1. Introduction to Graphs

2. Dependent variable is on the vertical axis (Y) Dependent variable is on the vertical axis (Y) Independent variable is on the horizontal axis (X) Independent variable is on the horizontal axis (X) Y X (“ Leppel ”)

3. STUDYING Grade v. Hours of Study Your course grade represents: A = 4, B = 3, C = 2, D = 1, and F = 0. (“ Leppel ”) Upward Sloping Lines

4. grade hrs. studied per week 43214321 0 2 4 6 8 study time grade 8 4 6 3 4 2 2 1 0 0 (“Introduction to Graphs”)

5. Your grade and the number of hours you study move in the same direction. Your grade and the number of hours you study move in the same direction. When you look left to right, you notice the line slopes upward. When you look left to right, you notice the line slopes upward. “This is called a positive or direct relation.” “This is called a positive or direct relation.” (“Introduction to Graphs”) grade hrs. studied per week 43214321 0 2 4 6 8

6. your grade hrs. studied your grade hrs. studied your grade hrs. studied your grade hrs. studied (“ Leppel ”)

7. What would your grade be, if you studied for two hours per week? grade hrs. studied per week 43214321 0 2 4 6 8 1 (D) 1 (D) (“ Leppel ”)

8. What would your grade be, if you studied for eight hours per week? grade hrs. studied per week 43214321 0 2 4 6 8 4 (A) 4 (A) (“ Leppel ”)

9. What would your grade be, if you studied for zero hours per week? grade hrs. studied per week 43214321 0 2 4 6 8 0 (F) 0 (F) (“ Leppel ”)

10. “At what number does the line intersect the vertical axis?” grade hrs. studied per week 43214321 0 2 4 6 8 0 (“ Leppel ”)

11. The number 0 is the value of the Y-intercept. y-intercept grade 43214321 0 2 4 6 8 The Y-intercept tells you the value of the Y variable (grade), when the value of the X variable (hrs. of study) is zero. (“ Leppel ”)

12. study time grade 8 4 6 3 4 2 2 1 0 0 If you normally study 2 hours per week, and decided to study an additional 2 hours per week. “By how much does your grade increase?” (2 – 1 = ?) 1 If you study 6 hours per week. Then, you decide to study an additional 2 hours per week. “By how much does your grade increase?” (4 – 3 = ?) 1 What would be the change in the Y variable (grade) divided by the change in the X variable (study time)? (“ Leppel ”) ∆Y/ ∆X The slope of the relation would be? (slope = ∆Y/ ∆X ) 1/2 1/2 =.5 21

13. grade changes 1 hrs. studied changes 2 grade changes 1 hrs. studied changes 2 change/change 1/2 change/change 1/2 slope 1/2 =.5 The slope = the change in the Y variable divided by the change in the X variable =  Y/  X = 1/2 =.5 The number.5 is the slope. (“ Leppel ”) RECAP!

14. Also expressed as the "rise" over the "run.” Also expressed as the "rise" over the "run.” “It is the distance the line “rises” in the vertical direction divided by the distance it “runs” in the horizontal direction.” “It is the distance the line “rises” in the vertical direction divided by the distance it “runs” in the horizontal direction.” The Slope Formula (“ Leppel ”)

15. slope = rise/run = 1/2 grade hrs. studied per week 43214321 0 2 4 6 8 rise = 1 run = 2 (“ Leppel ”)

16. RUNNING ILLUSTRATION Suppose the more rested you were the faster you could run. So, let’s use the relation between hours slept per day and the number of minutes it takes you to run a mile. Downward Sloping Lines http://images.google.com/images?gbv=2&hl=en&q=a+runner (“ Leppel ”)

17. min. per mile hrs. slept/day 0 1 2 3 4 5 6 7 8 9 10 8 7 6 5 4 hrs min/mi 6 8 7 7 8 6 9 5 10 4 (“ Leppel ”)

18. “What is the slope of the relation?” slope =  Y/  X slope =  Y/  X =  min/  hrs =  min/  hrs = -1/+1 = -1 = -1/+1 = -1 *An increase denotes a positive change. *A decrease denotes a negative change. hrs min/mi 6 8 7 7 8 6 9 5 10 4 (“ Leppel ”)

19. Amount of sleep minutes needed to run a mile. “The variables move in opposite directions. This type of relation is called a negative or inverse relation.” (“ Leppel ”)

20. Y X “Negative or inverse relations are downward sloping from left to right.” Y X (“ Leppel ”) negative slope positive slope “Positive or direct relations are upward sloping from left to right.”

21. What is the Y-intercept for this relation? What is the Y-intercept for this relation? (“ Leppel ”) 14 is the Y-intercept hours slept min/mile 6 8 5 9 4 10 3 11 2 12 1 13 0 14 “We know it takes 8 minutes to run a mile when you have had 6 hours of sleep.” Working down to zero for the number of hours slept, you will need 14 minutes to run the mile.

22. min./mile hrs. slept/day 0 1 2 3 4 5 6 7 8 9 10 15 12 9 6 3 y-intercept “The Y-intercept tells the value of the Y variable (minutes needed to run a mile) when the value of the X variable (hours slept) is zero.” “The Y-intercept tells the value of the Y variable (minutes needed to run a mile) when the value of the X variable (hours slept) is zero.” “You can also find the intercept by extending the line in the graph to the vertical axis.” (“ Leppel ”)

23. MEDICATION ILLUSTRATION Suppose you are taking medication for a virus. “The medication has the effect on the number of heartbeats per minute as indicated in the following graph.” Downward Sloping Lines (“ Leppel ”)

24. beats/min. medicine (mg.) 0 100 200 300 400 500 75 70 65 60 55 50 medication beats/min 0 75 100 70 200 65 300 60 400 55 500 50 (“ Leppel ”)

25. beats/min. medicine (mg.) 0 100 200 300 400 500 75 70 65 60 55 50 At what number does the line intersect the vertical axis? What would your heart rate be, if you didn’t take any medication? (“ Leppel ”) 75 What is the Y-intercept?75

26. - 5 (decreases by 5 beats/min.) - 5 (decreases by 5 beats/min.) If you increase your medication from 400 to 500 milligrams, by how much does your heart rate change? - 5 (decreases by 5 beats/min.) - 5 (decreases by 5 beats/min.) What is the change in the Y variable (beats/min) divided by the change in the X variable (medication)? What is the change in the Y variable (beats/min) divided by the change in the X variable (medication)? - 5/100 or -.05 - 5/100 or -.05 What is the slope of the relation? What is the slope of the relation? -.05 -.05 If you increase your medication from 200 to 300 milligrams, by how much does your heart rate change? (“ Leppel ”) medication beats/min 0 75 100 70 200 65 300 60 400 55 500 50

27. The slope is negative, because the variables are inversely related. The slope is negative, because the variables are inversely related. When the amount of medication When the amount of medication, the heart rate., the heart rate. When the amount of medication When the amount of medication, the heart rate., the heart rate. (“ Leppel ”) medication beats/min 0 75 100 70 200 65 300 60 400 55 500 50

28. “The negative slope is evident in the graph by the fact that the line slopes downward toward the right.” “The negative slope is evident in the graph by the fact that the line slopes downward toward the right.” beats mg. (“ Leppel ”)

29. Horizontal Lines Horizontal Lines DIETING ILLUSTRATION Let us suppose that no matter how hard you tried or how few calories you consumed, your weight remained the same. http://images.google.com/images?hl=en&q=scales&gbv=2 (“ Leppel ”)

30. weight 180 170 160 150 140 1000 1100 1200 1300 1400 calories calories weight 1000 180 1100 180 1200 180 1300 180 1400 180 (“ Leppel ”)

31. Y never changes, it stays constant slope = ∆Y/ ∆X = 0/∆X = 0 slope = ∆Y/ ∆X = 0/∆X = 0 “The slope of a horizontal line is zero. “The slope of a horizontal line is zero. Your weight would remain at 180, even if you consumed zero calories. Your weight would remain at 180, even if you consumed zero calories. So, the Y-intercept is 180.” So, the Y-intercept is 180.” Y 180 X (“ Leppel ”)

32. Vertical Lines Now let us suppose you consumed the same number of calories and your weight varied with exercise and stress. http://images.google.com/images?hl=en&q=scales&gbv=2 (“ Leppel ”)

33. weight 180 170 160 150 140 40 0 1000 1100 1200 1300 1400 calories calories weight 1100 140 1100 150 1100 160 1100 170 1100 180 (“ Leppel ”)

34. The Y variable, (weight) changes, but the X variable (calories) remains constant. The slope =  Y/  X In this case, a non-zero number divided by zero. The slope is infinity or undefined. The slope of a vertical line is infinity or undefined, because there is no Y-intercept. wgt 1100 calories (“ Leppel ”)

35. Let’s look at the vertical line graph using the study time and grade concept. Vertical Lines

36. grade hrs. studied per week 43214321 0 2 4 6 8 study time grade 6 4 6 4 6 3 6 3 6 2 6 2 6 1 6 1 6 0 6 0 (“ Leppel ”)

37. grade hrs. studied per week 43214321 0 2 4 6 8 6 How many hours did you study to get a grade of 2 (C)? (“ Leppel ”)

38. grade hrs. studied per week 43214321 0 2 4 6 8 6 How many hours did you study to get a grade of 3 (B)? (“ Leppel ”)

39. grade hrs. studied per week 43214321 0 2 4 6 8 6 How many hours did you study to get a grade of 4 (A)? (“ Leppel ”)

40. You always studied the same amount. So, why did your grade vary? The only reason had to be other factors, such as the amount of sleep you had and/or your diet. (“ Leppel ”)

41. study time grade 6 4 6 3 6 2 6 1 6 0 1/0 1/0 What is the slope of the relation? What is the slope of the relation? undefined or infinity undefined or infinity “What is the change in the Y variable (grade) divided by the change in the X variable (study time)?” (“ Leppel ”)

42. Nonlinear Relations Convex “If a curve looks like the letter U or part of a U, it is convex (from below).” “If a curve looks like the letter U or part of a U, it is convex (from below).” (“ Leppel ”)

43. “This curve is downward sloping and convex from below.” min. per mile hrs. slept per day (“ Leppel ”)

44. “This curve is upward sloping and convex from below. (It bulges toward some reference point, usually the horizontal axis or the origin of a diagram.) (It bulges toward some reference point, usually the horizontal axis or the origin of a diagram.) A curve is convex from below (or convex to something below it) if all straight lines connecting points on it lie on or above it.” calories wgt Convex Curve (“ Leppel ”)

45. “Suppose that you're trying to lose weight.” The chart below “represents your weight and the number of calories you consume per day.” “Suppose that you're trying to lose weight.” The chart below “represents your weight and the number of calories you consume per day.” calories weight 1000 142 1100 143 calories weight 1000 142 1100 143 1200 145 1200 145 1300 150 1300 150 1400 160 1400 160 http://images.google.com/images?hl=en&q=scales&gbv=2 (“ Leppel ”)

46. calories weight calories weight 1000 142 1100 143 1200 145 1200 145 1300 150 1300 150 1400 160 1400 160 “If you reduce your intake from 1400 to 1300 calories, your weight drops 10 pounds. When you reduce your intake from 1300 to 1200 calories, your weight only drops 5 pounds.” (“ Leppel ”)

47. calories weight calories weight 1000 142 1100 143 1000 142 1100 143 1200 145 1200 145 1300 150 1300 150 1400 160 1400 160 “When your reduce your intake from 1200 to 1100 calories, your weight drops just 2 pounds.” (“ Leppel ”)

48. weight 160 155 150 145 140 1000 1100 1200 1300 1400 calories The line is no longer a straight line (linear) relationship. Instead the relation is now curved. The line is no longer a straight line (linear) relationship. Instead the relation is now curved. Reflecting a changing slope. “The slope is the change in the Y-variable (wgt) divided by the change in the X-variable (calories).” (“ Leppel ”)

49. calories weight 1000 140 900 130 800 120 700 110 600 100 500 90 400 80 300 70 200 60 100 50 0 40 What would you weigh if your calories were zero? 40 pounds (“ Leppel ”)

50. Concave “Picture the opening of a cave. If a curve looks like this or part of this, it is concave (from below).” “Picture the opening of a cave. If a curve looks like this or part of this, it is concave (from below).” (“ Leppel ”) Nonlinear Relations

51. “This curve is upward sloping and concave from below.” “This curve is upward sloping and concave from below.” wgt calories

52. The thin person's perspective! Suppose you were trying to gain weight. http://images.google.com/images?hl=en&q=scales&gbv=2 (“ Leppel ”)

53. calories weight 1000 100 1000 100 1100 110 1100 110 1200 115 1200 115 1300 118 1300 118 1400 119 1400 119 By increasing your intake from 1000 to 1100 calories, your weight increased 10 pounds. But, when you increased your intake from 1100 to 1200 calories, your weight only increased 5 pounds. (“ Leppel ”)

54. weight 120 115 110 105 100 1000 1100 1200 1300 1400 calories calories weight 1000 100 1000 100 1100 110 1100 110 1200 115 1200 115 1300 118 1300 118 1400 119 1400 119 (“ Leppel ”)

55. calories weight  calories  wgt slope=  wt/  cal 1000 100 1000 100 100 10.10 100 10.10 1100 110 1100 110 100 5.05 100 5.05 1200 115 1200 115 100 3.03 100 3.03 1300 118 1300 118 100 1.01 100 1.01 1400 119 1400 119 “As calories increase, the slope decreases; the curve gets flatter.” (“ Leppel ”)

56. Recap Graphs!

57. Constant Opportunity Cost Graph Y X

58. Zero Opportunity Cost Graph Y X

59. Decreasing Opportunity Cost or Convex Graph Y X

60. Increasing Opportunity Cost or Concave Graph Y X

61. Works Cited Leppel, Professor Karen. “Introduction to Graphs.” Widener University. 25 Jul 2008. http://www.muse.widener.edu/~kleppel/EC 202_ppt/GRAPHS.PPT http://www.muse.widener.edu/~kleppel/EC 202_ppt/GRAPHS.PPT