Geometry Section 3-3 1112

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Slopes of Lines

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Geometry Section 3-3 1112

  1. 1. Section 3-3 Slopes of LinesMonday, December 19, 2011
  2. 2. Essential Questions How do you find slopes of lines? How do you use slope to identify parallel and perpendicular lines?Monday, December 19, 2011
  3. 3. Vocabulary 1. Slope: 2. Rate of Change:Monday, December 19, 2011
  4. 4. Vocabulary 1. Slope: The ratio of the vertical change to the horizontal change between two points 2. Rate of Change:Monday, December 19, 2011
  5. 5. Vocabulary 1. Slope: The ratio of the vertical change to the horizontal change between two points; Change in y over change in x 2. Rate of Change:Monday, December 19, 2011
  6. 6. Vocabulary 1. Slope: The ratio of the vertical change to the horizontal change between two points; Change in y over change in x rise over run 2. Rate of Change:Monday, December 19, 2011
  7. 7. Vocabulary 1. Slope: The ratio of the vertical change to the horizontal change between two points; Change in y over change in x rise over run 2. Rate of Change: A way to describe slopeMonday, December 19, 2011
  8. 8. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between the horizontal lines. Distance between vertical Distance between horizontalMonday, December 19, 2011
  9. 9. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical Distance between horizontal xMonday, December 19, 2011
  10. 10. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical Distance between horizontal x AMonday, December 19, 2011
  11. 11. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical B Distance between horizontal x AMonday, December 19, 2011
  12. 12. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical B Distance between horizontal x AMonday, December 19, 2011
  13. 13. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical B Distance between horizontal x AMonday, December 19, 2011
  14. 14. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical B Distance between horizontal x AMonday, December 19, 2011
  15. 15. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical B Distance between horizontal x AMonday, December 19, 2011
  16. 16. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical 5 Units B Distance between horizontal x AMonday, December 19, 2011
  17. 17. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical 5 Units B Distance between horizontal x 7 Units AMonday, December 19, 2011
  18. 18. Explore Graph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between y the horizontal lines. Distance between vertical 5 Units B Distance between horizontal x 7 Units A Up 7, Right 5Monday, December 19, 2011
  19. 19. Slope FormulaMonday, December 19, 2011
  20. 20. Slope Formula y 2 − y1 m= x 2 − x1 for points (x 1,y 1),(x 2 ,y 2 )Monday, December 19, 2011
  21. 21. Example 1 Find the slope of the line that goes through the following pairs of points. a. C(−3, 4) and D(8, 1)Monday, December 19, 2011
  22. 22. Example 1 Find the slope of the line that goes through the following pairs of points. a. C(−3, 4) and D(8, 1) (x 1,y 1)Monday, December 19, 2011
  23. 23. Example 1 Find the slope of the line that goes through the following pairs of points. a. C(−3, 4) and D(8, 1) (x 1,y 1) (x 2 ,y 2 )Monday, December 19, 2011
  24. 24. Example 1 Find the slope of the line that goes through the following pairs of points. a. C(−3, 4) and D(8, 1) (x 1,y 1) (x 2 ,y 2 ) y 2 − y1 m= x 2 − x1Monday, December 19, 2011
  25. 25. Example 1 Find the slope of the line that goes through the following pairs of points. a. C(−3, 4) and D(8, 1) (x 1,y 1) (x 2 ,y 2 ) y 2 − y1 1− 4 m= = x 2 − x 1 8 − (−3)Monday, December 19, 2011
  26. 26. Example 1 Find the slope of the line that goes through the following pairs of points. a. C(−3, 4) and D(8, 1) (x 1,y 1) (x 2 ,y 2 ) y 2 − y1 1− 4 −3 m= = = x 2 − x 1 8 − (−3) 11Monday, December 19, 2011
  27. 27. Example 1 Find the slope of the line that goes through the following pairs of points. a. C(−3, 4) and D(8, 1) (x 1,y 1) (x 2 ,y 2 ) y 2 − y1 1− 4 −3 m= = = x 2 − x 1 8 − (−3) 11 Down 3, Right 11Monday, December 19, 2011
  28. 28. Example 1 Find the slope of the line that goes through the following pairs of points. b. E(5, −1) and F(−3, 7)Monday, December 19, 2011
  29. 29. Example 1 Find the slope of the line that goes through the following pairs of points. b. E(5, −1) and F(−3, 7) y 2 − y1 m= x 2 − x1Monday, December 19, 2011
  30. 30. Example 1 Find the slope of the line that goes through the following pairs of points. b. E(5, −1) and F(−3, 7) y 2 − y 1 7 − (−1) m= = x 2 − x 1 −3 − 5Monday, December 19, 2011
  31. 31. Example 1 Find the slope of the line that goes through the following pairs of points. b. E(5, −1) and F(−3, 7) y 2 − y 1 7 − (−1) 8 m= = = x 2 − x 1 −3 − 5 −8Monday, December 19, 2011
  32. 32. Example 1 Find the slope of the line that goes through the following pairs of points. b. E(5, −1) and F(−3, 7) y 2 − y 1 7 − (−1) 8 m= = = = −1 x 2 − x 1 −3 − 5 −8Monday, December 19, 2011
  33. 33. Example 1 Find the slope of the line that goes through the following pairs of points. b. E(5, −1) and F(−3, 7) y 2 − y 1 7 − (−1) 8 m= = = = −1 x 2 − x 1 −3 − 5 −8 Down 1, Right 1Monday, December 19, 2011
  34. 34. Example 1 Find the slope of the line that goes through the following pairs of points. c. G(−1, 2) and H(−1, 7)Monday, December 19, 2011
  35. 35. Example 1 Find the slope of the line that goes through the following pairs of points. c. G(−1, 2) and H(−1, 7) y 2 − y1 m= x 2 − x1Monday, December 19, 2011
  36. 36. Example 1 Find the slope of the line that goes through the following pairs of points. c. G(−1, 2) and H(−1, 7) y 2 − y1 7−2 m= = x 2 − x 1 −1− (−1)Monday, December 19, 2011
  37. 37. Example 1 Find the slope of the line that goes through the following pairs of points. c. G(−1, 2) and H(−1, 7) y 2 − y1 7−2 5 m= = = x 2 − x 1 −1− (−1) 0Monday, December 19, 2011
  38. 38. Example 1 Find the slope of the line that goes through the following pairs of points. c. G(−1, 2) and H(−1, 7) y 2 − y1 7−2 5 m= = = x 2 − x 1 −1− (−1) 0 UndefinedMonday, December 19, 2011
  39. 39. Example 1 Find the slope of the line that goes through the following pairs of points. c. G(−1, 2) and H(−1, 7) y 2 − y1 7−2 5 m= = = x 2 − x 1 −1− (−1) 0 Undefined Up 5, Right 0Monday, December 19, 2011
  40. 40. Example 1 Find the slope of the line that goes through the following pairs of points. c. G(−1, 2) and H(−1, 7) y 2 − y1 7−2 5 m= = = x 2 − x 1 −1− (−1) 0 Undefined Up 5, Right 0 Vertical LineMonday, December 19, 2011
  41. 41. Example 1 Find the slope of the line that goes through the following pairs of points. d. J(3, 4) and K(−2, 4)Monday, December 19, 2011
  42. 42. Example 1 Find the slope of the line that goes through the following pairs of points. d. J(3, 4) and K(−2, 4) y 2 − y1 m= x 2 − x1Monday, December 19, 2011
  43. 43. Example 1 Find the slope of the line that goes through the following pairs of points. d. J(3, 4) and K(−2, 4) y 2 − y1 4−4 m= = x 2 − x 1 −2 − 3Monday, December 19, 2011
  44. 44. Example 1 Find the slope of the line that goes through the following pairs of points. d. J(3, 4) and K(−2, 4) y 2 − y1 4−4 0 m= = = x 2 − x 1 −2 − 3 −5Monday, December 19, 2011
  45. 45. Example 1 Find the slope of the line that goes through the following pairs of points. d. J(3, 4) and K(−2, 4) y 2 − y1 4−4 0 m= = = x 2 − x 1 −2 − 3 −5 Up 0, Left 5Monday, December 19, 2011
  46. 46. Example 1 Find the slope of the line that goes through the following pairs of points. d. J(3, 4) and K(−2, 4) y 2 − y1 4−4 0 m= = = x 2 − x 1 −2 − 3 −5 Up 0, Left 5 Horizontal LineMonday, December 19, 2011
  47. 47. Example 1 Find the slope of the line that goes through the following pairs of points. d. J(3, 4) and K(−2, 4) y 2 − y1 4−4 0 m= = = =0 x 2 − x 1 −2 − 3 −5 Up 0, Left 5 Horizontal LineMonday, December 19, 2011
  48. 48. Example 2 In 2000, the annual sales for one manufacturer of camping equipment was $48.9 million. In 2005, the total sales were $85.9 million. If sales increase at the same rate, what will the total sales be in 2015?Monday, December 19, 2011
  49. 49. Example 2 In 2000, the annual sales for one manufacturer of camping equipment was $48.9 million. In 2005, the total sales were $85.9 million. If sales increase at the same rate, what will the total sales be in 2015? 85.9 − 48.9 =Monday, December 19, 2011
  50. 50. Example 2 In 2000, the annual sales for one manufacturer of camping equipment was $48.9 million. In 2005, the total sales were $85.9 million. If sales increase at the same rate, what will the total sales be in 2015? 85.9 − 48.9 = 37Monday, December 19, 2011
  51. 51. Example 2 In 2000, the annual sales for one manufacturer of camping equipment was $48.9 million. In 2005, the total sales were $85.9 million. If sales increase at the same rate, what will the total sales be in 2015? 85.9 − 48.9 = 37Every 5 years, sales increase by $37 millionMonday, December 19, 2011
  52. 52. Example 2 In 2000, the annual sales for one manufacturer of camping equipment was $48.9 million. In 2005, the total sales were $85.9 million. If sales increase at the same rate, what will the total sales be in 2015? 85.9 − 48.9 = 37Every 5 years, sales increase by $37 million 85.9 + 2(37) =Monday, December 19, 2011
  53. 53. Example 2 In 2000, the annual sales for one manufacturer of camping equipment was $48.9 million. In 2005, the total sales were $85.9 million. If sales increase at the same rate, what will the total sales be in 2015? 85.9 − 48.9 = 37Every 5 years, sales increase by $37 million 85.9 + 2(37) = 159.9Monday, December 19, 2011
  54. 54. Example 2 In 2000, the annual sales for one manufacturer of camping equipment was $48.9 million. In 2005, the total sales were $85.9 million. If sales increase at the same rate, what will the total sales be in 2015? 85.9 − 48.9 = 37Every 5 years, sales increase by $37 million 85.9 + 2(37) = 159.9 In 2015, sales should be about $159.9 millionMonday, December 19, 2011
  55. 55. Postulates Slopes of parallel lines: Slopes of perpendicular lines:Monday, December 19, 2011
  56. 56. Postulates Slopes of parallel lines: Two lines will be parallel IFF they have the same slope. All vertical lines are parallel. Slopes of perpendicular lines:Monday, December 19, 2011
  57. 57. Postulates Slopes of parallel lines: Two lines will be parallel IFF they have the same slope. All vertical lines are parallel. Slopes of perpendicular lines: Two lines will be perpendicular IFF the product of their slopes is −1. Vertical and horizontal lines are perpendicular.Monday, December 19, 2011
  58. 58. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.Monday, December 19, 2011
  59. 59. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.   y −y m(FG ) = 2 1 x 2 − x1Monday, December 19, 2011
  60. 60. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.   y −y −1− (−3) m(FG ) = 2 1 = x 2 − x1 −2 − 1Monday, December 19, 2011
  61. 61. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.   y −y −1− (−3) 2 m(FG ) = 2 1 = = x 2 − x1 −2 − 1 −3Monday, December 19, 2011
  62. 62. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.   y −y −1− (−3) 2 m(FG ) = 2 1 = = x 2 − x1 −2 − 1 −3   y −y m(HJ ) = 2 1 x 2 − x1Monday, December 19, 2011
  63. 63. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.   y −y −1− (−3) 2 m(FG ) = 2 1 = = x 2 − x1 −2 − 1 −3   y −y 3−0 m(HJ ) = 2 1 = x 2 − x1 6 − 5Monday, December 19, 2011
  64. 64. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.   y −y −1− (−3) 2 m(FG ) = 2 1 = = x 2 − x1 −2 − 1 −3   y −y 3−0 3 m(HJ ) = 2 1 = = x 2 − x1 6 − 5 1Monday, December 19, 2011
  65. 65. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.   y −y −1− (−3) 2 m(FG ) = 2 1 = = x 2 − x1 −2 − 1 −3   y −y 3−0 3 m(HJ ) = 2 1 = = =3 x 2 − x1 6 − 5 1Monday, December 19, 2011
  66. 66. Example 3 Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3). Graph each line to verify your answer.   y −y −1− (−3) 2 m(FG ) = 2 1 = = x 2 − x1 −2 − 1 −3   y −y 3−0 3 m(HJ ) = 2 1 = = =3 x 2 − x1 6 − 5 1 NeitherMonday, December 19, 2011
  67. 67. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y xMonday, December 19, 2011
  68. 68. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 m= x 2 − x1 xMonday, December 19, 2011
  69. 69. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 m= = x 2 − x 1 −2 − 2 xMonday, December 19, 2011
  70. 70. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 3 m= = = x 2 − x 1 −2 − 2 −4 xMonday, December 19, 2011
  71. 71. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 3 m= = = x 2 − x 1 −2 − 2 −4 3 m=− 4 xMonday, December 19, 2011
  72. 72. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 3 m= = = x 2 − x 1 −2 − 2 −4 3 Q m=− 4 xMonday, December 19, 2011
  73. 73. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 3 m= = = x 2 − x 1 −2 − 2 −4 3 Q m=− 4 xMonday, December 19, 2011
  74. 74. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 3 m= = = x 2 − x 1 −2 − 2 −4 3 Q m=− 4 xMonday, December 19, 2011
  75. 75. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 3 m= = = x 2 − x 1 −2 − 2 −4 3 Q m=− 4 xMonday, December 19, 2011
  76. 76. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 3 m= = = x 2 − x 1 −2 − 2 −4 3 Q m=− 4 xMonday, December 19, 2011
  77. 77. Example 4 Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4) and N(2, 1). y y 2 − y1 4 −1 3 m= = = x 2 − x 1 −2 − 2 −4 M 3 Q m=− N 4 xMonday, December 19, 2011
  78. 78. Check Your Understanding Use problems 1-11 to check the ideas from this lessonMonday, December 19, 2011
  79. 79. Problem SetMonday, December 19, 2011
  80. 80. Problem Set p. 191 #13-39 odd “I have found power in the mysteries of thought.” - EuripidesMonday, December 19, 2011

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