1. Precalculus Warm-Up Graph the conic. Find center, vertices, and foci.

2. Copyright © 2010 Pearson Education, Inc. 9.1Parabolas 9.2Ellipses 9.3Hyperbolas Conic Sections 9

3. Slide 7.1 - 3 Copyright © 2010 Pearson Education, Inc. Introduction Conic Sections are named after the different ways a plane can intersect a cone.

4. Slide 7.1 - 4 Copyright © 2010 Pearson Education, Inc. Introduction The three basic conic sections are parabolas, ellipses, and hyperbolas. A circle is an example of an ellipse.

5. Copyright © 2010 Pearson Education, Inc. Hyperbolas ♦Find equations of hyperbolas ♦Graph hyperbolas ♦Learn the reflective property of hyperbolas ♦Translate hyperbolas 9.3

6. Hyperbola  The set of all co-planar points whose difference of the distances from two fixed points (foci) are constant.

7. Hyperbola Co-vertices endpoints of conjugate axis Center: (h, k)

8. Hyperbola Co-vertices endpoints of conjugate axis

9. Hyperbola c 2 = a 2 + b 2

10. Slide 7.3 - 10 Copyright © 2010 Pearson Education, Inc. Hyperbola A hyperbola is the set of points in a plane, the difference of whose distances from two fixed points is constant. Each fixed point is called a focus of the hyperbola.

11. Slide 7.3 - 11 Copyright © 2010 Pearson Education, Inc. Hyperbola The transverse axis is the line segment connecting the vertices, V 1 (a, 0) and V 2 (–a, 0), and its length equals 2a.

12. Slide 7.3 - 12 Copyright © 2010 Pearson Education, Inc. Hyperbola There are two branches–left and right–and two asymptotes (dashed lines). Conjugate axis is line segment connecting points (0, ±b). Dashed rectangle is fundamental rectangle.

13. Slide 7.3 - 13 Copyright © 2010 Pearson Education, Inc. Hyperbola Upper branch and lower branch. Two asymptotes (dashed lines). Conjugate axis is line segment connecting points (±b, 0). Dashed rectangle is fundamental rectangle.

14. Slide 7.3 - 14 Copyright © 2010 Pearson Education, Inc. Standard Equation for Hyperbolas Centered at (h, k) The hyperbola with center (h, k), and a horizontal transverse axis satisfies the following equation, where c 2 = a 2 + b 2. Vertices: (h ± a, k) Foci: (h ± c, k) Asymptotes:

15. Slide 7.3 - 15 Copyright © 2010 Pearson Education, Inc. Standard Equation for Hyperbolas Centered at (h, k) The hyperbola with center (h, k), and a vertical transverse axis satisfies the following equation, where c 2 = a 2 + b 2. Vertices: (h, k ± a) Foci: (h, k ± c) Asymptotes:

16. Slide 7.3 - 16 Copyright © 2010 Pearson Education, Inc. Example 1 Sketch the graph of Label vertices, foci, and asymptotes. Solution Equation is in standard form with a = 2 and b = 3. It has a horizontal transverse axis with vertices (±2, 0) Endpoints of conjugate axis are (0, ±3). Find c.

17. Slide 7.3 - 17 Copyright © 2010 Pearson Education, Inc. Example 1 Solution continued foci: asymptotes:

18. Slide 7.3 - 18 Copyright © 2010 Pearson Education, Inc. Example 2 Find the equation of the hyperbola centered at the origin with a vertical transverse axis of length 6 and focus (0, 5). Also find the equations of its asymptotes. Solution Since the hyperbola is centered at the origin with a vertical axis, its equation is

19. Slide 7.3 - 19 Copyright © 2010 Pearson Education, Inc. Example 2 Solution continued Transverse axis has length 6 = 2a, so a = 3. One focus is (0, 5) so c = 5. Find b. Standard equation is asymptotes are

20. Slide 7.3 - 20 Copyright © 2010 Pearson Education, Inc. Example 5 Graph the hyperbola whose equation is Label the vertices, foci, and asymptotes. Solution Vertical transverse axis. Center: (2, –2) a 2 = 9, b 2 = 16 c 2 = a 2 + b 2 = 9 + 16 = 25. a = 3, b = 4, c = 5

21. Slide 7.3 - 21 Copyright © 2010 Pearson Education, Inc. Example 5 Solution continued Center: (2,  2) a = 3, b = 4, c = 5 Vertices 3 units above and below center (2, 1), (2, –5) Foci 5 units above and below center: (2, 3), (2, –7) Asymptotes:

22. Slide 7.3 - 22 Copyright © 2010 Pearson Education, Inc. Example 6 Write 9x 2 – 18x – 4y 2 – 16y = 43 in the standard form for a hyperbola centered at (h, k). Identify the center, vertices and foci. Solution:

23. Slide 7.3 - 23 Copyright © 2010 Pearson Education, Inc. Example 6 Solution continued The center is (1, –2). Because a = 2 and the transverse axis is horizontal, the vertices are (1 ± 2, –2). The foci are (1 ±, –2).

24. Graph the following Hyperbola. Find the vertices, foci and asymptotes Center: (-1, 5) a = 4 in x direction b = 7 in y direction

25. Graph the following Hyperbola. Find the vertices, foci and asymptotes. Center: (-1, 5) a = 4 b = 7 a 2 + b 2 = c 2 4 2 + 7 2 = c 2 65 = c 2 16 + 49 = c 2

26. Graph the following Hyperbola. Find the vertices, foci and asymptotes Asymptotes

27. Graph the following Hyperbola. Find the vertices, foci and asymptotes Asymptotes

28. Graph the following Hyperbola. Find the vertices, foci and asymptotes Asymptotes: Center: (-1, 5) Vertices: (-5, 5) (3, 5) Co-Vertices: (-1, 12) (-1, -2) Foci:

29. Homework: pg. 656 1-41 odd

30. Precalculus Random Conics HWQ Find the standard form of the equation of a parabola with vertex (-2, 1) and directrix at x=1

31. Graph the following Hyperbola 4x 2 + 16x - 9y 2 + 72y - 5 = 87 4x 2 + 16x - 9y 2 + 72y = 87 + 5 4(x 2 + 4x + 2 2 ) - 9(y 2 - 8y + (-4) 2 ) = 92 + 16 - 144 4(x + 2) 2 - 9(y - 4) 2 = -36

32. Graph the following Hyperbola Center: (-2, 4) a = 2 in y direction b = 3 in x direction

33. Graph the following Hyperbola Center: (-2, 4) a = 2 b = 3 a 2 + b 2 = c 2 2 2 + 3 2 = c 2 13 = c 2 4 + 9 = c 2

34. Graph the following Hyperbola Asymptotes

35. Graph the following Hyperbola Asymptotes

36. Graph the following Hyperbola Asymptotes Center: (-2, 4) Vertices: (-2, 6) (-2, 2) Co-Vertices: (-5, 4) (1, 4) Length of Transverse axis: 4 Length of Conjugate axis: 6 Foci:

37. Slide 7.3 - 37 Copyright © 2010 Pearson Education, Inc. Trajectory of a Comet One interpretation of an asymptote relates to trajectories of comets as they approach the sun. Comets travel in parabolic, elliptic, or hyperbolic trajectories. If the speed of a comet is too slow, the gravitational pull of the sun will capture the comet in an elliptical orbit.

38. Slide 7.3 - 38 Copyright © 2010 Pearson Education, Inc. Trajectory of a Comet If the speed of the comet is too fast, the comet will pass by the sun once in a hyperbolic trajectory; farther from the sun, gravity becomes weaker and the comet will eventually return to a straight-line trajectory that is determined by the asymptote of the hyperbola.

39. Slide 7.3 - 39 Copyright © 2010 Pearson Education, Inc. Trajectory of a Comet Finally, if the speed is neither too slow nor too fast, the comet will travel in a parabolic path. In all three cases, the sun is located at a focus of the conic section.

40. Slide 7.3 - 40 Copyright © 2010 Pearson Education, Inc. Reflective Property of Hyperbolas Hyperbolas have an important reflective property. If a hyperbola is rotated about the x-axis, a hyperboloid is formed.

41. Slide 7.3 - 41 Copyright © 2010 Pearson Education, Inc. Reflective Property of Hyperbolas Any beam of light that is directed toward focus F 1 will be reflected by the hyperboloid toward focus F 2.

42. Homework: pg. Conics Worksheet