parabola of analytic geometry
DESCRIPTION
Parabola definition and equation is presentedTRANSCRIPT
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Conic Sections
©Mathworld
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Circle
©National Science Foundation
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Circle
• The Standard Form of a circle with a center at (0,0) and a
radius, r, is…….. 222
r y x
center (0,0)
radius = 2
Copyright ©1999-2004 Oswego City School District Regents Exam Prep Center
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Circles• The Standard Form of a circle with a center at (h,k) and
a radius, r, is……..
222
)()( r k yh x
center (3,3)
radius = 2
Copyright ©1999-2004 Oswego City School District Regents Exam Prep Center
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Parabolas
© Art Mayoff © Long Island Fountain Company
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What’s in a Parabola
• A parabola is the set of all points in a plane such
that each point in the set is equidistant from a
line called the directrix and a fixed point called
the focus.
Copyright © 1997-2004, Math Academy Online™ / Platonic Realms™.
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Why is the focus so important?
© Jill Britton, September 25, 2003
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Parabola
• The Standard Form of a Parabola that opens to the rightand has a vertex at (0,0) is……
px y 42
©1999 Addison Wesley Longman, Inc.
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Parabola
• The Parabola that opens to the right and has a vertex at
(0,0) has the following characteristics……
• p is the distance from the vertex of the parabola to thefocus or directrix
• This makes the coordinates of the focus (p,0)
• This makes the equation of the directrix x = -p• The makes the axis of symmetry the x-axis (y = 0)
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Parabola• The Standard Form of a Parabola that opens to the left
and has a vertex at (0,0) is……
px y 42
© Shelly Walsh
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Parabola
• The Parabola that opens to the left and has a vertex at
(0,0) has the following characteristics……
• p is the distance from the vertex of the parabola to thefocus or directrix
• This makes the coordinates of the focus(-p,0)
• This makes the equation of the directrix x = p• The makes the axis of symmetry the x-axis (y = 0)
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Parabola
• The Standard Form of a Parabola that opens up andhas a vertex at (0,0) is……
py x 4
2
©1999-2003 SparkNotes LLC, All Rights Reserved
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Parabola
• The Parabola that opens up and has a vertex at (0,0)
has the following characteristics……
• p is the distance from the vertex of the parabola to thefocus or directrix
• This makes the coordinates of the focus (0,p)
• This makes the equation of the directrix y = -p• This makes the axis of symmetry the y-axis (x = 0)
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Parabola
• The Standard Form of a Parabola that opens down and
has a vertex at (0,0) is……
py x 42
©1999 Addison Wesley Longman, Inc.
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Parabola
• The Parabola that opens down and has a vertex at (0,0)
has the following characteristics……
• p is the distance from the vertex of the parabola to thefocus or directrix
• This makes the coordinates of the focus (0,-p)
• This makes the equation of the directrix y = p• This makes the axis of symmetry the y-axis (x = 0)
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Parabola
• The Standard Form of a Parabola that opens to the right
and has a vertex at (h,k) is……
)(4)( 2 h x pk y
© Shel ly Walsh
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Parabola
• The Parabola that opens to the right and has a vertex at
(h,k) has the following characteristics……..
• p is the distance from the vertex of the parabola to thefocus or directrix
• This makes the coordinates of the focus (h+p, k)
• This makes the equation of the directrix x = h – p• This makes the axis of symmetry
a
b y
2
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Parabola
• The Standard Form of a Parabola that opens to the leftand has a vertex at (h,k) is……
)(4)(2
h x pk y
©June Jones, University of Georgia
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Parabola
• The Parabola that opens to the left and has a vertex at(h,k) has the following characteristics……
• p is the distance from the vertex of the parabola to the
focus or directrix
• This makes the coordinates of the focus (h – p, k)
• This makes the equation of the directrix x = h + p
• The makes the axis of symmetry
a
b y
2
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Parabola
• The Standard Form of a Parabola that opens up and has
a vertex at (h,k) is……
)(4)( 2 k y ph x
Copyright ©1999-2004 Oswego City School District Regents Exam Prep Center
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Parabola
• The Parabola that opens up and has a vertex at (h,k) has
the following characteristics……
• p is the distance from the vertex of the parabola to thefocus or directrix
• This makes the coordinates of the focus (h , k + p)
• This makes the equation of the directrix y = k – p
• The makes the axis of symmetrya
b x
2
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Parabola
• The Standard Form of a Parabola that opensdown and has a vertex at (h,k) is……
)(4)(2
k y ph x
Copyright ©1999-2004 Oswego City School District Regents Exam Prep Center
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Parabola
• The Parabola that opens down and has a vertex at (h,k) has thefollowing characteristics……
• p is the distance from the vertex of the parabola to the focus or directrix
• This makes the coordinates of the focus (h , k - p)
• This makes the equation of the directrix y = k + p
• This makes the axis of symmetrya
b x
2
Elli
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Ellipse
© Jill Britton, September 25, 2003
•Statuary Hall in the U.S. Capital building is elliptic. It was in this room that John Quincy Adams, while a member of the House of Representatives, discovered this acousticalphenomenon. He situated his desk at a focal point of the elliptical ceiling, easily
eavesdropping on the private conversations of other House members located near theother focal point.
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What is in an Ellipse?• The set of all points in the plane, the sum of whose
distances from two fixed points, called the foci , is a
constant. (“Foci” is the plural of “focus”, and is
pronounced FOH-sigh.)
•Copyright © 1997-2004, Math Academy Online™ / Platonic Realms™.
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Why are the foci of the ellipse
important?
• The ellipse has an important property that isused in the reflection of light and sound waves.Any light or signal that starts at one focus will
be reflected to the other focus. This principle isused in lithotripsy, a medical procedure for treating kidney stones. The patient is placed ina elliptical tank of water, with the kidney stone
at one focus. High-energy shock wavesgenerated at the other focus are concentratedon the stone, pulverizing it.
Wh th f i f th lli
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Why are the foci of the ellipse
important?
• St. Paul's Cathedral in London. If a personwhispers near one focus, he can be heard at the
other focus, although he cannot be heard at many
places in between.
© 1994-2004 Kevin Matthews and Artifice, Inc. All Rights Reserved.
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Ellipse
• General Rules – x and y are both squared
– Equation always equals(=) 1
– Equation is always plus(+) – a2 is always the biggest denominator
– c2 = a2 – b2
– c is the distance from the center to each foci
on the major axis
– The center is in the middle of the 2 vertices,
the 2 covertices, and the 2 foci.
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Ellipse
• General Rules
– a is the distance from the center to each
vertex on the major axis
– b is the distance from the center to eachvertex on the minor axis (co-vertices)
– Major axis has a length of 2a
– Minor axis has a length of 2b – Eccentricity(e): e = c/a (The closer e gets to
1, the closer it is to being circular)
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Ellipse
• The standard form of the ellipse with a center at (0,0)
and a horizontal axis is……
12
2
2
2
b
y
a
x
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Ellipse
• The ellipse with a center at (0,0) and a horizontal axis
has the following characteristics……
• Vertices ( a,0)
• Co-Vertices (0, b)
• Foci ( c,0)
1916
22
y x
© Cabalbag, Porter, Chadwick, and Liefting
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Ellipse
• The standard form of the ellipse with a center at (0,0)
and a vertical axis is……
12
2
2
2
a
y
b
x
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Ellipse
• The ellipse with a center at (0,0) and a vertical axis has
the following characteristics……
• Vertices (0, a)• Co-Vertices ( b,0)
• Foci (0, c)
1819
22
y x
© Cabalbag, Porter, Chadwick, and Liefting
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Ellipse
• The standard form of the ellipse with a center at (h,k)
and a horizontal axis is……
1)()(
2
2
2
2
b
k y
a
h x
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Ellipse
• The ellipse with a center at (h,k) and a horizontal axishas the following characteristics……
• Vertices (h a , k)
• Co-Vertices (h, k b)
• Foci (h c , k)
©Sellers, James
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Ellipse
• The standard form of the ellipse with a center at
(h,k) and a vertical axis is……
1
)()(
2
2
2
2
a
k y
b
h x
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Ellipse
• The ellipse with a center at (h,k) and a vertical axishas the following characteristics……
• Vertices (h, k a)
• Co-Vertices (h b , k)
• Foci (h, k c)
© Joan Bookbinder 1998 -2000
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Hyperbola
The huge chimney of a nuclear power plant has the shape of a hyperboloid, as does
the architecture of the James S. McDonnell Planetarium of the St. Louis Science Center.
© Jill Britton, September 25, 2003
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What is a Hyperbola?
• The set of all points in the plane, thedifference of whose distances from two fixed
points, called the foci, remains constant.
Copyright © 1997-2004, Math Academy Online™ / Platonic Realms™.
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Where are the Hyperbolas?
• A sonic boom shock wave has the
shape of a cone, and it intersects
the ground in part of a hyperbola.
It hits every point on this curve at
the same time, so that people in
different places along the curve onthe ground hear it at the same
time. Because the airplane is
moving forward, the hyperbolic
curve moves forward and
eventually the boom can be heard
by everyone in its path.
© Jill Britton, September 25, 2003
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Hyperbola
• General Rules – x and y are both squared
– Equation always equals(=) 1
– Equation is always minus(-) – a2 is always the first denominator
– c2 = a2 + b2
– c is the distance from the center to each foci
on the major axis – a is the distance from the center to each
vertex on the major axis
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Hyperbola
• General Rules – b is the distance from the center to each
midpoint of the rectangle used to draw theasymptotes. This distance runs
perpendicular to the distance (a). – Major axis has a length of 2a
– Eccentricity(e): e = c/a (The closer e gets to1, the closer it is to being circular
– If x2 is first then the hyperbola is horizontal
– If y2 is first then the hyperbola is vertical.
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Hyperbola
• General Rules – The center is in the middle of the 2 vertices and
the 2 foci.
– The vertices and the covertices are used to drawthe rectangles that form the asymptotes.
– The vertices and the covertices are the midpoints
of the rectangle
– The covertices are not labeled on the hyperbolabecause they are not actually part of the graph
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Hyperbola
• The standard form of the Hyperbola with a center at
(0,0) and a horizontal axis is……
12
2
2
2
b
y
a
x
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Hyperbola
• The Hyperbola with a center at (0,0) and a horizontal
axis has the following characteristics……
• Vertices ( a,0)
• Foci ( c,0)
• Asymptotes: x
a
b y
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Hyperbola
• The standard form of the Hyperbola with a center at
(0,0) and a vertical axis is……
12
2
2
2
b
x
a
y
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Hyperbola
• The Hyperbola with a center at (0,0) and a vertical
axis has the following characteristics……
• Vertices (0, a)
• Foci ( 0, c)
• Asymptotes:
xb
a y
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Hyperbola
• The standard form of the Hyperbola with a center at
(h,k) and a horizontal axis is……
1
)()(2
2
2
2
b
k y
a
h x
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Hyperbola
• The Hyperbola with a center at (h,k) and a horizontalaxis has the following characteristics……
• Vertices (h a, k)
• Foci (h c, k )
• Asymptotes:
)( h x
a
bk y
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Hyperbola
• The standard form of the Hyperbola with a center at
(h,k) and a vertical axis is……
1
)()(2
2
2
2
b
h x
a
k y
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Hyperbola
• The Hyperbola with a center at (h,k) and a vertical
axis has the following characteristics……
• Vertices (h, k a)• Foci (h, k c)
• Asymptotes:
)( h xb
a
k y
©Sellers, James
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Rotating the Coordinate Axis
022 F Ey DxCy Bxy Ax
© James Wilson
E ti f R t ti th C di t
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Equations for Rotating the Coordinate
Axes
sin'cos' y x x
cos'sin' y x y
B
C A 2cot C A
B
2tanor
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Resources
Bookbinder, John. Unit 8: Conic Sections (College AlgebraOnline). 2000. June 3, 2004<http://www.distancemath.com/unit8/ch8p1.htm>.
Britton, Jill. Occurrence of the Conics. September 25, 2003. June 3,
2004 <http://ccins.camosun.bc.ca/~jbritton/jbconics.htm>.
Cabalbag, Christain, and Porter, Amanda and Chadwick, Justin andLiefting. Nick. Graphing Conic Sections (Microsoft Power PointPresentation 1997). 2001. June3, 2004
<http://www.granite.k12.ut.us/Hunter_High/StaffPages/Olsen_P/ClassWebSite/2003%20student%20projects/27circlesandelipse.ppt
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Resources
Finney, Ross, et. al. Calculus: Graphical, Numerical, Algebraic. Scott Foresman-Addison Wesley, 1999.
Jones, June. Instructional Unit on Conic Sections. University of Georgia.
June 3, 2004http://jwilson.coe.uga.edu/emt669/Student.Folders/Jones.June/conics/conics.html
Mathews, Kevin. Great Buildings Online. Great Buildings. une 3,
2004 <http://www.GreatBuildings.com/buildings/Saint_Pauls_Cathedral.html
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ResourcesMayoff, Art. San Francisco and the Golden Gate Bridge.
June 3, 2004
http://mathworld.wolfram.com/ConicSection.html>.
Mueller, William. Modeling Periodicity .
June 3, 2004
<http://www.wmueller.com/precalculus/funcdata/1_10.html>.
PRIME Articles. Platomic Realms.
June 3, 2004
<http://www.mathacademy.com/pr/prime/index.asp>.
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Resources
Quadratics. Spark Notes from Barnes and Noble.
June 3, 2004
<http://www.sparknotes.com/math/algebra1/quadratics/section1.html
Roberts, Donna. Mathematics A . Oswego City School District
Regents Exam Prep.June, 3, 2004 <http://regentsprep.org/Regents/math/math-
topic.cfm?TopicCode=conics>.
Seek One Web Services, Long Island Fountain
Company. <http://www.lifountain.com/fountainideas.html>.
Sellers, James, Introduction to Conics, June 8, 2004.
http://www.krellinst.org/UCES/archive/resources/conics/newconics.ht
ml
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Resources
Walsh, Shelly. Chapter 9 (Precalculus).
June 3, 2004
http://faculty.ed.umuc.edu/~swalsh/UM/M108Ch9.html
Weissteing, Eric W. "Conic Section." From MathWorld --A Wolfram WebResource. http://mathworld.wolfram.com/ConicSection.html
Wilson, James W. CURVE BUILDING. An Exploration with
Algebraic Relations University of Georgia.
June 3, 2004http://jwilson.coe.uga.edu/Texts.Folder/cb/curve.building.html