4.1 stem hyperbolas
- 1. Hyperbolas
- 2. Hyperbolas Just as all the other conic sections, hyperbolas are defined by distance relations.
- 3. Hyperbolas Just as all the 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 all the 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 all the 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 all the 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 all the 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 has a “center”,
- 9. Hyperbolas A hyperbola has a “center”, and two straight lines that cradle the hyperbolas which are called asymptotes.
- 10. Hyperbolas A hyperbola has a “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 has a “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 has a “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 is defined by the x-radius a, and y-radius b as shown. b a
- 14. Hyperbolas The center-box is defined 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 is defined 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 is defined 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 is defined 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 of hyperbolas have the form Ax2 + By2 + Cx + Dy = E where A and B are opposite signs.
- 19. Hyperbolas The equations of hyperbolas 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 of hyperbolas 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 of hyperbolas 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 of hyperbolas 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 of hyperbolas 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 of hyperbolas 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 of hyperbolas 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 the steps for graphing a hyperbola.
- 27. Hyperbolas Following are the steps for graphing a hyperbola. 1. Put the equation into the standard form.
- 28. Hyperbolas Following 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. Hyperbolas Following 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. Hyperbolas Following 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. Hyperbolas Following 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. Hyperbolas Following 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. Hyperbolas Example 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. Hyperbolas Example 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 Center: (3, -1)
- 35. Hyperbolas Example 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 Center: (3, -1) x-rad = 4 y-rad = 2
- 36. Hyperbolas Example 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 Center: (3, -1) 2 x-rad = 4 4 y-rad = 2 (3, -1)
- 37. Hyperbolas Example 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 Center: (3, -1) 2 x-rad = 4 4 y-rad = 2 (3, -1)
- 38. Hyperbolas Example 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 22 Center: (3, -1) 2 x-rad = 4 4 y-rad = 2 (3, -1) The hyperbola opens left-rt
- 39. Hyperbolas Example 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 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. 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 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. 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 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. 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 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. Put 4y2 – 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. Put 4y2 – 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. Put 4y2 – 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. Put 4y2 – 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. Put 4y2 – 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. Put 4y2 – 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. Put 4y2 – 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. Put 4y2 – 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. Put 4y2 – 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. Put 4y2 – 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. Put 4y2 – 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. Put 4y2 – 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. Put 4y2 – 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. Put 4y2 – 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. Put 4y2 – 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. Put 4y2 – 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. Put 4y2 – 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. Put 4y2 – 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. 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. Hyperbolas Exercise B. Given the 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. Given the 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