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04 Introduction to Analytic Geometry

04 Introduction to Analytic Geometry. College Algebra. 4.1 Coordinate Systems. Origin Line Plane Point Coordinate (a,b) same as (x,y) Units Three Space (a,b,c). z. y. a. 0. b. y. x. x. Dimensions. 1-D 2-D 3-D. z. y. a. 0. b. y. x. x. Dimensions. 1-D 2-D 3-D.

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04 Introduction to Analytic Geometry

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  1. 04 Introduction to Analytic Geometry College Algebra

  2. 4.1 Coordinate Systems • Origin • Line • Plane • Point • Coordinate (a,b) same as (x,y) • Units • Three Space (a,b,c)

  3. z y a 0 b y x x Dimensions • 1-D • 2-D • 3-D

  4. z y a 0 b y x x Dimensions • 1-D • 2-D • 3-D

  5. Coordinate System Review y II I x Origin III IV

  6. Line vs Line Segment y y = mx + b x

  7. Line vs Line Segment y AB B x A

  8. b a Ordered Pairs Review : (a,b) II I (-a,b) (a,b) III IV (-a,-b) (a,-b)

  9. Transformations & Coordinate Geometry

  10. Transformations – Model Motion • Translation – Glide or Slide • Rotation – (about an axis) • Reflection – Mirror image • Dilation – larger or smaller

  11. Terminology • Image – final image after transformation Labeled with “Prime” • Pre-image – image before transformation Labeled with Capital Letters B B’ Horizontal Translation A A’ Pre-Image Image

  12. Translation A Pre-Image Image Slide Arrow C B A’ B’ C’

  13. Rotation – 90° 180° 270° 45° ? ° Pre-Image Image 270° Image 90° Image 180° Note: Example Rotation is Clockwise

  14. Reflection Pre-Image Image Mirror Line

  15. ( x , y ) ( x , y ) ( x , y ) ( x , y ) ( x , y ) ( x , y ) ( x + h, y ) ( x , y + v ) ( x , -y ) ( -x , y ) ( -x , -y ) ( nx, ny ) Graphing Motion Pre-Image Image Horizontal Translation Vertical Translation Reflection through x-axis Reflection through y-axis 180 Rotation about Origin Dilation

  16. Back to the text… • Distance Formula • 1-D • 2-D • 3-D

  17. a 0 b 1-D • | b-a | or • | a-b |

  18. 2-D: “THE” Distance formula • what do you know about the distance formula???

  19. 2-D: “THE” Distance formula B A

  20. 2-D: “THE” Distance formula B A

  21. 2-D: “THE” Distance formula (5,7) (-3,2) d = sqrt((5- -3)2 + (7-2)2)

  22. 2-D: “THE” Distance formula (5,7) (-3,2) d = (5- -3)2 + (7-2)2

  23. 2-D: “THE” Distance formula (5,7) (-3,2) d = (8)2 + (5)2

  24. 2-D: “THE” Distance formula (5,7) (-3,2) d = 64 + 25

  25. 2-D: “THE” Distance formula (5,7) (-3,2) d = 89 = 9.434

  26. Who can do the Pythagorean Theorem? (5,7) d (-3,2) ( 5,2)

  27. Who can do the Pythagorean Theorem? (5,7) d 5 (-3,2) ( 5,2) 8 d = 82 + 52

  28. Who can do the Pythagorean Theorem? (5,7) d 5 (-3,2) ( 5,2) 8 d = 64 + 25

  29. The distance formula and the Pythagorean Theorem are very similar.

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