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MTH065 Elementary Algebra II

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.

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MTH065 Elementary Algebra II

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  1. MTH065Elementary Algebra II Chapter 13 Conic Sections Introduction Parabolas (13.1) Circles (13.1) Ellipses (13.2) Hyperbolas (13.3) Summary

  2. 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.

  3. 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.

  4. What’s missing … Note: Quadratic equations may also include an xy term, but the study of such equations requires trigonometry. 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

  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 > 0 • Open downward: a < 0 • Width: • Narrow: |a| > 1 • Wide: |a| < 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 > 0 • Open left: a < 0 • Width: • Narrow: |a| > 1 • Wide: |a| < 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 > 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.

  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 b a (h,k) or b a (h,k) 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 Parabola Hyperbola 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/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

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