more parabolas ; hyperbolas

1. Hyperbolas and More Parabolas (optional)

2. Hyperbolas and More Parabolas (optional)

3. Hyperbolas and More Parabolas (optional)
Using similar methods to analyze Ax2 + By2 + Cx + Dy = E
we obtain the graphs of hyperbolas and parabolas.

4. Hyperbolas and More Parabolas (optional)
Using similar methods to analyze Ax2 + By2 + Cx + Dy = E
we obtain the graphs of hyperbolas and parabolas.
In the special case B = 0 (or A = 0),

5. Hyperbolas and More Parabolas (optional)
Using similar methods to analyze Ax2 + By2 + Cx + Dy = E
we obtain the graphs of hyperbolas and parabolas.
In the special case B = 0 (or A = 0), after dividing by A (by B),
the equation becomes 1x2 + #x + #y = # or 1y2 + #x + #y = #,

6. Hyperbolas and More Parabolas (optional)
Using similar methods to analyze Ax2 + By2 + Cx + Dy = E
we obtain the graphs of hyperbolas and parabolas.
In the special case B = 0 (or A = 0), after dividing by A (by B),
the equation becomes 1x2 + #x + #y = # or 1y2 + #x + #y = #,

7. Hyperbolas and More Parabolas (optional)
Using similar methods to analyze Ax2 + By2 + Cx + Dy = E
we obtain the graphs of hyperbolas and parabolas.
In the special case B = 0 (or A = 0), after dividing by A (by B),
the equation becomes 1x2 + #x + #y = # or 1y2 + #x + #y = #,
their graphs are parabolas. Assuming both variables x and y
remained in the equation
(
(

8. Hyperbolas and More Parabolas (optional)
Using similar methods to analyze Ax2 + By2 + Cx + Dy = E
we obtain the graphs of hyperbolas and parabolas.
In the special case B = 0 (or A = 0), after dividing by A (by B),
the equation becomes 1x2 + #x + #y = # or 1y2 + #x + #y = #,
their graphs are parabolas.
1x2 + #x + #y = # or
1y2 + #x + #y = #
Parabolas:
Assuming both variables x and y
remained in the equation
(
(

9. Hyperbolas and More Parabolas (optional)
Using similar methods to analyze Ax2 + By2 + Cx + Dy = E
we obtain the graphs of hyperbolas and parabolas.
In the special case B = 0 (or A = 0), after dividing by A (by B),
the equation becomes 1x2 + #x + #y = # or 1y2 + #x + #y = #,
their graphs are parabolas.
1x2 + #x + #y = # or
1y2 + #x + #y = #
If the equation Ax2 + By2 + Cx + Dy = E
has A and B of opposite signs,
Parabolas:
Assuming both variables x and y
remained in the equation
(
(

10. Hyperbolas and More Parabolas (optional)
Using similar methods to analyze Ax2 + By2 + Cx + Dy = E
we obtain the graphs of hyperbolas and parabolas.
In the special case B = 0 (or A = 0), after dividing by A (by B),
the equation becomes 1x2 + #x + #y = # or 1y2 + #x + #y = #,
their graphs are parabolas.
1x2 + #x + #y = # or
1y2 + #x + #y = #
If the equation Ax2 + By2 + Cx + Dy = E
has A and B of opposite signs, after dividing by A, we have 1x2
+ ry2 + #x + #y = #, with r < 0.
Parabolas:
Assuming both variables x and y
remained in the equation
(
(

11. Hyperbolas and More Parabolas (optional)
Using similar methods to analyze Ax2 + By2 + Cx + Dy = E
we obtain the graphs of hyperbolas and parabolas.
In the special case B = 0 (or A = 0), after dividing by A (by B),
the equation becomes 1x2 + #x + #y = # or 1y2 + #x + #y = #,
their graphs are parabolas.
1x2 + #x + #y = # or
1y2 + #x + #y = #
If the equation Ax2 + By2 + Cx + Dy = E
has A and B of opposite signs, after dividing by A, we have 1x2
+ ry2 + #x + #y = #, with r < 0. These are hyperbolas.
Parabolas:
Assuming both variables x and y
remained in the equation
(
(

12. Hyperbolas and More Parabolas (optional)
1x2 + #x + #y = # or
1y2 + #x + #y = #
(r < 0)
Hyperbolas: 1x2 + ry2 + #x + #y = #
If the equation Ax2 + By2 + Cx + Dy = E
has A and B of opposite signs, after dividing by A, we have 1x2
+ ry2 + #x + #y = #, with r < 0. These are hyperbolas.
Parabolas:
Using similar methods to analyze Ax2 + By2 + Cx + Dy = E
we obtain the graphs of hyperbolas and parabolas.
In the special case B = 0 (or A = 0), after dividing by A (by B),
the equation becomes 1x2 + #x + #y = # or 1y2 + #x + #y = #,
their graphs are parabolas. Assuming both variables x and y
remained in the equation
(
(
(in general)

13. Conic Sections
After dividing Ax2 + By2 + Cx + Dy = E by A, we obtain
1x2 + ry2 + #x + #y = #.

14. Conic Sections
After dividing Ax2 + By2 + Cx + Dy = E by A, we obtain
1x2 + ry2 + #x + #y = #. Let’s use the simpler cases 1x2 + ry2 = 1
to study the geometry.

15. Conic Sections
After dividing Ax2 + By2 + Cx + Dy = E by A, we obtain
1x2 + ry2 + #x + #y = #. Let’s use the simpler cases 1x2 + ry2 = 1
to study the geometry. Starting with r = 1, a circle,
circle
r = 1

16. Conic Sections
After dividing Ax2 + By2 + Cx + Dy = E by A, we obtain
1x2 + ry2 + #x + #y = #. Let’s use the simpler cases 1x2 + ry2 = 1
to study the geometry. Starting with r = 1, a circle,
as r becomes smaller = ¼,
circle
r = 1

17. Conic Sections
After dividing Ax2 + By2 + Cx + Dy = E by A, we obtain
1x2 + ry2 + #x + #y = #. Let’s use the simpler cases 1x2 + ry2 = 1
to study the geometry. Starting with r = 1, a circle,
as r becomes smaller = ¼, 1/9, 1/16… ,
circle
r = 1

18. Conic Sections
After dividing Ax2 + By2 + Cx + Dy = E by A, we obtain
1x2 + ry2 + #x + #y = #. Let’s use the simpler cases 1x2 + ry2 = 1
to study the geometry. Starting with r = 1, a circle,
as r becomes smaller = ¼, 1/9, 1/16… ,
circle
r = 1

19. Conic Sections
After dividing Ax2 + By2 + Cx + Dy = E by A, we obtain
1x2 + ry2 + #x + #y = #. Let’s use the simpler cases 1x2 + ry2 = 1
to study the geometry. Starting with r = 1, a circle,
as r becomes smaller = ¼, 1/9, 1/16… ,
1x2 + y2 = 1
1
16
4
1
circle
r = 1

20. Conic Sections
After dividing Ax2 + By2 + Cx + Dy = E by A, we obtain
1x2 + ry2 + #x + #y = #. Let’s use the simpler cases 1x2 + ry2 = 1
to study the geometry. Starting with r = 1, a circle,
as r becomes smaller = ¼, 1/9, 1/16… ,
the circle is elongated to taller and taller ellipses.
circle ellipses
1x2 + y2 = 1
1
16
4
1
r = 1

21. Conic Sections
After dividing Ax2 + By2 + Cx + Dy = E by A, we obtain
1x2 + ry2 + #x + #y = #. Let’s use the simpler cases 1x2 + ry2 = 1
to study the geometry. Starting with r = 1, a circle,
as r becomes smaller = ¼, 1/9, 1/16… ,
the circle is elongated to taller and taller ellipses.
When r = 0, we’ve 1x2 + 0y2 = 1,
circle ellipses
. … r → 0
1x2 + y2 = 1
1
16
r = 1/16
4
1
r = 1

22. Conic Sections
After dividing Ax2 + By2 + Cx + Dy = E by A, we obtain
1x2 + ry2 + #x + #y = #. Let’s use the simpler cases 1x2 + ry2 = 1
to study the geometry. Starting with r = 1, a circle,
as r becomes smaller = ¼, 1/9, 1/16… ,
the circle is elongated to taller and taller ellipses.
When r = 0, we’ve 1x2 + 0y2 = 1, which is a pair of lines x = ±1,
circle ellipses
. … r → 0
1x2 + y2 = 1
1
16
r = 1/16
4
1
r = 1

23. Conic Sections
After dividing Ax2 + By2 + Cx + Dy = E by A, we obtain
1x2 + ry2 + #x + #y = #. Let’s use the simpler cases 1x2 + ry2 = 1
to study the geometry. Starting with r = 1, a circle,
as r becomes smaller = ¼, 1/9, 1/16… ,
the circle is elongated to taller and taller ellipses.
When r = 0, we’ve 1x2 + 0y2 = 1, which is a pair of lines x = ±1,
circle ellipses
. … r → 0
1x2 + y2 = 1
1
16
r = 1/16
4
1
1
1x2 = 1
or
x = ±1
r = 1 r = 0
two lines

24. Conic Sections
After dividing Ax2 + By2 + Cx + Dy = E by A, we obtain
1x2 + ry2 + #x + #y = #. Let’s use the simpler cases 1x2 + ry2 = 1
to study the geometry. Starting with r = 1, a circle,
as r becomes smaller = ¼, 1/9, 1/16… ,
the circle is elongated to taller and taller ellipses.
When r = 0, we’ve 1x2 + 0y2 = 1, which is a pair of lines x = ±1,
i.e. the ellipses are elongated
into two parallel lines.
circle ellipses
. … r → 0
1x2 + y2 = 1
1
16
r = 1/16
4
1
1
1x2 = 1
or
x = ±1
r = 1 r = 0
two lines

25. Hyperbolas
Just as all the other conic sections, hyperbolas are defined
by distance relations.

26. Hyperbolas
Given two fixed points, called foci, a hyperbola is the set
of points whose difference of the distances to the foci is
a constant.
Just as all the other conic sections, hyperbolas are defined
by distance relations.

27. A
If A, B and C are points on a hyperbola as shown
Hyperbolas
Given two fixed points, called foci, a hyperbola is the set
of points whose difference of the distances to the foci is
a constant.
B
C
Just as all the other conic sections, hyperbolas are defined
by distance relations.

28. A
a2
a1
If A, B and C are points on a hyperbola as shown then
a1 – a2
Hyperbolas
Given two fixed points, called foci, a hyperbola is the set
of points whose difference of the distances to the foci is
a constant.
B
C
Just as all the other conic sections, hyperbolas are defined
by distance relations.

29. A
a2
a1
b2
b1
If A, B and C are points on a hyperbola as shown then
a1 – a2 = b1 – b2
Hyperbolas
Given two fixed points, called foci, a hyperbola is the set
of points whose difference of the distances to the foci is
a constant.
B
C
Just as all the other conic sections, hyperbolas are defined
by distance relations.

30. A
a2
a1
b2
b1
If A, B and C are points on a hyperbola as shown then
a1 – a2 = b1 – b2 = c2 – c1 = constant.
c1
c2
Hyperbolas
Given two fixed points, called foci, a hyperbola is the set
of points whose difference of the distances to the foci is
a constant.
B
C
Just as all the other conic sections, hyperbolas are defined
by distance relations.

31. Hyperbolas
A hyperbola has a “center”,

32. Hyperbolas
A hyperbola has a “center”, and two straight lines that
cradle the hyperbolas which are called asymptotes.

33. 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.

34. 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 rectangle with the vertices of the
hyperbola touching the rectangle.

35. 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 rectangle with the vertices of the
hyperbola touching the rectangle.

36. Hyperbolas
The center-rectangle is defined by the x-radius a, and y-
radius b as shown.
a
b

37. Hyperbolas
The center-rectangle 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-rectangle first.
a
b

38. Hyperbolas
The center-rectangle 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-rectangle first.
a
b

39. Hyperbolas
The center-rectangle 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-rectangle first. Draw the
diagonals of the rectangle which are the asymptotes.
a
b

40. Hyperbolas
The center-rectangle 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-rectangle first. Draw the
diagonals of the rectangle which are the asymptotes. Label
the vertices and trace the hyperbola along the asymptotes.
a
b

41. Hyperbolas
a
b
The location of the center, the x-radius a, and y-radius b may
be obtained from the equation.
The center-rectangle 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-rectangle first. Draw the
diagonals of the rectangle which are the asymptotes. Label
the vertices and trace the hyperbola along the asymptotes.

42. Hyperbolas
The equations of hyperbolas have the form
Ax2 + By2 + Cx + Dy = E
where A and B are opposite signs.

43. 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 into the standard forms below.

44. (x – h)2 (y – k)2
a2 b2
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 into the standard forms below.
– = 1 (x – h)2
(y – k)2
a2
b2 – = 1

45. (x – h)2 (y – k)2
a2 b2
(h, k) is the center.
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 into the standard forms below.
– = 1 (x – h)2
(y – k)2
a2
b2 – = 1

46. (x – h)2 (y – k)2
a2 b2
x-rad = a, y-rad = b
(h, k) is the center.
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 into the standard forms below.
– = 1 (x – h)2
(y – k)2
a2
b2 – = 1

47. (x – h)2 (y – k)2
a2 b2
x-rad = a, y-rad = b
(h, k) is the center.
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 into the standard forms below.
– = 1 (x – h)2
(y – k)2
a2
b2
y-rad = b, x-rad = a
– = 1

48. (x – h)2 (y – k)2
a2 b2
x-rad = a, y-rad = b
(h, k) is the center.
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 into the standard forms below.
– = 1 (x – h)2
(y – k)2
a2
b2
y-rad = b, x-rad = a
– = 1
(h, k)
Open in the x direction

49. (x – h)2 (y – k)2
a2 b2
x-rad = a, y-rad = b
(h, k) is the center.
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 into the standard forms below.
– = 1 (x – h)2
(y – k)2
a2
b2
y-rad = b, x-rad = a
– = 1
(h, k)
Open in the x direction
(h, k)
Open in the y direction

50. Hyperbolas
Following are the steps for graphing a hyperbola.

51. Hyperbolas
Following are the steps for graphing a hyperbola.
1. Put the equation into the standard form.

52. 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-rectangle.

53. 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-rectangle.
3. Draw the diagonals of the rectangle, which are the
asymptotes.

54. 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-rectangle.
3. Draw the diagonals of the rectangle, which are the
asymptotes.
4. Determine the direction of the hyperbolas and label the
vertices of the hyperbola.

55. 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-rectangle.
3. Draw the diagonals of the rectangle, 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-rectangle.

56. 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-rectangle.
3. Draw the diagonals of the rectangle, 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-rectangle.
5. Trace the hyperbola along the asymptotes.

57. Hyperbolas
Example A. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
(x – 3)2 (y + 1)2
42 22
– = 1

58. Center: (3, -1)
Hyperbolas
Example A. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
(x – 3)2 (y + 1)2
42 22
– = 1

59. Center: (3, -1)
x-rad = 4
y-rad = 2
Hyperbolas
Example A. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
(x – 3)2 (y + 1)2
42 22
– = 1

60. Center: (3, -1)
x-rad = 4
y-rad = 2
Hyperbolas
(3, -1)
4
2
Example A. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
(x – 3)2 (y + 1)2
42 22
– = 1

61. Center: (3, -1)
x-rad = 4
y-rad = 2
Hyperbolas
(3, -1)
4
2
Example A. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
(x – 3)2 (y + 1)2
42 22
– = 1

62. Center: (3, -1)
x-rad = 4
y-rad = 2
The hyperbola opens
left-rt
Hyperbolas
Example A. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
(x – 3)2 (y + 1)2
42 22
– = 1
(3, -1)
4
2

63. Center: (3, -1)
x-rad = 4
y-rad = 2
The hyperbola opens
left-rt and the vertices
are (7, -1), (-1, -1) .
Hyperbolas
Example A. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
(x – 3)2 (y + 1)2
42 22
– = 1
(3, -1)
4
2

64. Center: (3, -1)
x-rad = 4
y-rad = 2
The hyperbola opens
left-rt and the vertices
are (7, -1), (-1, -1) .
Hyperbolas
(3, -1)
(7, -1)
(-1, -1) 4
2
Example A. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
(x – 3)2 (y + 1)2
42 22
– = 1

65. Center: (3, -1)
x-rad = 4
y-rad = 2
The hyperbola opens
left-rt and the vertices
are (7, -1), (-1, -1) .
Hyperbolas
(3, -1)
(7, -1)
(-1, -1) 4
2
Example A. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
(x – 3)2 (y + 1)2
42 22
– = 1

66. Center: (3, -1)
x-rad = 4
y-rad = 2
The hyperbola opens
left-rt and the vertices
are (7, -1), (-1, -1) .
Hyperbolas
(3, -1)
(7, -1)
(-1, -1) 4
2
Example A. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
(x – 3)2 (y + 1)2
42 22
– = 1
When we use completing the square to get to the standard
form of the hyperbolas, depending on the signs,
we add a number or subtract a number from both sides.

67. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
Hyperbolas

68. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
Group the x’s and the y’s:
Hyperbolas

69. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
Group the x’s and the y’s:
4y2 – 16y – 9x2 – 18x = 29
Hyperbolas

70. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
Group the x’s and the y’s:
4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients
Hyperbolas

71. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
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
Hyperbolas

72. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
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
Hyperbolas

73. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
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
Hyperbolas

74. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
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
Hyperbolas

75. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
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
16
Hyperbolas

76. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
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
Hyperbolas

77. 4(y – 2)2 – 9(x + 1)2 = 36
Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
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
Hyperbolas

78. 4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1
Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
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
Hyperbolas

79. 9(x + 1)2
4(y – 2)2
36 36
4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1
– = 1
Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
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
Hyperbolas

80. 9(x + 1)2
4(y – 2)2
36 36
4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1
– = 1
Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
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
Hyperbolas
9

81. 9(x + 1)2
4(y – 2)2
36 36
4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1
– = 1
Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
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
Hyperbolas
9 4

82. 9(x + 1)2
4(y – 2)2
36 36
4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1
– = 1
Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
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
Hyperbolas
(y – 2)2 (x + 1)2
32 22
– = 1
9 4

83. 9(x + 1)2
4(y – 2)2
36 36
4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1
– = 1
Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard
form. List the center, the x-radius, the y-radius.
Draw the rectangle, the asymptotes, and label the vertices.
Trace the hyperbola.
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
Hyperbolas
(y – 2)2 (x + 1)2
32 22
– = 1
Center: (-1, 2), x-rad = 2, y-rad = 3
The hyperbola opens up and down.
9 4

84. (-1, 2)
Hyperbolas
Center: (-1, 2),
x-rad = 2,
y-rad = 3

85. (-1, 2)
(-1, 5)
(-1, -1)
Hyperbolas
Center: (-1, 2),
x-rad = 2,
y-rad = 3
The hyperbola opens up and down.
The vertices are (-1, -1) and (-1, 5).

86. (-1, 2)
(-1, 5)
(-1, -1)
Hyperbolas
Center: (-1, 2),
x-rad = 2,
y-rad = 3
The hyperbola opens up and down.
The vertices are (-1, -1) and (-1, 5).

87. Hyperbolas

88. More Graphs of Parabolas

89. The graphs of the equations of the form
y = ax2 + bx + c and x = ay2 + bx + c
are parabolas.
More Graphs of Parabolas

90. Each parabola has a vertex and the center line that contains
the vertex.
The graphs of the equations of the form
y = ax2 + bx + c and x = ay2 + bx + c
are parabolas.
More Graphs of Parabolas

91. Each parabola has a vertex and the center line that contains
the vertex.
The graphs of the equations of the form
y = ax2 + bx + c and x = ay2 + bx + c
are parabolas.
More Graphs of Parabolas

92. Each parabola has a vertex and the center line that contains
the vertex. Suppose we know another point on the parabola,
The graphs of the equations of the form
y = ax2 + bx + c and x = ay2 + bx + c
are parabolas.
More Graphs of Parabolas

93. Each parabola has a vertex and the center line that contains
the vertex. Suppose we know another point on the parabola,
The graphs of the equations of the form
y = ax2 + bx + c and x = ay2 + bx + c
are parabolas.
More Graphs of Parabolas

94. Each parabola has a vertex and the center line that contains
the vertex. Suppose we know another point on the parabola,
then the reflection of the point across the center line is also
on the parabola.
The graphs of the equations of the form
y = ax2 + bx + c and x = ay2 + bx + c
are parabolas.
More Graphs of Parabolas

95. Each parabola has a vertex and the center line that contains
the vertex. Suppose we know another point on the parabola,
then the reflection of the point across the center line is also
on the parabola.
The graphs of the equations of the form
y = ax2 + bx + c and x = ay2 + bx + c
are parabolas.
More Graphs of Parabolas

96. Each parabola has a vertex and the center line that contains
the vertex. Suppose we know another point on the parabola,
then the reflection of the point across the center line is also
on the parabola. There is exactly one parabola that goes
through these three points.
The graphs of the equations of the form
y = ax2 + bx + c and x = ay2 + bx + c
are parabolas.
More Graphs of Parabolas

97. Each parabola has a vertex and the center line that contains
the vertex. Suppose we know another point on the parabola,
then the reflection of the point across the center line is also
on the parabola. There is exactly one parabola that goes
through these three points.
The graphs of the equations of the form
y = ax2 + bx + c and x = ay2 + bx + c
are parabolas.
More Graphs of Parabolas

98. When graphing parabolas, we must also give the x-intercepts
and the y-intercept.
More Graphs of Parabolas

99. When graphing parabolas, we must also give the x-intercepts
and the y-intercept.
The y-intercept is obtained by setting x = 0 and solve for y.
More Graphs of Parabolas

100. When graphing parabolas, we must also give the x-intercepts
and the y-intercept.
The y-intercept is obtained by setting x = 0 and solve for y.
The x-intercept is obtained by setting y = 0 and solve for x.
More Graphs of Parabolas

101. When graphing parabolas, we must also give the x-intercepts
and the y-intercept.
The y-intercept is obtained by setting x = 0 and solve for y.
The x-intercept is obtained by setting y = 0 and solve for x.
More Graphs of Parabolas
The graphs of y = ax2 + bx = c are up-down parabolas.

102. When graphing parabolas, we must also give the x-intercepts
and the y-intercept.
The y-intercept is obtained by setting x = 0 and solve for y.
The x-intercept is obtained by setting y = 0 and solve for x.
More Graphs of Parabolas
The graphs of y = ax2 + bx = c are up-down parabolas.
If a > 0, the parabola opens up.
a > 0

103. When graphing parabolas, we must also give the x-intercepts
and the y-intercept.
The y-intercept is obtained by setting x = 0 and solve for y.
The x-intercept is obtained by setting y = 0 and solve for x.
More Graphs of Parabolas
The graphs of y = ax2 + bx = c are up-down parabolas.
If a > 0, the parabola opens up.
If a < 0, the parabola opens down.
a > 0 a < 0

104. When graphing parabolas, we must also give the x-intercepts
and the y-intercept.
The y-intercept is obtained by setting x = 0 and solve for y.
The x-intercept is obtained by setting y = 0 and solve for x.
More Graphs of Parabolas
The graphs of y = ax2 + bx = c are up-down parabolas.
If a > 0, the parabola opens up.
If a < 0, the parabola opens down.
a > 0 a < 0
Vertex Formula (up-down parabolas) The x-coordinate of
the vertex of the parabola y = ax2 + bx + c is at x = .
-b
2a

105. More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.

106. More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
-b
2a

107. More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
-b
2a

108. More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
3. Locate its reflection across the center line, these three
points form the tip of the parabola. Trace the parabola.
-b
2a

109. More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
3. Locate its reflection across the center line, these three
points form the tip of the parabola. Trace the parabola.
4. Set y = 0 and solve to find the x intercept.
-b
2a

110. Example A. Graph y = –x2 + 2x + 15
More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
3. Locate its reflection across the center line, these three
points form the tip of the parabola. Trace the parabola.
4. Set y = 0 and solve to find the x intercept.
-b
2a

111. The vertex is at x = = 1
Example A. Graph y = –x2 + 2x + 15
–(2)
2(–1)
More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
3. Locate its reflection across the center line, these three
points form the tip of the parabola. Trace the parabola.
4. Set y = 0 and solve to find the x intercept.
-b
2a

112. The vertex is at x = = 1y = 16.
Example A. Graph y = –x2 + 2x + 15
–(2)
2(–1)
More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
3. Locate its reflection across the center line, these three
points form the tip of the parabola. Trace the parabola.
4. Set y = 0 and solve to find the x intercept.
-b
2a

113. The vertex is at x = = 1y = 16.
Example A. Graph y = –x2 + 2x + 15 (1, 16)
–(2)
2(–1)
More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
3. Locate its reflection across the center line, these three
points form the tip of the parabola. Trace the parabola.
4. Set y = 0 and solve to find the x intercept.
-b
2a

114. The vertex is at x = = 1y = 16.
The y-intercept is at (0, 15)
Example A. Graph y = –x2 + 2x + 15
–(2)
2(–1)
More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
3. Locate its reflection across the center line, these three
points form the tip of the parabola. Trace the parabola.
4. Set y = 0 and solve to find the x intercept.
-b
2a
(1, 16)

115. The vertex is at x = = 1y = 16.
The y-intercept is at (0, 15)
Example A. Graph y = –x2 + 2x + 15
–(2)
2(–1)
More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
3. Locate its reflection across the center line, these three
points form the tip of the parabola. Trace the parabola.
4. Set y = 0 and solve to find the x intercept.
-b
2a
(1, 16)
(0, 15)

116. The vertex is at x = = 1y = 16.
The y-intercept is at (0, 15)
Plot its reflection (2, 15).
Example A. Graph y = –x2 + 2x + 15
–(2)
2(–1)
More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
3. Locate its reflection across the center line, these three
points form the tip of the parabola. Trace the parabola.
4. Set y = 0 and solve to find the x intercept.
-b
2a
(1, 16)
(0, 15)

117. The vertex is at x = = 1y = 16.
The y-intercept is at (0, 15)
Plot its reflection (2, 15).
Example A. Graph y = –x2 + 2x + 15
–(2)
2(–1)
More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
3. Locate its reflection across the center line, these three
points form the tip of the parabola. Trace the parabola.
4. Set y = 0 and solve to find the x intercept.
-b
2a
(1, 16)
(0, 15) (2, 15)

118. The vertex is at x = = 1y = 16.
The y-intercept is at (0, 15)
Plot its reflection (2, 15)
Draw,
Example A. Graph y = –x2 + 2x + 15 (1, 16)
(0, 15) (2, 15)
–(2)
2(–1)
More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
3. Locate its reflection across the center line, these three
points form the tip of the parabola. Trace the parabola.
4. Set y = 0 and solve to find the x intercept.
-b
2a

119. The vertex is at x = = 1y = 16.
The y-intercept is at (0, 15)
Plot its reflection (2, 15)
Draw, set y = 0 to get x-int:
Example A. Graph y = –x2 + 2x + 15 (1, 16)
(0, 15) (2, 15)
–(2)
2(–1)
More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
3. Locate its reflection across the center line, these three
points form the tip of the parabola. Trace the parabola.
4. Set y = 0 and solve to find the x intercept.
-b
2a

120. The vertex is at x = = 1y = 16.
The y-intercept is at (0, 15)
Plot its reflection (2, 15)
Draw, set y = 0 to get x-int:
–x2 + 2x + 15 = 0
Example A. Graph y = –x2 + 2x + 15 (1, 16)
(0, 15) (2, 15)
–(2)
2(–1)
More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
3. Locate its reflection across the center line, these three
points form the tip of the parabola. Trace the parabola.
4. Set y = 0 and solve to find the x intercept.
-b
2a

121. The vertex is at x = = 1y = 16.
The y-intercept is at (0, 15)
Plot its reflection (2, 15)
Draw, set y = 0 to get x-int:
–x2 + 2x + 15 = 0
x2 – 2x – 15 = 0
Example A. Graph y = –x2 + 2x + 15 (1, 16)
(0, 15) (2, 15)
–(2)
2(–1)
More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
3. Locate its reflection across the center line, these three
points form the tip of the parabola. Trace the parabola.
4. Set y = 0 and solve to find the x intercept.
-b
2a

122. The vertex is at x = = 1y = 16.
The y-intercept is at (0, 15)
Plot its reflection (2, 15)
Draw, set y = 0 to get x-int:
–x2 + 2x + 15 = 0
x2 – 2x – 15 = 0
(x + 3)(x – 5) = 0
x = –3, x = 5
Example A. Graph y = –x2 + 2x + 15 (1, 16)
(0, 15) (2, 15)
–(2)
2(–1)
More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
3. Locate its reflection across the center line, these three
points form the tip of the parabola. Trace the parabola.
4. Set y = 0 and solve to find the x intercept.
-b
2a

123. The vertex is at x = = 1y = 16.
The y-intercept is at (0, 15)
Plot its reflection (2, 15)
Draw, set y = 0 to get x-int:
–x2 + 2x + 15 = 0
x2 – 2x – 15 = 0
(x + 3)(x – 5) = 0
x = –3, x = 5
Example A. Graph y = –x2 + 2x + 15 (1, 16)
(0, 15) (2, 15)
(-3, 0) (5, 0)
–(2)
2(–1)
More Graphs of Parabolas
Following are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
3. Locate its reflection across the center line, these three
points form the tip of the parabola. Trace the parabola.
4. Set y = 0 and solve to find the x intercept.
-b
2a

124. The graphs of the equations
x = ay2 + by + c
are parabolas that open sideways.
More Graphs of Parabolas

125. The graphs of the equations
x = ay2 + by + c
are parabolas that open sideways.
More Graphs of Parabolas
If a>0, the parabola opens
to the right.

126. The graphs of the equations
x = ay2 + by + c
are parabolas that open sideways.
More Graphs of Parabolas
If a>0, the parabola opens
to the right.
If a<0, the parabola opens
to the left.

127. The graphs of the equations
x = ay2 + by + c
are parabolas that open sideways.
Each sideways parabola is symmetric to a horizontal center
line.
More Graphs of Parabolas
If a>0, the parabola opens
to the right.
If a<0, the parabola opens
to the left.

128. The graphs of the equations
x = ay2 + by + c
are parabolas that open sideways.
Each sideways parabola is symmetric to a horizontal center
line. The vertex of the parabola is on this line.
More Graphs of Parabolas
If a>0, the parabola opens
to the right.
If a<0, the parabola opens
to the left.

129. The graphs of the equations
x = ay2 + by + c
are parabolas that open sideways.
Each sideways parabola is symmetric to a horizontal center
line. The vertex of the parabola is on this line. If we know the
location of the vertex and another point on the parabola, the
parabola is completely determined.
More Graphs of Parabolas
If a>0, the parabola opens
to the right.
If a<0, the parabola opens
to the left.

130. The graphs of the equations
x = ay2 + by + c
are parabolas that open sideways.
Each sideways parabola is symmetric to a horizontal center
line. The vertex of the parabola is on this line. If we know the
location of the vertex and another point on the parabola, the
parabola is completely determined. The vertex formula is the
same as before except it's for the y coordinate.
More Graphs of Parabolas
If a>0, the parabola opens
to the right.
If a<0, the parabola opens
to the left.

131. More Graphs of Parabolas
Vertex Formula (sideways parabolas)
The y coordinate of the vertex of the parabola
x = ay2 + by + c
is at y = .
–b
2a

132. Following are steps to graph the parabola x = ay2 + by + c.
More Graphs of Parabolas
Vertex Formula (sideways parabolas)
The y coordinate of the vertex of the parabola
x = ay2 + by + c
is at y = .
–b
2a

133. Following are steps to graph the parabola x = ay2 + by + c.
1. Set y = in the equation to find the x coordinate of the
vertex.
–b
2a
More Graphs of Parabolas
Vertex Formula (sideways parabolas)
The y coordinate of the vertex of the parabola
x = ay2 + by + c
is at y = .
–b
2a

134. Following are steps to graph the parabola x = ay2 + by + c.
1. Set y = in the equation to find the x coordinate of the
vertex.
2. Find another point; use the x intercept (c, 0) if it's not the
vertex.
–b
2a
More Graphs of Parabolas
Vertex Formula (sideways parabolas)
The y coordinate of the vertex of the parabola
x = ay2 + by + c
is at y = .
–b
2a

135. Following are steps to graph the parabola x = ay2 + by + c.
1. Set y = in the equation to find the x coordinate of the
vertex.
2. Find another point; use the x intercept (c, 0) if it's not the
vertex.
3. Locate the reflection of the point across the horizontal
center line, these three points form the tip of the parabola.
Trace the parabola.
–b
2a
More Graphs of Parabolas
Vertex Formula (sideways parabolas)
The y coordinate of the vertex of the parabola
x = ay2 + by + c
is at y = .
–b
2a

136. Following are steps to graph the parabola x = ay2 + by + c.
1. Set y = in the equation to find the x coordinate of the
vertex.
2. Find another point; use the x intercept (c, 0) if it's not the
vertex.
3. Locate the reflection of the point across the horizontal
center line, these three points form the tip of the parabola.
Trace the parabola.
4. Set x = 0 to find the y intercept.
–b
2a
More Graphs of Parabolas
Vertex Formula (sideways parabolas)
The y coordinate of the vertex of the parabola
x = ay2 + by + c
is at y = .
–b
2a

137. Example B. Graph x = –y2 + 2y + 15
More Graphs of Parabolas

138. Example B. Graph x = –y2 + 2y + 15
Vertex: set y = =1
–(2)
2(–1)
More Graphs of Parabolas

139. Example B. Graph x = –y2 + 2y + 15
Vertex: set y = =1 then x = –(1)2 + 2(1) + 15 = 16
–(2)
2(–1)
More Graphs of Parabolas

140. Example B. Graph x = –y2 + 2y + 15
Vertex: set y = =1 then x = –(1)2 + 2(1) + 15 = 16
–(2)
2(–1)
so v = (16, 1).
More Graphs of Parabolas

141. Example B. Graph x = –y2 + 2y + 15
Vertex: set y = =1 then x = –(1)2 + 2(1) + 15 = 16
–(2)
2(–1)
so v = (16, 1).
Another point:
Set y = 0 then x = 15
or (15, 0).
More Graphs of Parabolas

142. Example B. Graph x = –y2 + 2y + 15
Vertex: set y = =1 then x = –(1)2 + 2(1) + 15 = 16
–(2)
2(–1)
so v = (16, 1).
Another point:
Set y = 0 then x = 15
or (15, 0).
More Graphs of Parabolas
(16, 1)

143. Example B. Graph x = –y2 + 2y + 15
Vertex: set y = =1 then x = –(1)2 + 2(1) + 15 = 16
–(2)
2(–1)
so v = (16, 1).
Another point:
Set y = 0 then x = 15
or (15, 0).
(15, 0)
More Graphs of Parabolas
(16, 1)

144. Example B. Graph x = –y2 + 2y + 15
Vertex: set y = =1 then x = –(1)2 + 2(1) + 15 = 16
–(2)
2(–1)
so v = (16, 1).
Another point:
Set y = 0 then x = 15
or (15, 0).
Plot its reflection.
(15, 0)
More Graphs of Parabolas
(16, 1)

145. Example B. Graph x = –y2 + 2y + 15
Vertex: set y = =1 then x = –(1)2 + 2(1) + 15 = 16
–(2)
2(–1)
so v = (16, 1).
Another point:
Set y = 0 then x = 15
or (15, 0).
Plot its reflection.
It's (15, 2).
(15, 0)
More Graphs of Parabolas
(16, 1)
(15, 2)

146. Example B. Graph x = –y2 + 2y + 15
Vertex: set y = =1 then x = –(1)2 + 2(1) + 15 = 16
–(2)
2(–1)
so v = (16, 1).
Another point:
Set y = 0 then x = 15
or (15, 0).
Plot its reflection.
It's (15, 2).
Draw. (15, 0)
More Graphs of Parabolas
(16, 1)
(15, 2)

147. Example B. Graph x = –y2 + 2y + 15
Vertex: set y = =1 then x = –(1)2 + 2(1) + 15 = 16
–(2)
2(–1)
so v = (16, 1).
Another point:
Set y = 0 then x = 15
or (15, 0).
Plot its reflection.
It's (15, 2)
Draw. (15, 0)
(15, 2)
More Graphs of Parabolas
(16, 1)

148. Example B. Graph x = –y2 + 2y + 15
Vertex: set y = =1 then x = –(1)2 + 2(1) + 15 = 16
–(2)
2(–1)
so v = (16, 1).
Another point:
Set y = 0 then x = 15
or (15, 0).
Plot its reflection.
It's (15, 2)
Draw. Get y-int:
–y2 + 2y + 15 = 0
(15, 0)
(15, 2)
More Graphs of Parabolas
(16, 1)

149. Example B. Graph x = –y2 + 2y + 15
Vertex: set y = =1 then x = –(1)2 + 2(1) + 15 = 16
–(2)
2(–1)
so v = (16, 1).
Another point:
Set y = 0 then x = 15
or (15, 0).
Plot its reflection.
It's (15, 2)
Draw. Get y-int:
–y2 + 2y + 15 = 0
y2 – 2y – 15 = 0
(y – 5) (y + 3) = 0
y = 5, -3
(15, 0)
(15, 2)
More Graphs of Parabolas
(16, 1)

150. Example B. Graph x = –y2 + 2y + 15
Vertex: set y = =1 then x = –(1)2 + 2(1) + 15 = 16
–(2)
2(–1)
so v = (16, 1).
Another point:
Set y = 0 then x = 15
or (15, 0).
Plot its reflection.
It's (15, 2)
Draw. Get y-int:
–y2 + 2y + 15 = 0
y2 – 2y – 15 = 0
(y – 5) (y + 3) = 0
y = 5, -3
(15, 0)
(15, 2)
More Graphs of Parabolas
(16, 1)
(0, -3)

151. When graphing parabolas, we must also give the x-intercepts
and the y-intercept.
The y-intercept is (o, c) obtained by setting x = 0.
The x-intercept is obtained by setting y = 0 and solve the
equation 0 = ax2 + bx + c which may or may not have real
number solutions. Hence there might not be any x-intercept.
More Graphs of Parabolas
Following are the steps to graph a parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.
2. Find another point, use the y-intercept (0, c) if possible.
3. Locate its reflection across the center line, these three
points form the tip of the parabola. Trace the parabola.
4. Set y = 0 and solve to find the x intercept.
-b
2a
The graph of y = ax2 + bx = c are up-down parabolas.
If a > 0, the parabola opens up.
If a < 0, the parabola opens down.

152. More Graphs of Parabolas
Exercise A. Graph the following parabolas. Identify which
direction the parabolas face, determine the vertices using
the vertex method. Label the x and y intercepts, if any.
1. x = –y2 – 2y + 15 2. y = x2 – 2x – 15
3. y = x2 + 2x – 15 4. x = –y2 + 2y + 15
5. x = –y2 – 4y + 12 6. y = x2 – 4x – 21
7. y = x2 + 4x – 12 8. x = –y2 + 4y + 21
9. x = –y2 + 4y – 4 10. y = x2 – 4x + 4
11. x = –y2 + 4y – 4 12. y = x2 – 4x + 4
13. y = –x2 – 4x – 4 14. x = –y2 – 4y – 4
15. x = –y2 + 6y – 40 16. y = x2 – 6x – 40
17. y = –x2 – 8x + 48 18. x = y2 – 8y – 48
19. x = –y2 + 4y – 10 20. y = x2 – 4x – 2
21. y = –x2 – 4x – 8 22. x = –y2 – 4y – 5

153. Ellipses
B. Complete the square of the following hyperbola-equations
if needed. Find the centers and the radii of the hyperbolas.
Draw and label the vertices.
1. x2 – 4y2 = 1 2. 9x2 – 4y2 = 1
3. 4x2 – y2/9 = 1 4. x2/4 – y2/9 = 1
5. 0.04x2 – 0.09y2 = 1 6. 2.25x2 – 0.25y2 = 1
7. x2 – 4y2 = 100 8. x2 – 49y2 = 36
9. 4x2 – y2/9 = 9 10. x2/4 – 9y2 = 100
13. (x – 4)2 – 9(y + 1)2 = 25
17. y2 – 8x – 4x2 + 24y = 29
15. x2 – 6x – 25y2 = 27
18. 9y2 – 18y – 25x2 + 100x = 116
11. x2 – 4y2 + 8y = 5 12. y2 – 8x – 4x2 + 24y = 29
14. 9(x + 2)2 – 4(y + 1)2 = 36
16. x2 – 25y2 – 100y = 200

154. More Graphs of Parabolas
1.
(0, 3)
(0, -5)
(16, –1)
3.
(3, 0)
(-5, 0)
(-1, -16)
5.
(0, 2)
(0, -6)
(16, -2)
(2, 0)
(-6, 0)
(-2, -16)
7.
(Answers to odd problems) Exercise A.

155. More Graphs of Parabolas
9.
(-16, -2)
(-16, 6)
(0, 2)
11.
(-2, 0)
(0, -4)
13.
(-16, -2)
(-16, 6)
(0, 2)
15.
(-31, 3)
(-40, 0)

156. More Graphs of Parabolas
17. 19.
21.
(-4, 64)
(-12, 0) (4, 0)
(-10, 0)
(-6, 2)
(-2, -4)
(0, -8)

157. 1. Center: (0, 0)
x radius: 1
y radius: 1/2
Exercise B.
(-1, 0) (1, 0)
3. Center: (0, 0)
x radius: 1/2
y radius: 3
(0.5, 0)
(-0.5, 0)
Ellipses
5. Center: (0, 0)
x radius: 5
y radius: 3.33
(5, 0)
(-5, 0)
7. Center: (0, 0)
x radius: 10
y radius: 5
(10, 0)
(-10, 0)

158. 9. Center: (0, 0)
x radius: 1.5
y radius: 9
11. Center: (0, 1)
x radius: 1
y radius: 1/2
Ellipses
13. Center: (4, -1)
x radius: 5
y radius: 9/3
15. Center: (3, 0)
x radius:
y radius:
(-1.5, 0) (1.5, 0) (1, 1)
(-1, 1)
(5, 0)
(-5, 0) (9, 0)
(-3, 0)

159. 17. Center: (-1,-12)
x radius: 6.5
y radius: 13
Ellipses
(-1, 1)
(-1, -25)