Transcript 13.1 - Conic Sections: Parabolas and Circles

MTH065
Elementary Algebra II
Chapter 13
Conic Sections
Introduction
Parabolas (13.1)
Circles (13.1)
Ellipses (13.2)
Hyperbolas (13.3)
Summary
Where we’ve 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.
y  2x  2
Where we’ve 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.
y  12 x2  x  52
What’s 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.
Parabolas
y = ax2 + bx + c
• Graphing (complete the square): y = a(x - h)2 + k
• Vertex: (h, k)
5
1 2
• h = -b/(2a)
• Orientation:
• Open upward: a > 0
• Open downward: a < 0
y  2 x x 2
y  ( x  1)  3
1
2
• Width:
• Narrow: |a| > 1
• Wide: |a| < 1
• Graphing: Vertex & One Other Point
2
Parabolas
x = ay2 + by + c
• Graphing (complete the square): x = a(y - k)2 + h
• Vertex: (h, k)
2
• k = -b/(2a)
• Orientation:
• Open right: a > 0
• Open left: a < 0
x  2 y  12 y  19
x  2( y  3)  1
• Width:
• Narrow: |a| > 1
• Wide: |a| < 1
• Graphing: Vertex & One Other Point
2
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.
1
p
4a
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
( x  h) ( y  k )

1
2
2
a
b
• Center: (h,k)
• 4 Vertices: (h ± a, k), (h, k ± b)
2
2
Ellipses – Special Properties
Foci
• The two points c units from the center along the
major axis where c2 = a2 – b2 if a > b or c2 = b2 – a2
if a < 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.
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
Hyperbolas
Ax2 + By2 + Cx + Dy + E = 0
where A & B have opposite signs.
• Graphing form: Complete the squares & set equal to 1
( x  h) 2 ( y  k ) 2

1
2
2
a
b
or
( x  h) 2 ( y  k ) 2


1
2
2
a
b
• Center: (h,k)
• 2 Vertices:
• 1st form: (h ± a, k)
• 2nd form: (h, k ± b)
• Asymptotes: y   ba ( x  h)  k
b
(h,k) a
Hyperbolas – Special Properties
Foci
• The two points c units from the center inside each
branch, where c2 = a2 + b2
Parabola
• Reflective property:
Hyperbola
• 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.
Conic Sections – Summary
Ax2 + 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.
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/Winchester/jhhs
/math/lessons/calc2004/appellip.html
• Hyperbolas
• http://www.doe.virginia.gov/Div/Winchester/jhhs
/math/lessons/calc2004/apphyper.html