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Topics in Analytic Geometry

Topics in Analytic Geometry

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Topics in Analytic Geometry

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  1. Topics in Analytic Geometry Pre Calc Chapter 9

  2. Parabolas—9.1

  3. Parabolas • Vertex • The lowest or highest point of the graph (based on which way it opens) • Axis of symmetry

  4. Geometric Definition • A parabola is the set of points in the plane equidistant from a fixed point F (the focus) and a fixed line l called the directrix

  5. Parabolas

  6. Analytic Geometry • Concerned with shapes, not necessarily functionality • Parabolas can open up • Parabolas can open down • Inverse

  7. Parabola with Vertical Axis The Graph of the equation:is a parabola with the following: Vertex Focus Directrix Parabola opens up if or down if

  8. Parabola with Horizontal Axis The Graph of the equation:is a parabola with the following: Vertex Focus Directrix Parabola opens right if or left if

  9. Focal Diameter • Distance across the parabola along the line parallel to the directrix

  10. Examples

  11. Examples

  12. Examples

  13. p.730#1-10,25-27,37-40

  14. Ellipses—9.2

  15. Geometric Definition • An ellipse is the set of all points in the plane the sum of whose distances from two fixed points and is a constant. • These points are the foci

  16. Ellipse Equation Vertices Major Axis Horizontal, length 2a Vertical, 2a Minor Axis Vertical, length 2b Horizontal, 2b Foci

  17. Ellipses

  18. Ellipses

  19. Ellipses

  20. Eccentricity • For the ellipse or the eccentricity, e, is the numberwhere and the eccentricity of every ellipse satisfies

  21. Ellipses • Find the equation of the ellipse with foci and eccentricity

  22. p.741#1-8,19-22,29-31

  23. Hyperbolas—9.3

  24. Geometric Definition • A hyperbola is the set of all points in the plane, the difference of whose distances from two fixed points and is a constant. • These are the foci of the hyperbola

  25. Hyperbolas Equations Vertices Transverse Axis Horizontal, length 2a Asymptotes Foci

  26. Hyperbolas Equations Vertices Transverse Axis Vertical, length 2b Asymptotes Foci

  27. Sketching Hyperbolas • Sketch the Central Box • Sketch the asymptotes • Plot the vertices • Sketch the hyperbola • Smile 

  28. Hyperbolas

  29. Hyperbolas

  30. Hyperbolas • Find the equation of the hyperbola with vertices and foci

  31. p.751#1-8,17-19

  32. Shifted Conics—9.4

  33. Shifting Conics

  34. Shifted Conics

  35. Shifted Conics

  36. Shifted Conics

  37. General Equation of a Conic • The graph of the equationWhere A and C are not both 0, is a conic or degenerate conic where the graph is: • A parabola if A or C is 0 • An ellipse if A and C have the same sign • Circle if A = C • A hyperbola is A and C have opposite signs

  38. Conics

  39. Degenerate Conic • Conic which simplifies to only 2 lines

  40. p.760#1,2,5,6,9,10, 19-23 (10 pt)

  41. Rotation of Axes—9.5

  42. Rotation of Axes • Recall… • Now…

  43. Rotation of Axes Formula • Suppose the x- and y-axes in a coordinate plane are rotated through the acute angle to produce the X- and Y-axes. Then the coordinates (x,y) and (X,Y) of a point in the xy- and XY-panes are related as follows: x=Xcosφ-Ysinφ X=xcosφ+ysinφ y=Xsinφ+Ycosφ Y=-xsinφ+ycosφ

  44. Rotation of Axes • If the coordinates are rotated 30 degrees, find the XY-coordinates of the point with xy-coordinates (2, -4)

  45. Rotation of Axes • Rotate the coordinate axes through 45 degrees to show that the graph of the equation xy = 2 is a hyperbola

  46. Identifying Conics by the Discriminant

  47. Polar Coordinates—9.6

  48. Polar Coordinates • Uses distances and directions to specify locations on the plane • Origin (Pole) • Polar Axis • Polar Coordinates