2.7 more parabolas a& hyperbolas (optional) x
Transcript of 2.7 more parabolas a& hyperbolas (optional) x
Hyperbolas and More Parabolas (optional)
Hyperbolas and More Parabolas (optional)
Hyperbolas and More Parabolas (optional)Using similar methods to analyze Ax2 + By2 + Cx + Dy = E
we obtain the graphs of hyperbolas and parabolas.
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),
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 = #,
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 = #,
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(
(
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(
(
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(
(
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(
(
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(
(
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)
Conic SectionsAfter dividing Ax2 + By2 + Cx + Dy = E by A, we obtain
1x2 + ry2 + #x + #y = #.
Conic SectionsAfter 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.
Conic SectionsAfter 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
Conic SectionsAfter 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
Conic SectionsAfter 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
Conic SectionsAfter 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
Conic SectionsAfter 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 = 11
16
4
1
circle
r = 1
Conic SectionsAfter 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 = 11
16
4
1
r = 1
Conic SectionsAfter 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 = 11
16
r = 1/16
4
1
r = 1
Conic SectionsAfter 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 = 11
16
r = 1/16
4
1
r = 1
Conic SectionsAfter 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 = 11
16
r = 1/16
4
1
1
1x2 = 1or
x = ±1
r = 1 r = 0two lines
Conic SectionsAfter 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 = 11
16
r = 1/16
4
1
1
1x2 = 1or
x = ±1
r = 1 r = 0two lines
HyperbolasJust as all the other conic sections, hyperbolas are defined
by distance relations.
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.
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.
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.
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.
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.
HyperbolasA hyperbola has a “center”,
HyperbolasA hyperbola has a “center”, and two straight lines that
cradle the hyperbolas which are called asymptotes.
HyperbolasA hyperbola has a “center”, and two straight lines that
cradle the hyperbolas which are called asymptotes.
There are two vertices, one for each branch.
HyperbolasA 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.
HyperbolasA 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.
HyperbolasThe center-rectangle is defined by the x-radius a, and y-
radius b as shown.
a
b
HyperbolasThe 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
HyperbolasThe 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
HyperbolasThe 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
HyperbolasThe 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
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.
HyperbolasThe equations of hyperbolas have the form
Ax2 + By2 + Cx + Dy = E
where A and B are opposite signs.
HyperbolasThe 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.
(x – h)2 (y – k)2
a2 b2
HyperbolasThe 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
a2b2 – = 1
(x – h)2 (y – k)2
a2 b2
(h, k) is the center.
HyperbolasThe 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
a2b2 – = 1
(x – h)2 (y – k)2
a2 b2
x-rad = a, y-rad = b
(h, k) is the center.
HyperbolasThe 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
a2b2 – = 1
(x – h)2 (y – k)2
a2 b2
x-rad = a, y-rad = b
(h, k) is the center.
HyperbolasThe 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
a2b2
y-rad = b, x-rad = a
– = 1
(x – h)2 (y – k)2
a2 b2
x-rad = a, y-rad = b
(h, k) is the center.
HyperbolasThe 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
a2b2
y-rad = b, x-rad = a
– = 1
(h, k)
Open in the x direction
(x – h)2 (y – k)2
a2 b2
x-rad = a, y-rad = b
(h, k) is the center.
HyperbolasThe 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
a2b2
y-rad = b, x-rad = a
– = 1
(h, k)
Open in the x direction
(h, k)
Open in the y direction
HyperbolasFollowing are the steps for graphing a hyperbola.
HyperbolasFollowing are the steps for graphing a hyperbola.
1. Put the equation into the standard form.
HyperbolasFollowing are the steps for graphing a hyperbola.
1. Put the equation into the standard form.
2. Read off the center, the x-radius a, the y-radius b, and
draw the center-rectangle.
HyperbolasFollowing are the steps for graphing a hyperbola.
1. Put the equation into the standard form.
2. Read off the center, the x-radius a, the y-radius b, and
draw the center-rectangle.
3. Draw the diagonals of the rectangle, which are the
asymptotes.
HyperbolasFollowing are the steps for graphing a hyperbola.
1. Put the equation into the standard form.
2. Read off the center, the x-radius a, the y-radius b, and
draw the center-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.
HyperbolasFollowing are the steps for graphing a hyperbola.
1. Put the equation into the standard form.
2. Read off the center, the x-radius a, the y-radius b, and
draw the center-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.
HyperbolasFollowing are the steps for graphing a hyperbola.
1. Put the equation into the standard form.
2. Read off the center, the x-radius a, the y-radius b, and
draw the center-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.
HyperbolasExample 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
Center: (3, -1)
HyperbolasExample 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
Center: (3, -1)
x-rad = 4
y-rad = 2
HyperbolasExample 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
Center: (3, -1)
x-rad = 4
y-rad = 2
Hyperbolas
(3, -1)
42
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
Center: (3, -1)
x-rad = 4
y-rad = 2
Hyperbolas
(3, -1)
42
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
Center: (3, -1)
x-rad = 4
y-rad = 2
The hyperbola opens
left-rt
HyperbolasExample 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)
42
Center: (3, -1)
x-rad = 4
y-rad = 2
The hyperbola opens
left-rt and the vertices
are (7, -1), (-1, -1) .
HyperbolasExample 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)
42
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
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
9(x + 1)24(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(x + 1)24(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
9(x + 1)24(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
9(x + 1)24(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
9(x + 1)24(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
(-1, 2)
Hyperbolas
Center: (-1, 2),
x-rad = 2,
y-rad = 3
(-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).
(-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).
Hyperbolas
More Graphs of Parabolas
The graphs of the equations of the form
y = ax2 + bx + c and x = ay2 + bx + c
are parabolas.
More Graphs of Parabolas
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
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
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
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
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
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
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
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
When graphing parabolas, we must also give the x-intercepts
and the y-intercept.
More Graphs of Parabolas
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
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
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.
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
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
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 = .-b2a
More Graphs of ParabolasFollowing are the steps to graph the parabola y = ax2 + bx + c.
More Graphs of ParabolasFollowing are the steps to graph the parabola y = ax2 + bx + c.
1. Set x = in the equation to find the vertex.-b2a
More Graphs of ParabolasFollowing 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.
-b2a
More Graphs of ParabolasFollowing 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.
-b2a
More Graphs of ParabolasFollowing 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.
-b2a
Example A. Graph y = –x2 + 2x + 15
More Graphs of ParabolasFollowing 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.
-b2a
The vertex is at x = = 1
Example A. Graph y = –x2 + 2x + 15 –(2)
2(–1)
More Graphs of ParabolasFollowing 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.
-b2a
The vertex is at x = = 1y = 16.
Example A. Graph y = –x2 + 2x + 15 –(2)
2(–1)
More Graphs of ParabolasFollowing 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.
-b2a
The vertex is at x = = 1y = 16.
Example A. Graph y = –x2 + 2x + 15 (1, 16)–(2)
2(–1)
More Graphs of ParabolasFollowing 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.
-b2a
The vertex is at x = = 1y = 16.
The y-intercept is at (0, 15)
Example A. Graph y = –x2 + 2x + 15 –(2)
2(–1)
More Graphs of ParabolasFollowing 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.
-b2a
(1, 16)
The vertex is at x = = 1y = 16.
The y-intercept is at (0, 15)
Example A. Graph y = –x2 + 2x + 15 –(2)
2(–1)
More Graphs of ParabolasFollowing 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.
-b2a
(1, 16)
(0, 15)
The vertex is at x = = 1y = 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 ParabolasFollowing 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.
-b2a
(1, 16)
(0, 15)
The vertex is at x = = 1y = 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 ParabolasFollowing 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.
-b2a
(1, 16)
(0, 15) (2, 15)
The vertex is at x = = 1y = 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 ParabolasFollowing 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.
-b2a
The vertex is at x = = 1y = 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 ParabolasFollowing 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.
-b2a
The vertex is at x = = 1y = 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 ParabolasFollowing 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.
-b2a
The vertex is at x = = 1y = 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 ParabolasFollowing 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.
-b2a
The vertex is at x = = 1y = 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 ParabolasFollowing 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.
-b2a
The vertex is at x = = 1y = 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 ParabolasFollowing 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.
-b2a
The graphs of the equations
x = ay2 + by + c
are parabolas that open sideways.
More Graphs of Parabolas
The graphs of the equations
x = ay2 + by + c
are parabolas that open sideways.
More Graphs of Parabolas
If a>0, the parabola opensto the right.
The graphs of the equations
x = ay2 + by + c
are parabolas that open sideways.
More Graphs of Parabolas
If a>0, the parabola opensto the right.
If a<0, the parabola opensto the left.
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 opensto the right.
If a<0, the parabola opensto the left.
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 opensto the right.
If a<0, the parabola opensto the left.
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 opensto the right.
If a<0, the parabola opensto the left.
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 opensto the right.
If a<0, the parabola opensto the left.
More Graphs of ParabolasVertex Formula (sideways parabolas)
The y coordinate of the vertex of the parabola x = ay2 + by + c
is at y = .–b2a
Following are steps to graph the parabola x = ay2 + by + c.
More Graphs of ParabolasVertex Formula (sideways parabolas)
The y coordinate of the vertex of the parabola x = ay2 + by + c
is at y = .–b2a
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.
–b2a
More Graphs of ParabolasVertex Formula (sideways parabolas)
The y coordinate of the vertex of the parabola x = ay2 + by + c
is at y = .–b2a
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.
–b2a
More Graphs of ParabolasVertex Formula (sideways parabolas)
The y coordinate of the vertex of the parabola x = ay2 + by + c
is at y = .–b2a
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.
–b2a
More Graphs of ParabolasVertex Formula (sideways parabolas)
The y coordinate of the vertex of the parabola x = ay2 + by + c
is at y = .–b2a
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.
–b2a
More Graphs of ParabolasVertex Formula (sideways parabolas)
The y coordinate of the vertex of the parabola x = ay2 + by + c
is at y = .–b2a
Example B. Graph x = –y2 + 2y + 15
More Graphs of Parabolas
Example B. Graph x = –y2 + 2y + 15
Vertex: set y = =1 –(2)2(–1)
More Graphs of Parabolas
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
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
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
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)
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)
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)
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)
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)
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)
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)
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)
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)
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.
-b2a
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.
More Graphs of ParabolasExercise 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
EllipsesB. 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
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.
More Graphs of Parabolas9.
(-16, -2)
(-16, 6)
(0, 2)
11.
(-2, 0)
(0, -4)
13.
(-16, -2)
(-16, 6)
(0, 2)
15.
(-31, 3)
(-40, 0)
More Graphs of Parabolas17. 19.
21.
(-4, 64)
(-12, 0) (4, 0)(-10, 0)
(-6, 2)
(-2, -4)
(0, -8)
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)
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)
17. Center: (-1,-12)
x radius: 6.5
y radius: 13
Ellipses
(-1, 1)
(-1, -25)