Pre-Cal 40S May 25, 2009

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The hyperbola.

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Pre-Cal 40S May 25, 2009

  1. 1. The Anatomy of the Hyperbola 12.3 Hyperbolas
  2. 2. What does a quot;sonic boomquot; have to do with hyperbolas? sonicboomplane_navy.jpg
  3. 3. Throw 2 stones in a pond. The resulting concentric ripples meet in a hyperbola shape.
  4. 4. Hyperbolic Light Casting by flickr user harry harris
  5. 5. Hyperbolic Cooling Towers Sheffield Cooling Towers (HDR) by flickr user Paul Denton Cocker
  6. 6. Generating Hyperbolas HOMEWORK
  7. 7. Constructing Hyperbolas CONJUGATE AXIS VERTEX VERTEX TRANSVERSE AXIS ASYMPTOTES
  8. 8. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k).
  9. 9. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k). A1A2 is the length of the transverse axis. It's length is 2a.
  10. 10. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k). A1A2 is the length of the B1 B2 is the length of the transverse axis. It's length is 2a. conjugate axis. It's length is 2b.
  11. 11. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k). A1A2 is the length of the B1 B2 is the length of the transverse axis. It's length is 2a. conjugate axis. It's length is 2b. F 1 & F 2 are called the foci. They are c units from the centre.
  12. 12. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k). A1A2 is the length of the B1 B2 is the length of the transverse axis. It's length is 2a. conjugate axis. It's length is 2b. F 1 & F 2 are called the foci. They PF1 & PF 2 are the focal are c units from the centre. radii of the hyperbola.
  13. 13. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k). A1A2 is the length of the B1 B2 is the length of the transverse axis. It's length is 2a. conjugate axis. It's length is 2b. F 1 & F 2 are called the foci. They PF1 & PF 2 are the focal are c units from the centre. radii of the hyperbola. OA1 = OA2 is the length of the semitransverse axis with length a.
  14. 14. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k). A1A2 is the length of the B1 B2 is the length of the transverse axis. It's length is 2a. conjugate axis. It's length is 2b. F 1 & F 2 are called the foci. They PF1 & PF 2 are the focal are c units from the centre. radii of the hyperbola. OA1 = OA2 is the length of the A1 & A 2 are called the semitransverse axis with length a. vertices of the hyperbola.
  15. 15. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k). A1A2 is the length of the B1 B2 is the length of the transverse axis. It's length is 2a. conjugate axis. It's length is 2b. F 1 & F 2 are called the foci. They PF1 & PF 2 are the focal are c units from the centre. radii of the hyperbola. OA1 = OA2 is the length of the A1 & A 2 are called the semitransverse axis with length a. vertices of the hyperbola. OB1 = OB2 is the length of the semiconjugate axis with length b.
  16. 16. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k). A1A2 is the length of the B1 B2 is the length of the transverse axis. It's length is 2a. conjugate axis. It's length is 2b. F 1 & F 2 are called the foci. They PF1 & PF 2 are the focal are c units from the centre. radii of the hyperbola. OA1 = OA2 is the length of the A1 & A 2 are called the semitransverse axis with length a. vertices of the hyperbola. OB1 = OB2 is the length of the B1 & B2 are called the endpoints semiconjugate axis with length b. of the conjugate axis.
  17. 17. The Standard Form for the Equation of a Hyperbola Horizontal Orientation Vertical Orientation Similarities Differences • Both equal one • a^2 with x^2 on left, • have (x-h)^2 and (y-k)^2 a^2 with y^2 on right. • have a^2 and b^2 in • b^2 with y^2 on left, denominator. a^2 with x^2 on right • b^2 denominator in negative • when y (+) is vertical, term. when y(-) is horizontal. • a^2 denominator in positive • when x (-) is vertical, term. x(+) is horizontal.
  18. 18. Horizontal Hyperbola Vertical Hyperbola a b a O O b
  19. 19. The Pythagorean Property B1 b a c F1 F2 A2 O c A 1 B2 2 2 2 c = a +b Horizontal Hyperbola Vertical Hyperbola
  20. 20. Conics Animations Source http://tinyurl.com/conicAni
  21. 21. Conics Animations Source http://tinyurl.com/conicAni
  22. 22. Conics Animations Source http://tinyurl.com/conicAni
  23. 23. Conics Animations Source http://tinyurl.com/conicAni
  24. 24. For the hyperbola whose equation is given below. (i) Write the equation in standard form (ii) Determine the lengths of the transverse and conjugate axes, the coordinates of the verticies and foci, and the equations of the asymptotes. HOMEWORK (iii) Sketch a graph of the hyperbola.

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