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MTH065 Elementary Algebra II

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More Applications of Conics Parabolas http://www.doe.virginia.gov/Div/Winchester/jhhs/math/lessons/calc2004/appparab.html Ellipses http://www.doe.virginia.gov/Div ... – PowerPoint PPT presentation

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Title: MTH065 Elementary Algebra II


1
MTH065Elementary Algebra II
  • Chapter 13
  • Conic Sections

Introduction Parabolas (13.1) Circles
(13.1) Ellipses (13.2) Hyperbolas (13.3) Summary
2
Where weve been
  • MTH 060 Linear Functions Equations
  • Single Variable ax b 0
  • Solution A single real number.
  • Two Variables ax by c
  • y mx b
  • f(x) mx b
  • Solutions Many ordered pairs of real
    numbers.
  • Graph A line.

3
Where weve been
  • MTH 065 Quadratic Functions Equations
  • Single Variable ax2 bx c 0
  • Solutions 0, 1, or 2 real numbers
  • Two Variables y ax2 bx c
  • f(x) ax2 bx
    c
  • f(x) a(x h)2
    k
  • Solutions Many ordered pairs of real
    numbers.
  • Graph A parabola.

4
Whats missing
  • Quadratic Equations that may also include a y2
    term (not all functions).
  • Ax2 By2 Cx Dy E 0
  • A, B, C, D, E are constants
  • A and B not both 0

Note Quadratic equations may also include an xy
term, but the study of such equations requires
trigonometry.
5
Parabolas
  • y ax2 bx c
  • Graphing (complete the square) y a(x - h)2 k
  • Vertex (h, k)
  • h -b/(2a)
  • Orientation
  • Open upward a gt 0
  • Open downward a lt 0
  • Width
  • Narrow a gt 1
  • Wide a lt 1
  • Graphing Vertex One Other Point

6
Parabolas
  • x ay2 by c
  • Graphing (complete the square) x a(y - k)2 h
  • Vertex (h, k)
  • k -b/(2a)
  • Orientation
  • Open right a gt 0
  • Open left a lt 0
  • Width
  • Narrow a gt 1
  • Wide a lt 1
  • Graphing Vertex One Other Point

7
Parabolas Special Properties
  • Focus
  • The point 1/(4a) units from the vertex along the
    axis of symmetry and inside the parabola.
  • Reflective property
  • Light or any other wave emitted from the focus
    will be reflected in a beam parallel to the axis
    of symmetry.
  • A satellite dish, for example, uses this property
    in reverse.

8
Ellipses
  • Ax2 By2 Cx Dy E 0
  • where A B are both positive or both negative.
  • Graphing form Complete the squares set equal
    to 1
  • Center (h,k)
  • 4 Vertices (h a, k), (h, k b)

9
Ellipses Special Properties
  • Foci
  • The two points c units from the center along the
    major axis where c2 a2 b2 if a gt b or c2 b2
    a2 if a lt b.
  • Reflective property
  • Sound or any other wave emitted from one focus
    will be reflected to the other focus.
  • Satellites have elliptical orbits with the object
    being orbited at one of the foci.

10
Circles Special Ellipses
  • A circle is just an ellipse with a b and a
    single focus at the center (since c2 a2 b2
    0).
  • Ax2 Ay2 Cx Dy E 0
  • (x h)2 (y k)2 r2
  • Center (h, k)
  • Radius r

11
Hyperbolas
  • Ax2 By2 Cx Dy E 0
  • where A B have opposite signs.
  • Graphing form Complete the squares set equal
    to 1
  • Center (h,k)
  • 2 Vertices
  • 1st form (h a, k)
  • 2nd form (h, k b)
  • Asymptotes

12
Hyperbolas Special Properties
  • Foci
  • The two points c units from the center inside
    each branch, where c2 a2 b2
  • Reflective property
  • Light or any other wave emitted from one focus
    towards the other branch will be reflected
    directly away from the other focus (or vice
    versa).
  • Hyperbolic mirrors are used in reflector
    telescopes.
  • Lampshades cast hyperbolic shadows on a wall.

13
Conic Sections SummaryAx2 By2 Cx Dy E
0
  • A ? 0 B 0
  • Up/Down Parabola
  • A 0 B ? 0
  • Left/Right Parabola
  • A B w/ same sign
  • Ellipse
  • A B gives a circle
  • A B w/ opposite signs
  • Hyperbola

To graph complete the squares.
14
More Applications of Conics
  • Parabolas
  • http//www.doe.virginia.gov/Div/Winchester/jhhs/ma
    th/lessons/calc2004/appparab.html
  • Ellipses
  • http//www.doe.virginia.gov/Div/Winchester/jhhs/ma
    th/lessons/calc2004/appellip.html
  • Hyperbolas
  • http//www.doe.virginia.gov/Div/Winchester/jhhs/ma
    th/lessons/calc2004/apphyper.html
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