1. 12/7/2015Math 120 - KM1 Chapter 3 3.1 Cartesian Coordinates3.1 3.2 Linear Functions: Graphs and Slope3.2 3.3 More on Graphing Linear Equations 3.4 Finding Equations of Lines; Applications

2. 12/7/2015Math 120 - KM2 The Cartesian Plane Rene Descartes (1596 - 1650) 2.1

3. 12/7/2015Math 120 - KM3 Coordinates of Points 2.1

4. 12/7/2015Math 120 - KM4 Plotting Points 2.1

5. 12/7/2015Math 120 - KM5 Plot: y = 2x - 1 xy 0 1/20 11 2.1

6. 12/7/2015Math 120 - KM6 Plot: y = -2x xy 00 00 1-2 2-4 2.1

7. 12/7/2015Math 120 - KM7 2.2 Linear Functions: Graphs and Slope 2.4

8. 12/7/2015Math 120 - KM8 Linear Function: f(x) = mx + b Remember y is the same as f(x). f(x) = mx + b or y = mx + b (x, y)(x, f(x)) 2.4

9. 12/7/2015Math 120 - KM9 Linear Function: f(x) = mx + b What does b do? y = x y = x + 3 y = x - 4 (0,3) (0,0) (0,-4) 2.4

10. 12/7/2015Math 120 - KM10 Linear Function: f(x) = mx + b What does m do? y = x y = 3x y = -4x (1,3) (1,1) (1,-4) 2.4

11. 12/7/2015Math 120 - KM11 y = mx+b (0, b) is the y-intercept m is the slope y = mx + b (0,b) RUN 1 RISE m 2.4

12. 12/7/2015Math 120 - KM12 EquationSlopey-intercept y = 2x - 5 y = -x + 1 y = x y = 3 x = 2 3x- 2y - 4 = 0 y = mx+b m is the slope (0, b) is the y-intercept (0,-5) (0, 1) (0,0) (0,3) no y-intercept m = 2 m = -1 m = 1 m = 0 no slope (0,-2) m = 3/2 2.4

13. 12/7/2015Math 120 - KM13 SLOPE BASICS  POSITIVE SLOPE  The line rises from left to right.  ZERO SLOPE  The line is HORIZONTAL (zero rise)  NEGATIVE SLOPE  The line falls from left to right.  UNDEFINED SLOPE  The line is VERTICAL (zero run) 2.4

14. 12/7/2015Math 120 - KM14 Computing the Slope: (x 1,y 1 ) and (x 2, y 2 ) are coordinates of two points 2.4

15. 12/7/2015Math 120 - KM15 A Slope Triangle P 1 (x 1, y 1 ) RUN x 2 - x 1 RISE y 2 - y 1 P 2 (x 2, y 2 ) RISE RUN 2.4

16. 12/7/2015Math 120 - KM16 Compute the Slope (-2, -3) (4, -1) Upwards + Slope 2.4

17. 12/7/2015Math 120 - KM17 Another Slope (-4, -3) (0, 5) Upwards + Slope 2.4

18. 12/7/2015Math 120 - KM18 Compute this Slope (-5, 2) (4, -1) DOWNWARDS - Slope 2.4

19. 12/7/2015Math 120 - KM19 Compute the Slope (-5, -1) (4, -1) HORIZONTAL O 2.4

20. 12/7/2015Math 120 - KM20 Compute the Slope (-4, -5) (-4, 0) VERTICAL no slope 2.4

21. 12/7/2015Math 120 - KM21 Practice Problem 1a Determine the slope of the line containing points P 1 and P 2. RUN = 3 RISE = 2 The line rises from left to right. 2.4

22. 12/7/2015Math 120 - KM22 Practice Problem 1b Determine the slope of the line containing points P 1 and P 2. The slope is undefined. Vertical Line 2.4

23. 12/7/2015Math 120 - KM23 Practice Problem 1c Determine the slope of the line containing points P 1 and P 2. RUN = 2 RISE = -1 The line falls from left to right. 2.4

24. 12/7/2015Math 120 - KM24 Practice Problem 1d Determine the slope of the line containing points P 1 and P 2. RUN = 6 RISE = 0 The line is Horizontal. 2.4

25. 12/7/2015Math 120 - KM25 2.5 More on Graphing Linear Equations 2.5

26. 12/7/2015Math 120 - KM26 Intercepts x-intercept (, 0) Y – intercept (0, ) 2.5

27. 12/7/2015Math 120 - KM27 Sketch the Graph x-intercept (, 0) Y – intercept (0, ) (3, 0) (0, 6) 2.5

28. 12/7/2015Math 120 - KM28 Sketch the Graph x-intercept (, 0) Y – intercept (0, ) (1, 0) (1, 3) 2.5

29. 12/7/2015Math 120 - KM29 Sketch the Graph x-intercept (, 0) Y – intercept (0, ) (3, 0) (0, -2) 2.5

30. 12/7/2015Math 120 - KM30 Sketch the Graph x-intercept (, 0) Y – intercept (0, ) (3, 2) (0, 2) 2.5

31. 12/7/2015Math 120 - KM31 Sketch the Graph x-intercept (, 0) Y – intercept (0, ) (3, 2) (0, 0) 2.5

32. 12/7/2015Math 120 - KM32 Graph a Line using a Point and the Slope P 1 (x 1, y 1 ) P 2 (x 1 + RUN, y 1 + RISE) RUN RISE 2.5

33. 12/7/2015Math 120 - KM33 Graph (0, - 4) is the y-intercept 2/5 is the slope Run 5 then rise 2 (0+5, - 4+2) or (5, -2) is the second point. (0, -4) (5, -2) 2.5

34. 12/7/2015Math 120 - KM34 Graph (0, 4) (2, 3) (0, 4) is the y-intercept -1/2 is the slope Run 2 then rise -1 (0+2, 4+-1) or (2, 3) is the second point. 2.5

35. 12/7/2015Math 120 - KM35 Graph (0, 0) (3, -2) Solve for y (0, 0) is the y-intercept -2/3 is the slope Run 3 then rise -2 (0 + 3, 0 + -2) or (3, -2) is the second point. 2.5

36. 12/7/2015Math 120 - KM36 Uh oh, no y ! (-4, 0) The x-coordinate is always - 4, so this is a vertical line! Plot (-4, #) 2.5

37. 12/7/2015Math 120 - KM37 Uh Oh, no x? (0, 2) The y-coordinate is always 2, so this is a horizontal line! Plot (#, 2) 2.5

38. 12/7/2015Math 120 - KM38 Special Pairs of Lines 2.5

39. 12/7/2015Math 120 - KM39 Parallel Lines Are in the same plane And never intersect! Parallel Lines have the same slope! 2.5

40. 12/7/2015Math 120 - KM40 Perpendicular Lines Intersect to form a right angle (90 degrees)! Perpendicular lines have slopes that are negative reciprocals The slopes multiply to -1. (as long as they are not vertical or horizontal.) 2.5

41. 12/7/2015Math 120 - KM41 mm || mm What’s My Slope? 2.5

42. 12/7/2015Math 120 - KM42 mm || mm What’s My Slope? 2.5

43. 12/7/2015Math 120 - KM43 mm || mm What’s My Slope? 2.5

44. 12/7/2015Math 120 - KM44 mm || mm What’s My Slope? 2.5

45. 12/7/2015Math 120 - KM45 mm || mm What’s My Slope? 2.5

46. 12/7/2015Math 120 - KM46 Solve this Mystery! Are these lines parallel 2.5

47. 12/7/2015Math 120 - KM47 Solve this Mystery! Are these lines parallel 2.5

48. 12/7/2015Math 120 - KM48 OK … here it comes! Determine the equation of the line through the point (4,1) that is perpendicular to y = -3x+7 2.5

49. 12/7/2015Math 120 - KM49 2.6 Finding Equations of Lines; Applications 2.6

50. 12/7/2015Math 120 - KM50 Linear Equations 2.6

51. 12/7/2015Math 120 - KM51 What’s my line? Clues ! 2.6

52. 12/7/2015Math 120 - KM52 Look Closely! Aha! 2.6

53. 12/7/2015Math 120 - KM53 Where’s m ? Uh oh! 2.6

54. 12/7/2015Math 120 - KM54 Look ! 2.6

55. 12/7/2015Math 120 - KM55 I bet I can’t fool you ! 2.6

56. 12/7/2015Math 120 - KM56 A Chemistry Problem? In the Fahrenheit system, water freezes at 32 o F and boils at 212 o F. In the Celsius system, water freezes at 0 o C and boils at 100 o C. Using (F, C), (32 o, 0 o ), and (212 o, 100 o ) determine a formula to calculate C if F is known! 2.6

57. 12/7/2015Math 120 - KM57 A Chemistry Problem - clues? (F, C) (32 o, 0 o ) (212 o, 100 o ) 2.6

58. 12/7/2015Math 120 - KM58 A Chemistry Problem - clues? (F, C) (32 o, 0 o ) (212 o, 100 o ) (F 1, C 1 ) 2.6

59. 12/7/2015Math 120 - KM59 Up in the Air? An ultralight crop duster is climbing at a rate of 800 ft/min. Write an equation which gives the height of the plane in terms of the time after take off. What is the height 3 minutes after take off? 2.6

60. 12/7/2015Math 120 - KM60 Up in the Air – Clues! An ultralight crop duster is climbing at a rate of 800 ft/min. 2.6

61. 12/7/2015Math 120 - KM61 Up in the Air – Clues! The crop duster takes off from ground level. So his height at the start is 0 feet. 2.6

62. 12/7/2015Math 120 - KM62 Up in the Air – Clues! What is the height 3 minutes after take off? t = 3 2.6

63. 12/7/2015Math 120 - KM63 That’s All For Now!

64. 12/7/2015Math 120 - KM64 Where We Left Off Last Class