Hyperbolas are defined by the difference between distances to two fixed points called foci. A hyperbola consists of all points where this difference is a constant. It has two branches, two vertices, and two asymptotes which are the diagonals of an invisible box defined by the hyperbola's x-radius and y-radius. To graph a hyperbola, one puts its equation into standard form to determine the center, radii, and direction of opening, then draws the corresponding box and curves.

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