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Math III

The. Math III. Sections. TITD. Midpoint Distance Completing the Square. Conic Sections. The intersection of a plane and a cone. http://www.math.odu.edu/cbii/calcanim/consec.avi. STANDARDS. MGSE9-12.A.REI.7

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Math III

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  1. The Math III Sections

  2. TITD Midpoint Distance Completing the Square

  3. Conic Sections The intersection of a plane and a cone. http://www.math.odu.edu/cbii/calcanim/consec.avi

  4. STANDARDS • MGSE9-12.A.REI.7 • Solve a simple system consisting of a linear equation and quadratic equation in two variables algebraically and graphically.

  5. STANDARDS • MGSE9-12.G.GPE.2 • Derive the equation of a parabola given a focus and directrix. • MGSE9-12.G.GPE.3 • Derive the equation of ellipses and hyperbolas given to foci for the ellipse, and two directrices for the hyperbolas

  6. ESSENTIAL QUESTIONS • How do I indentify the characteristics of circles from equations? • What characteristics of circles are necessary to graph and write the equations of circles?

  7. KEY VOCABULARY • Cone • Coplanar • Focus • Directrix • Circle • Equidistance • Center • Radius • General form • Standard form

  8. The Conic Sections Index The Conics Translations Completing the Square Classifying Conics

  9. The Conics Circle Ellipse Click on a Photo Hyperbola Parabola Back to Index

  10. The Parabola A parabola is formed when a plane intersects a cone and the base of that cone Back to Conics Back to Index

  11. Parabolas Around Us Back to Conics

  12. Parabolas • A Parabola is a set of points equidistant from a fixed point and a fixed line. • The fixed point is called the focus. • The fixed line is called the directrix. Back to Conics

  13. Parabolas Parabola FOCUS Directrix Back to Conics

  14. Standard form of the equation of a parabola with vertex (0,0) Back to Conics

  15. To Find p Example: x2=24y 4p=24 p=6 4p is equal to the term in front of x or y. Then solve for p. Back to Conics

  16. Examples for ParabolasFind the Focus and Directrix Example 1 y = 4x2 x2= (1/4)y 4p = 1/4 p = 1/16 FOCUS (0, 1/16) Directrix Y = - 1/16 Back to Conics

  17. Examples for ParabolasFind the Focus and Directrix Example 2 x = -3y2 y2= (-1/3)x 4p = -1/3 p = -1/12 FOCUS (-1/12, 0) Directrix x = 1/12 Back to Conics

  18. Examples for ParabolasFind the Focus and Directrix Example 3 (try this one on your own) y = -6x2 FOCUS ???? Directrix ???? Back to Conics

  19. Examples for ParabolasFind the Focus and Directrix FOCUS (0, -1/24) Example 3 y = -6x2 Directrix y = 1/24 Back to Conics

  20. Examples for ParabolasFind the Focus and Directrix Example 4 (try this one on your own) x = 8y2 FOCUS ???? Directrix ???? Back to Conics

  21. Examples for ParabolasFind the Focus and Directrix FOCUS (2, 0) Example 4 x = 8y2 Directrix x = -2 Back to Conics

  22. Parabola Examples Now write an equation in standard form for each of the following four parabolas Back to Conics

  23. Write in Standard Form Example 1 Focus at (-4,0) Identify equation y2 =4px p = -4 y2 = 4(-4)x y2 = -16x Back to Conics

  24. Write in Standard Form Example 2 With directrix y = 6 Identify equation x2 =4py p = -6 x2 = 4(-6)y x2 = -24y Back to Conics

  25. Write in Standard Form Example 3 (Now try this one on your own) With directrix x = -1 y2 = 4x Back to Conics

  26. Write in Standard Form Example 4 (On your own) Focus at (0,3) x2 = 12y Back to Conics

  27. Circles A Circle is formed when a plane intersects a cone parallel to the base of the cone. Back to Index Back to Conics

  28. Circles in real life Back to Conics

  29. Standard Equation of a Circle with Center (0,0) Back to Conics

  30. Circles & Points of Intersection Distance formula used to find the radius Back to Conics

  31. CirclesExample 1 Write the equation of the circle with the point (4,5) on the circle and the origin as it’s center. Back to Conics

  32. Example 1 Point (4,5) on the circle and the origin as it’s center. Back to Conics

  33. Example 2Find the intersection points on the graph of the following two equations Back to Conics

  34. Now what??!!??!!?? Back to Conics

  35. Example 2Find the intersection points on the graph of the following two equations Substitute these in for x. Back to Conics

  36. Example 2Find the intersection points on the graph of the following two equations Back to Conics

  37. An ellipses is formed when a plane intersects a cone without being parallel or perpendicular to the base of the cone. Ellipses Back to Conics Back to Index

  38. Ellipses Examples of Ellipses Back to Conics

  39. Ellipses Horizontal Major Axis Back to Conics

  40. FOCI (-c,0) & (c,0) CO-VERTICES (0,b)& (0,-b) The Equation Vertices (-a,0) & (a,0) CENTER (0,0)

  41. Ellipses Vertical Major Axis Back to Conics

  42. FOCI (0,-c) & (0,c) CO-VERTICES (b, 0)& (-b,0) The Equation Vertices (0,-a) & (0, a) CENTER (0,0) Back to Conics

  43. Ellipse Notes • Length of major axis = a (vertex & larger #) • Length of minor axis = b (co-vertex & smaller#) • To Find the foci (c) use: c2 = a2 - b2 Back to Conics

  44. Ellipse ExamplesFind the Foci and Vertices Back to Conics

  45. Ellipse ExamplesFind the Foci and Vertices Back to Conics

  46. Write an equation of an ellipse whose vertices are (-5,0) & (5,0) and whose co-vertices are (0,-3) & (0,3). Then find the foci. Back to Conics

  47. Write the equation in standard form and then find the foci and vertices. Back to Conics

  48. The Hyperbola An hyperbola is formed when a plane intersects a cone parallel to the axis of the cone. Back to Conics Back to Index

  49. Hyperbola Examples Back to Conics

  50. Asymptotes Vertices (a,0) & (-a,0) Foci (c,0) & (-c, 0) Hyperbola NotesHorizontal Transverse Axis Center (0,0) Back to Conics

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