8.1-8.3: Parabolas, Hyperbolas
and Ellipses
Name: ___________________
Objectives:
Students will be able to:
-graph parabolas, hyperbolas and ellipses and answer
characteristic questions about these graphs.
-write equations of conic sections
Dec 1­6:05 PM
B
Ellipses
-Ellipse:
C
D
F1
-Foci:
A
F2
E
-Center:
-Minor Axis:
-Major Axis:
-Vertices:
Apr 4­11:18 AM
1
Standard Form of
the Equation of Orientation
and Ellipse
Description
(x - h)2 + (y - k)2 = 1,
-Center: (h,k)
a2
b2 -Foci: (h±c, k)
where c2 = -Major
a2 - b2. Axis: y = k
(h,k)
-Major Axis
Vertices: (h±a,k)
a is bigger than b.
-Minor Axis: x = h
-Minor Axis
Vertices: (h,k±b)
y=k
x=h
(y - k)2 + (x - h)2 = 1,
a2
Foci:
b2 (h,k±c)
where c2 = -Major
a2 - b2. Axis: x = h
-Major Axis
Vertices: (h,k±a)
a is bigger than b.
-Minor Axis: y = k
-Minor Axis
Vertices: (h±b,k)
-Center: (h,k)
(h,k)
y=k
x=h
Apr 4­11:56 AM
The noun eccentricity comes from the adjective eccentric, which
means off-center. Mathematically, the eccentricity is the ratio of c
to a. The larger c is, compared to a, the more off-center the foci
are. The eccentricity of an ellipse is between 0 and 1.
e = c = √a2 - b2
a
a
Note: c = distance from the center of the ellipse to either focus.
Feb 18­2:17 PM
2
Examples:
1.) Consider the ellipse graphed
at the right.
(2,­4)
(8,­4)
(2,­7)
a.) Write the equation of the ellipse in standard form.
b.) Find the coordinates of the foci.
Apr 5­9:09 AM
2.) For the equation (y - 3) 2 + (x + 4)2 = 1, find the coordinates
25
9
of the center, foci, vertices and eccentricity of the ellipse.
Then graph.
Apr 5­9:15 AM
3
3.) Find the coordinates of the center, foci, vertices
and eccentricity of the ellipse with the equation
4x2 + 9y2 - 40x + 36y + 100 = 0. Then graph the ellipse.
Apr 5­9:17 AM
4.) Find the equation in standard form of the ellipse whose
endpoints of axes are (±7,0) and (0,±4).
Feb 18­2:42 PM
4
asy
Hyperbolas
mp
to
te
center
asy
tote
mp
transverse axis
-Hyperbola:
F1
F2
vertices
-Foci:
conjugate axis
-Center:
-Vertex:
-Asymptotes:
-Transverse Axis:
-Conjugate Axis:
Apr 5­9:21 AM
Standard Form of
the Equation of a OrientationDescription
Hyperbola
(x - h)2 - (y - k)2 = 1
a2
b2
y = k
-Center: (h,k)
-Foci: (h±c,k)
-Vertices: (h±a,k)
-Equation of transverse
axis: y = k
-Asymptotes:
y - k = ±(b/a)(x - h)
(h,k)
a is not necessarily
bigger than b.
x = h
(y - k)2 - (x - h)2 = 1
a2
-Center: (h,k)
b2
a is not necessarily
(h,k)
bigger than b.
-Foci: (h,k±c)
-Vertices: (h,k±a)
y = k
-Equation of transverse
axis: x = h
-Asymptotes:
y - k = ±(a/b)(x - h)
x = h
Apr 5­10:22 AM
5
The eccentricity of a hyperbola is:
e = c = √a2 + b2
a
a
Note: c = distance from the center of the hyperbola to either
focus.
Feb 18­2:34 PM
Examples:
1.) Find the coordinates of the center, the foci, the vertices and
eccentricity and the equa tions of the asymptotes of the hyperbola
whose equation is x2 - y2 = 1. Then graph.
25 4
Apr 5­10:45 AM
6
2.) Find the coordinates of the center, foci, vertices, eccentricity
and the equations of the asymptotes of the graph of
9x2 - 4y2 - 54x - 40y - 55 = 0. Then graph.
Apr 5­11:17 AM
4.) Write the equation in standard form for the hyperbola whose
transverse axis endpoints are (5,3) and (-7,3) and conjugate axis is
length 10.
Feb 18­2:43 PM
7
Parabolas
vertex
focus
axis of
symmetry
-Parabola:
-Focus:
directrix
-Directrix:
-Axis of symmetry:
-Vertex:
Apr 5­11:21 AM
Standard Form of the
Equation of a Parabola Orientation
(y - k)2 = 4p(x - h)
Vertex: (h,
Description
Vertex: (h,k)
Focus: (h + p,
Focus: (h + p, k)
(h, k)
Axis of symmetry:
(h + p, k)
y=k
y=k
Directrix: x = h - p
Opening: Right if p > 0
Left if p < 0
x=h-p
(x - h)2 = 4p(y - k)
Vertex: (h,k)
Focus: (h, k + p)
)
,k
+p
Axis of symmetry:
(h
x=h
Directrix: y = k - p
(h, k)
y=k-p
x=h
Opening: Up if p > 0
Down if p < 0
Apr 5­11:45 AM
8
Examples: For the equation of each parabola, find the
coordinates of the vertex and focus and the equations of the
directrix and axis of symmetry. Then graph.
1.) x2 = 12(y - 1)
Apr 5­12:02 PM
2.) y2 - 4x + 2y + 5 = 0
Apr 5­12:29 PM
9
Examples: Write the equation of the parabola that meets each
set of conditions. Then graph.
1.) The vertex is at (-5,1) and the focus is at (2,1).
2.) The axis of symmetry is y = 6, the focus is at (0,6) and p = -3.
Apr 5­12:31 PM
Feb 18­2:55 PM
10
8.1-8.3 Homework Name: ____________________
For each equation of the ellipse, find the coordinates of the
center, foci, eccentricity and vertices. Then graph each equation.
1.) x2 + (y - 4)2 = 1
81
49
2.) 9x2 + 4y2 - 18x + 16y = 11
Apr 5­12:57 PM
Write the equation of each ellipse in standard form. Then find
the coordinates of the foci.
3.)
4.)
Apr 5­1:01 PM
11
5.) Write the equation of the hyperbola below.
=1
=1
=1
=1
6.) Write the equation of a hyperbola centered at the origin,
with a = 8, b = 5 and transverse axis on the y-axis.
Apr 5­1:08 PM
For the equation the hyperbola, find the coordinates of the
center, the foci, eccentricity, vertices and the equations of the
asymptotes. Then graph.
7.) (y - 3)2 - (x - 2)2 = 1
16
4
Apr 5­1:27 PM
12
For the equation of each parabola, find the coordinates of the
vertex and focus, and the equations of the directrix and axis of
symmetry. Then graph the equation.
8.) x2 + 8x + 4y + 8 = 0
Apr 5­1:32 PM
9.)
(y - 6)2 = 4x
10.) Explain a way in which you might distinguish the equation
of a parabola from the equation of a hyperbola.
Apr 5­1:36 PM
13
Write the equation in standard form for the conic with the given
characteristics.
11.) Parabola: Focus: (0,5), Directrix: y = -5
12.) Parabola: Focus (-2,-4), Vertex: (-4,-4)
13.) Ellipse: Major axis length 6 on y-axis, minor axis length 4
Feb 18­2:55 PM
14.) Ellipse: Minor axis endpoints: (0,±4), Major axis length 10
15.) Hyperbola: Foci (±5, 0), Transverse axis length 3
16.) Hyperbola: Transverse axis endpoints (-1,3) and (5,3), slope of
one asymptote is 4/3.
Feb 18­2:58 PM
14