standard forms below.
- 20. Hyperbolas The equations ofhyperbolas have the form Ax2 + By2 + Cx + Dy = E where 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. Hyperbolas The equations ofhyperbolas have the form Ax2 + By2 + Cx + Dy = E where 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. Hyperbolas The equations ofhyperbolas have the form Ax2 + By2 + Cx + Dy = E where 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. Hyperbolas The equations ofhyperbolas have the form Ax2 + By2 + Cx + Dy = E where 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. Hyperbolas The equations ofhyperbolas have the form Ax2 + By2 + Cx + Dy = E where 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. Hyperbolas The equations ofhyperbolas have the form Ax2 + By2 + Cx + Dy = E where 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. Hyperbolas Following are thesteps for graphing a hyperbola.
- 27. Hyperbolas Following are thesteps for graphing a hyperbola. 1. Put the equation into the standard form.
- 28. Hyperbolas Following are thesteps 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. Hyperbolas Following are thesteps 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. Hyperbolas Following are thesteps 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. Hyperbolas Following are thesteps 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. Hyperbolas Following are thesteps 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. Hyperbolas Example A. Listthe 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. Hyperbolas Example A. Listthe 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 Center: (3, -1)
- 35. Hyperbolas Example A. Listthe 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 Center: (3, -1) x-rad = 4 y-rad = 2
- 36. Hyperbolas Example A. Listthe 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 Center: (3, -1) 2 x-rad = 4 4 y-rad = 2 (3, -1)
- 37. Hyperbolas Example A. Listthe 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 Center: (3, -1) 2 x-rad = 4 4 y-rad = 2 (3, -1)
- 38. Hyperbolas Example A. Listthe 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 22 Center: (3, -1) 2 x-rad = 4 4 y-rad = 2 (3, -1) The hyperbola opens left-rt
- 39. Hyperbolas Example A. Listthe 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 22 Center: (3, -1) 2 x-rad = 4 4 y-rad = 2 (3, -1) The hyperbola opens left-rt and the vertices are (7, -1), (-1, -1) .
- 40. Hyperbolas Example A. Listthe 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 22 Center: (3, -1) 2 x-rad = 4 (-1, -1) 4 (7, -1) y-rad = 2 (3, -1) The hyperbola opens left-rt and the vertices are (7, -1), (-1, -1) .
- 41. Hyperbolas Example A. Listthe 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 22 Center: (3, -1) 2 x-rad = 4 (-1, -1) 4 (7, -1) y-rad = 2 (3, -1) The hyperbola opens left-rt and the vertices are (7, -1), (-1, -1) .
- 42. Hyperbolas Example A. Listthe 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 22 Center: (3, -1) 2 x-rad = 4 (-1, -1) 4 (7, -1) y-rad = 2 (3, -1) The hyperbola opens left-rt and the vertices are (7, -1), (-1, -1) . When we use completing the square to get to the standard form of the hyperbolas, because the signs, we add a number and subtract a number from both sides.
- 43. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points.
- 44. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points. Group the x’s and the y’s:
- 45. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29
- 46. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29
- 47. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square
- 48. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x ) = 29
- 49. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29
- 50. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 16
- 51. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 16 –9
- 52. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9
- 53. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9 4(y – 2)2 – 9(x + 1)2 = 36
- 54. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9 4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1
- 55. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9 4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1 4(y – 2)2 – 9(x + 1)2 = 1 36 36
- 56. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9 4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1 4(y – 2)2 – 9(x + 1)2 = 1 36 9 36 4
- 57. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9 4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1 4(y – 2)2 – 9(x + 1)2 = 1 36 9 36 4 (y – 2)2 – (x + 1)2 = 1 32 22
- 58. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9 4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1 4(y – 2)2 – 9(x + 1)2 = 1 36 9 36 4 (y – 2)2 – (x + 1)2 = 1 32 22 Center: (-1, 2),
- 59. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9 4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1 4(y – 2)2 – 9(x + 1)2 = 1 36 9 36 4 (y – 2)2 – (x + 1)2 = 1 32 22 Center: (-1, 2), x-rad = 2, y-rad = 3
- 60. Hyperbolas Example B. Put4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, and left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9 4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1 4(y – 2)2 – 9(x + 1)2 = 1 36 9 36 4 (y – 2)2 – (x + 1)2 = 1 32 22 Center: (-1, 2), x-rad = 2, y-rad = 3 The hyperbola opens up and down.
- 61. Hyperbolas Center: (-1, 2), x-rad= 2, y-rad = 3 (-1, 2)
- 62. Hyperbolas Center: (-1, 2), x-rad= 2, (-1, 5) y-rad = 3 The hyperbola opens up and down. The vertices are (-1, -1) and (-1, 5). (-1, 2) (-1, -1)
- 63. Hyperbolas Center: (-1, 2), x-rad= 2, (-1, 5) y-rad = 3 The hyperbola opens up and down. The vertices are (-1, -1) and (-1, 5). (-1, 2) (-1, -1)
- 64. Hyperbolas Exercise A. Writethe 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. Hyperbolas Exercise B. Giventhe equations of the hyperbolas find the center and radii. Draw and label the center and the vertices. 7. 1 = x2 – y2 8. 16 = y2 – 4x2 9. 36 = 4y2 – 9x2 10. 100 = 4x2 – 25y2 11. 1 = (y – 2)2 – (x + 3)2 12. 16 = (x – 5)2 – 4(y + 7)2 13. 36 = 4(y – 8)2 – 9(x – 2)2 14. 100 = 4(x – 5)2 – 25(y + 5)2 15. 225 = 25(y + 1)2 – 9(x – 4)2 16. –100 = 4(y – 5)2 – 25(x + 3)2
- 66. Hyperbolas Exercise C. Giventhe equations of the hyperbolas find the center and radii. Draw and label the center and the vertices. 17. x –4y +8y = 5 2 2 18. x2–4y2+8x = 20 20. y –2x–x +4y = 6 2 2 19. 4x –y +8y = 52 2 2 21. x –16y +4y +16x = 16 2 2 22. 4x2–y2+8x–4y = 4 23. y +54x–9x –4y = 86 2 2 24. 4x2+18y–9y2–8x = 41