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Algebra II
Conic Sections
www.njctl.org
2015-04-21
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Table of Contents
Review of Midpoint and Distance Formulas
Parabolas
Circles
Ellipses
Hyperbolas
Recognizing Conic Sections from General Form
Introduction to Conic Sections
click on the topic to go to that section
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Midpoint and DistanceFormula
Return to Tableof Contents
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A (-3, 6)
B (5, 2)
What is the midpoint of segment AB?
Look at this segment - the midpoint is halfway. To find the coordinates of the midpoint, find the average of the x-values and the average of the y-values.
(x,y)
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The Midpoint FormulaGiven points A(x1,y1) and B (x2,y2), the midpoint of AB is
Examples: Find the midpoint of the segment with the given endpoints.
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1 Find the midpoint of K(1,8) & L(5,2).
A (2,3)
B (3,5)
C (-2,-3)
D (-3,-5)
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2 Find the midpoint of H(-4 , 8) & L(6, 10).
A (5,9)
B (-1,9)
C (1,9)
D (5,1)
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3 If the midpoint of a segment is (4,9) and one endpoint is (-3,10), find the other endpoint.
A (-10,8)
B (11,8)
C (-10,11)
D (.5,9.5)
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A (1, 4)
B (7, -2)C (1,-2)
How far apart are points A and B?
The Distance Formula is derived from the Pythagorean Theorem, a2 + b2 = c2.
In this example,
AC2 + CB2=AB2
62+62 = AB2
72 = AB2
AB = =
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4 What is the distance between (2, 4) and (-1, 8)?
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5 What is the distance between (0, 7) and (5, -5)?
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6 Given A( 4, 5) and B(x, 1) and AB=5, find all of the possible values of x.
A -7
B -5
C -3
D -1
E 0
F 1
G 3
H 5
I 7
J 9
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7 If the distance between (4,5) and (x,-2) is 10, what are the possible values of x?
A
B
C
D
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Introduction to Conic
Sections
Return to Tableof Contents
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Conic Sections are created by intersecting a set of double cones with a plane.
Discussion Question: Which conic sections are functions?
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Click on the title below to take you to a webpage for more background information about conic sections:
"The Occurrence of the Conics", by Dr. Jill Britton
Click the link below for a YouTube video that demonstrates the cutting of the cones.
SalMathGuy Conics Video
More Info About Conics
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A Circle comes from cutting parallel to the "base".
The term base is misleading because like lines and planes, conic sections continue on forever.
The Circle
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An Ellipse comes from cutting skew to the "base".
The Ellipse
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A Parabola comes from intersecting the cone with a plane that is parallel to a side of the cone.
The Parabola
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A Hyperbola comes from cutting the cones perpendicular to the "bases".
This is the only cross section that intersects both cones.
The Hyperbola
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Parabolas
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This is the graph of y = x2.Complete the table below:
x y-3-2-10123
Discuss the patterns that you observe.
A graph that has this shape is called a parabola.
y = x2 is the "parent function".
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The equation of a parabola can be written in two forms: y = ax2 + bx + c (the General Form)
y = a(x - h)2 + k (Standard Form)where (h,k) is the vertex. This is also called Vertex Form.
Example: Name the vertex of each equation: A) y= -3(x - 4)2 + 5
B) y= 2(x + 7)2 + 2 C) y= (x -3)2
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Match each equation to its parabola - drag the number of the graph to its equation.
f(x) = (x - 3)² - 2
g(x) = -2(x + 1)²
h(x) = 2 / 3 (x + 5)² - 7
1 2
3
2
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9 What is the vertex of ?
A (3, 2)
B (-3, -2)
C (2, 3)
D (-2, -3)
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10 What is the vertex of ?
A (2, -3)
B (-3, -2)
C (2, 3)
D (-2, -3)
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11 What is the vertex of ?
A (-3, 2)
B (-3, -2)
C (2, 3)
D (-2, -3)
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12 What is the vertex of ?
A (3, 2)
B (-3, -2)
C (2, 3)
D (-2, -3)
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Fill in the blank to complete the square:Half of 6 is 3, 32 = 9
Converting from General Form to Standard Form
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Converting from General Form to Standard Form
y = x2 - 8x + 5
y = (x2 - 8x +___) + 5 - _____
What number completes the square in the parenthesis above?
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18
A (4, 5)
B (-4, 5)
C (-5, 4)
D (5, 4)
What is the vertex of x = y2 - 10y + 29?
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19 What is the vertex of y= x2 - 8x +21?
A (4, 5)
B (-4, 5)
C (-5, 4)
D (5, 4)
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Converting from General Form to Standard Form
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20 What should be factored out of
x = (4y2 - 8y + ___)+ 9 - ___ ?
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21 What value completes the square of
x = 4(y2 - 2y + ___)+ 9 - ___ ?
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22 What value should follow "+ 9" in
x = 4(y2 - 2y + ___) + 9 ___ ?
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23 Which is the correct standard form of
x = (4y2 - 8y + ___)+ 9 - ___ ?
A x = 4(y - 1)2 + 8 B x = 4(y + 1)2 + 8
C x = 4(y - 1)2 + 5
D x = 4(y + 1)2 + 5
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24 What should be factored out of y = (-5x2 - 20x + ___)+ 7 - ___ ?
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25 What value completes the square of
y = -5(x2 + 4x + ___)+ 7 - ___ ?click to reveal
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26 What value should follow "+7" in
y = -5(x2 + 4x + ___)+ 7 ___ ?click to reveal
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27 Which is the correct standard form of
y = (-5x2 - 20x + ___)+ 7 - ___ ?
A y = -5(x - 2)2 + 3
B y = -5(x + 2)2 + 27
C y = -5(x - 2)2 -13
D y = -5(x - 2)2 + 27
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A parabola is a locus* of points equidistant from a fixed point, the focus, and a fixed line, the directrix.
*locus is just a fancy word for set.
Geometric Definition
Every parabola is symmetric with respect to a line through the focus and perpendicular to the directrix. The vertex of the parabola is the "turning point" and is on the axis of symmetry.
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Focus and Directrix of a Parabola
Axis of Symmetry
Directrix
Focus
Every point on the parabola is the same distance from the directrix and the focus.
L1
L2
L1=L2
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Directrix
Focus
L1
L2
L1=L2
Eccentricity of a Parabola
All parabolas have an eccentricity of 1.
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Parts of a ParabolaParts are the same for all parabolas, regardless of the direction in which they open.
Directrix
Axis of Symmetry
Vertex
FocusVertex
Focus
Directrix
Axis of Symmetry
x=ay2+by+cy=ax2+bx+c
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Compare the graphs below: What makes the graph more "narrow" or "wide"?
y = x2
y = 2x2
y = .5x2
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28 Which of the parabolas below are narrower than their parent functions?
A
B
C
D
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Focal Distance
To calculate:
focal distance =
The distance from the vertex to the focus is 1.
The distance from the vertex to the directrix is 1.
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Genera l Form y= ax2 + bx + c x= ay2 +by + c
S tandard Form y= a(x - h)2 +k x= a (y - k)2 + h
Opens a>0 opens upa<0 opens down
a>0 opens to the righta<0 opens to the le ft
Axis of Symmetry x = h y = k
Vertex (h , k) (h , k)
Foca l Dis tance
Directrix
Focus
Eccentricity 1 1
Parabola Summary
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Slide 65 / 181Graph the equation from the last example.
Dire
ctrix
Focus Axis of Symmetry
Slide 66 / 181Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity?
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Graph
Slide 68 / 181Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity?
Step 1: Convert the equation from general to standard form.
Slide 69 / 181Step 2: Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity?
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Graph
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29 Given the following equation, which direction does it open?
A Up
B Down
C Left
D Right
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30 How does the following equation compare to the parent function
A Is narrower B Is widerC Is the same width
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31 Where is the vertex for the following equation?
A (-3 , 4)
B (3 , 4)
C (4 , 3)
D (4 , -3)
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32 What is the equation of the axis of symmetry for the following equation?
A y = 3
B y = -3
C x = 4
D x = -4
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33 What is the focal distance in the following equation?
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34 What is the equation of the directrix for the following equation?
A y = 2
B y = -4
C x = 3
D x = -5
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35 Where is the focus for the following equation?
A (-3 , 5)
B (3 , 5)
C (5 , 3)
D (5 , -3)
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36 What is the eccentricity of the following conic section?
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51 What is the equation of the parabola with vertex (2,3) and directrix y = 4?
A y = 4(x - 2)2 + 3
y = -1/4(x - 2)2 + 3
x = 4(y - 2)2 + 3
x = 1/4(y - 2)2 + 3
B
C
D
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The St. Louis Arch is 630 feet tall and 630 feet wide at the base. Write an equation to represent the shape of the arch.
Challenge Problem
Answer on next page...
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Circles
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A circle is a locus of points in a plane that are equidistant from a given point.
Radius
Center (h,k)
(x,y)
The distance from the center to a point on the circle is
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52 Write the equation of the circle with center (5 , 2) and radius 6
A
B
C
D
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53 Write the equation of the circle with center (-5,0) and radius 7
A
B
C
D
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54 Write the equation of the circle with center (-2,1) and radius
A
B
C
D
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55 What is the center and radius of the following equation?
A
B
C
D
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57 What is the center and radius of the following equation?
A
B
C
D
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58 What is eccentricity of a circle?
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Write the equations for each part of this unfortunate snowman.
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Write the equation of the circle that meets the following criteria:
Center (1 , -2) and passes through (4 , 6)
Since we know the center we only need to find the radius. The radius is the distance from the center to the point.
The equation of the circle is:
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Write the equation of the circle that meets the following criteria:
Diameter with endpoints (4 , 7) and (-2 , -1).
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Write the equation of the circle that meets the following criteria:
Center at (-5 , 6) and tangent to the y-axis.
"Tangent to the y-axis" means the circle only touches the y-axis at one point. Look at the graph.
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Write the equation of the circle in standard form that meets the following criteria:
Complete the square for the x's. (Remember, the y-term is 0y.)
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59 What is the equation of the circle that has a diameter with endpoints (0,0) and (16,12)?
A
B
C
D
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60 What is the equation of the circle with center (-3,5) that contains the point (1,3)?
A
B
C
D
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61 What is the equation of the circle with center (7,-3) and tangent to the x-axis?
A
B
C
D
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Challenge Question: What is the equation of a circle that passes through the three points (2,3), (2,-2), and (5,-3)?
Remember that the distance from the radius to the circle is the same for every radius. Let (x,y) be the center and use the distance formula twice.
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Ellipses
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An ellipse is a focus of points in a plane that are each the same total distance from 2 fixed points, called the foci (plural of focus).
F1 F2
P2P1
For example, P1F1 + P1F2 = P2F1 + P2F2
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Eccentricity of an Ellipse0 < e < 1
The eccentricity of an ellipse is a number between 0 and 1. The more elongated the ellipse the closer the eccentricity is to 1. The closer an ellipse is to being a circle, the closer the eccentricity is to 0.
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A
D
B
C
O
The major axis, AB, is the segment through both foci whose endpoints are on the ellipse.The minor axis, CD, is perpendicular to the major axis through the center, O. The vertices of an ellipse are the endpoints of the major axis, points A and B. The co-vertices are the endpoints of the minor axis, points C and D.
Slide 124 / 181Parts of an Ellipse
Major axis
Maj
or a
xis
Minor axisMin
or a
xis
Vertex
Co-vertex
VertexCo-vertex
Vertex
Co-vertex
Focus
Horizontal ellipse Vertical ellipse
The length of the major axis is 2a.
The length of the minor axis is 2b.
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64 What letter or letters corresponds with ellipse's center?
A
B
C
D
E
F
G
A
D
B
C
FE G
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65 What letter or letters corresponds with ellipse's foci?
A
B
C
D
E
A
D
B
C
FE G
FG
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66 What letter or letters corresponds with ellipse's major axis?
A
B
C
D
E
F
G
H
I
ABCD G
E F
H
I
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67 Which choice best describes an ellipse's eccentricity?
A e = 0
B 0< e < 1
C e = 1
D e > 1
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68 Which of the ellipses has the greater eccentricity?
AB A B
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a
ab
c
a
Finding the foci:
In this ellipse, a = 5 and b = 4, so c = 3.
The coordinates of the foci are (3-3,2) and (3+3,2) or (0,2) and (6,2)
(Note that in this case, a represents the hypotenuse of the triangle.)
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69 What is the center of
A (9 , 4)
B (5 , 6)
C (-5 , -6)
D (3 , 2)
?
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70 How long is the major axis of
A 9
B 6
C 3
D 2
?
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71 How long is the minor axis of
A 9
B 4
C 3
D 2
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72 Name one foci of
A
B
C
D
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73 Name one foci of
A
B
C
D
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Graphing an Ellipse
· Find and graph the center · Find the length and direction of the major and minor axes· Draw the major and minor axes· Draw the ellipse
The center is (4 , -2)The major axis is 6 units and horizontalThe minor axis is 4 units and vertical
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74 Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, what is the center of the ellipse?
A (8 , 2)
B (0 , 2)
C (0 , 1)
D (-8 , 1)
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75 Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, in which direction is the ellipse elongated?
A horizontally
B vertically
C obliquely
D it is not elongated
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76 Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, how far is it from the center to an endpoint of the major axis?
A 10
B 100
C 5
D 25
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77 Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, which equation would be used to find the distance from the center to an endpoint of the minor axis?
A
B
C
D
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78 Given that an ellipse has foci (4 , 1) and (-4 , 1) and major axis of length 10, find b.
A
B
C
D
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80 Given that an ellipse has foci (4,-4) and (4,2) and
minor axis of length 8, which is the equation of the
ellipse?
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Steps for Converting the ellipse from General Form to Standard Form
· factor the x's and y's
· divide by the constant
· complete the square for x and/or y
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80 Convert the following ellipses to standard form.
A
B
C
D
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81 Convert the following ellipses to standard form.
A
B
C
D
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Hyperbolas
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Like the ellipse, the hyperbola is a set of points at a given distance from two foci. In the case of the hyperbola, the absolute value of the difference of the distances from a point to the foci is constant.
a
F1
dc
b
|a - b| = |c - d|F2
(Don't worry so much about this definition - it is just to put things in perspective.)
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Vertex FocusFocus
Vertex Center
AsymptoteAsymptote
a ab
b
Horizontal Hyperbola
Equation:
Vertices: move a units to the left and right of the center
Foci: move c units to the left and right of the center, where
Asymptotes: slope = ± b/a (The asymptotes are lines that pass through the vertices of the rectangle between the vertices with length 2a and width 2b. An asymptote is a line that the graph approaches but never touches.)
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Vertex Focus
FocusVertex Center
Asym
ptot
eAsymptote
a
a
bb
Vertical Hyperbola
Equation:
Vertices: move a units up and down from the center
Foci: move c units up and down from the center, where
Asymptotes: slope = ± a/b
Slide 158 / 181To graph a hyperbola in standard form:
· Find and graph the center· Plot points a right and left of the center, and b up and down for
horizontal, or b right and left, and a up and down for vertical· Make a rectangle through the four points from previous step· Draw asymptotes that contain the diagonals of the rectangle · Sketch the graph of the hyperbola
Center: (-1,2)
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Example: Graph
The cente r of the rectangle is ( -5 , 4 )
From the cente r move le ft/right 2
From the cente r move up/down 3
The hyperbola opens up and down
What are the slopes of the asymptotes?
How does this relate to a and b? Why?
Ans
wer
click
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Example: Graph
The cente r of the rectangle is ( 6 , 0 )
From the cente r move le ft/right 4
From the cente r move up/down 5
The hyperbola opens le ft and right
click
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90 What is the equation of a hyperbola that has vertices (±6,0) and foci (±10,0)?
A
B
C
D
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Convert to standard form:
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Recognizing Conic Sections
fromGeneral Form
Return to Tableof Contents
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General Form: ax2 + bx + cy2 + dy + e = 0
This form could represent any conic under the following conditions:
In a parabola, either a=0 or b=0.
ax2 + bx + dy +e =0
cy2 + dy + bx + e=0
In a circle, a=c and both a and c are positive. ax2 + bx + cy2 + dy + e = 0
In an ellipse, a and c are both positive, and a≠c. * ax2 + bx + cy2 + dy + e = 0
In a hyperbola, either a<0 and c>0 or a>0 and c<0.
ax2 + bx - cy2 + dy + e = 0
cy2 + dy - ax2 + bx + e = 0
* A circle is a special type of ellipse in which a = c.
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Eccentricity of Conic Sections
This picture depicts the comparative eccentricity of conic sections. Eccentricity (e) is a measure of "unroundness". A circle is round, so has e=0. For an ellipse, as the ellipse becomes more elongated, e increases from 0 to 1, not-including 1. A parabola has e=1, and for a hyperbola e>1.
Circle
e=0
Ellipse
0<e<1
Parabola
e = 1
Hyperbola
e > 1
