mth065 elementary algebra ii
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
MTH065Elementary Algebra II
Chapter 13Conic Sections
IntroductionParabolas (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.
2 2y x
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.
2 512 2y x x
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 constantsA 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)• h = -b/(2a)
• Orientation:• Open upward: a > 0• Open downward: a < 0
• Width:• Narrow: |a| > 1• Wide: |a| < 1
• Graphing: Vertex & One Other Point
2 512 2y x x
212 ( 1) 3y x
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 > 0• Open left: a < 0
• Width:• Narrow: |a| > 1• Wide: |a| < 1
• Graphing: Vertex & One Other Point
22 12 19x y y 22( 3) 1x y
Parabolas – Special PropertiesFocus• 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
4p
a
Ellipses
Ax2 + By2 + Cx + Dy + E = 0where 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)
2 2
2 2
( ) ( )1
x h y k
a b
Ellipses – Special PropertiesFoci• 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
HyperbolasAx2 + 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:
2 2
2 2
( ) ( )1
x h y k
a b
2 2
2 2
( ) ( )1
x h y k
a b
or
( )bay x h k
ba(h,k)
ba(h,k)
Hyperbolas – Special PropertiesFoci• 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.
Parabola
Hyperbola
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.
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