1. Professor Karen Leppel Economics 202
GRAPHS
Professor Karen Leppel
Economics 202

2. Upward-sloping lines Example 1: DIETING
Consider your weight and the number of calories you consume per day. Suppose the following relation holds.

3. calories weight

4. Graph of Weight and Calories
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calories

5. Graph of Weight and Calories
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calories weight
calories

6. Graph of Weight and Calories
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calories weight
calories

7. Graph of Weight and Calories
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calories weight
calories

8. Graph of Weight and Calories
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calories weight
calories

9. Graph of Weight and Calories
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calories weight
calories

10. Graph of Weight and Calories
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calories weight
calories

11. Graph of Weight and Calories
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calories weight
calories

12. Graph of Weight and Calories
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calories weight
calories

13. Your weight depends on the number of calories you consume.
Your weight is the dependent variable, and the number of calories consumed is the independent variable.
The dependent variable, generally denoted by Y, is on the vertical axis.
The independent variable, generally denoted by X, is on the horizontal axis.

14. your calories your weight

15. your calories your weight

16. The number of calories and your weight move in the same direction.
So when looking from left to right, we see a line that slopes upward.
This is called a positive or direct relation.

17. calories weight
calories / weight /100 calories weight /10 = .1

18. The number .1 is the slope.
The slope is calculated as the change in the Y variable divided by the change in the X variable = D Y/ D X = 10/100 = .1

19. The slope formula is also sometimes 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.

20. slope = rise/run = 10/100 = 1/10 weight 180 170 160 150 140 rise = 10
calories

21. According to the pattern, your weight would be 40 pounds.
calories weight
Theoretically, we can determine what you would weigh if your calories were zero.
According to the pattern, your weight would be 40 pounds.

22. The number 40 is the value of the Y-intercept.
You can also find this number, by drawing the graph and extending the line to the vertical axis.
The Y-intercept tells you the value of the Y variable (weight) when the value of the X variable (calories) is zero.

23. weight
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140 40
y-intercept
calories

24. Equation of a Line: Slope-Intercept Form
Recall that the equation of a line can be written as Y= mX + b, where X is the independent variable, Y is the dependent variable, m is the slope of the line, and b is the vertical intercept.
In our example, the independent variable (X) is calories, the dependent variable (Y) is weight, the slope (m) is 0.1, and the vertical intercept (b) is 40.
So the equation of this line is weight = * calories .

25. Downward Sloping Lines Example 2: RUNNING Suppose that the more rested you are, the faster you can run.
So the more hours you sleep, the fewer minutes it takes you to run a mile.
Suppose the relation between hours slept per day and the number of minutes it takes you to run a mile is as follows.

26. hours slept minutes per mile

27. min./mile hrs min/mi 6 8 7 7 8 6 9 5 10 4 hrs. slept/day 8 7 6 5 4 8 7 6 5 4
hrs. slept/day

28. What is the slope of the relation?
hrs min/mi
slope = D Y/ D X
= D min/ D hrs
= -1/+1 = -1
A positive change denotes an increase. A negative change denotes a decrease.

29. When the amount of sleep increases, minutes needed to run a mile decrease.
When the amount of sleep decreases, minutes needed to run a mile increase.
The variables move in opposite directions.
This type of relation is called a negative or inverse relation.

30. Positive or direct relations are upward sloping from left to right.
Negative or inverse relations are downward sloping from left to right. Downward sloping lines have a negative slope.
Positive or direct relations are upward sloping from left to right.
Upward sloping lines have a positive slope. Y X Y X

31. What is the Y-intercept for this relation?
It is the number of minutes needed to run a mile, when the amount of sleep is zero. You need one more minute to run the mile, for each hour less of sleep you get.

32. We know it takes 8 minutes to run a mile when you have had 6 hours of sleep. We can work down from there.
So when the number of hours slept is zero, you need 14 minutes to run the mile.
The number 14 is the Y-intercept.
hours slept min/mile

33. hrs. slept/day min./mile
You can also find the intercept by extending the line in the graph to the vertical axis. 15 12 9 6 3
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.
y-intercept
hrs. slept/day

34. Given that for this example, the independent variable is hrs slept, the dependent variable is minutes per mile, and we found that the slope is -1 and the intercept is 14, what is the equation of the relation?
min per mile = 14 + (-1) * hrs slept or min per mile = * hrs slept
Remember that multiplication and division take precedence over addition and subtraction. So you multiply first and then subtract. So the right side of this equation is not 13 * hrs slept .

35. Horizontal Lines Example 3: DIETING
Suppose that no matter how many or how few calories you consumed, your weight stayed the same. Suppose, in particular, the following relation holds.

36. calories weight
Answers:
(6) 1 unit
(7) 1 unit

37. calories weight 1000 180 1100 180 1200 180 1300 180 1400 180 weight
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calories weight
calories

38. Notice that Y never changes.
180
Notice that Y never changes. X
slope = D Y/ D X = 0/D X = 0
The slope of a horizontal line is zero.
In this relation, your weight would remain at 180 even if you consumed zero calories.
So the Y-intercept is 180.

39. Given that for this example, the independent variable is calories, the dependent variable is weight, and we found that the slope is 0 and the intercept is 180, what is the equation of the relation?
weight = * calories or weight = 180

40. Vertical Lines Example 4: DIETING
Suppose that you always consumed the same number of calories. Your weight varied with other factors, such as exercise and stress. Suppose, in particular, the following relation holds.
Answers:
(6) 1 unit
(7) 1 unit

41. calories weight
Answers:
(4) 2
(5) 2

42. calories weight 1100 140 1100 150 1100 160 1100 170 1100 180 weight
calories

43. The slope, which is D Y/ D X, is a non-zero number divided by zero.
wgt
Even though we don't change calories (the X variable), weight (the Y variable) does change.
calories
The slope, which is D Y/ D X, is a non-zero number divided by zero.
Thus, the slope is infinity or undefined.
The slope of a vertical line is infinity or undefined.
There is no Y-intercept.

44. Since for a vertical line, the slope is undefined and there is either no intercept or an infinite number of intercepts, the equation of a vertical line is not written in the slope-intercept form.
Instead it is written as: X = X0 , where X0 is the constant value of the independent variable.
For our example, the equation is calories =

45. We will next consider Nonlinear Relations
We will not be putting these relations in the form Y = mX + b.
That equation only applies to straight lines. For curves, the slope is not constant; instead it changes from point to point.

46. Example 5: DIETING - It keeps getting tougher
Example 5: DIETING - It keeps getting tougher The heavy person's perspective
Consider your weight and the number of calories you consume per day. Suppose that you're trying to lose weight.

47. calories weight
If you reduce your intake from 1400 to calories, your weight drops 10 pounds.

48. calories weight
When you reduce your intake from 1300 to 1200 calories, your weight only drops 5 pounds.

49. calories weight
When your reduce your intake from 1200 to 1100 calories, your weight drops just 2 pounds.

50. weight
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calories

51. We now do not have a straight line (linear) relationship
We now do not have a straight line (linear) relationship. Instead the relation is curved.
This reflects a changing slope.
Recall, the slope is the change in the Y-variable (wgt) divided by the change in the X-variable (calories).

52. calories wgt D wgt 1 2 5 10

53. calories wgt D wgt slope=Dwgt/Dcal

54. calories wgt D wgt slope=Dwgt/Dcal 1000 162 1 .01 1100 163 2 .02
As calories increase, the slope increases; the curve gets steeper. The slope tells you the rate at which Y (weight) changes as X (number of calories) changes.

55. This curve is upward sloping and convex from below.
The Y variable (wgt) is increasing at an increasing rate.
Since we don't know exactly what the relationship looks like as we get near zero calories, we can't determine precisely what the Y-intercept would be.
wgt
calories

56. Example 6: DIETING - It keeps getting tougher
Example 6: DIETING - It keeps getting tougher The thin person's perspective
Consider your weight and the number of calories you consume per day. Suppose that you're trying to gain weight.

57. calories weight
If you increase your intake from 1000 to calories, your weight increases 10 pounds.

58. calories weight
When you increase your intake from 1100 to 1200 calories, your weight only increases 5 pounds.

59. calories weight
When your increase your intake from 1200 to 1300 calories, your weight increases just 3 pounds.

60. weight
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calories

61. The curve is upward sloping and concave from below.
The Y variable (wgt) is increasing at a decreasing rate.
wgt
calories

62. Example 7: RUNNING
Suppose again that the more rested you are, the faster you can run.
For every extra hour of sleep you get, you shave some time off the number of minutes it takes to run a mile.
Now, however, the amount you shave off gets smaller and smaller.

63. hours slept minutes per mile

64. min./mile hrs min/mi 6 8.0 7 7.0 8 6.4 9 6.1 10 6.0 hrs. slept/day 8.0
8.0
7.8
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7.2
7.0
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6.0
hrs. slept/day

65. min. per mile hrs. slept per day
The curve is downward sloping and convex from below. Here the Y variable is decreasing at a decreasing rate.
min. per mile
hrs. slept per day

66. Example 8: MEDICINE
Suppose that you're taking medication for a virus that you've contracted. The medication has the effect on the number of heartbeats per minute as indicated in the following graph.
Answers:
(6) 1 unit
(7) 1 unit

67. beats/min. medicine (mg.) med. beats/min 0 75 100 74 200 72 300 69 75 74 72 69 64 56
medicine (mg.)

68. beats/min. medicine (mg.)
The curve is downward sloping and concave from below. Here the Y variable is decreasing at an increasing rate.
beats/min.
medicine (mg.)

69. Concave
Picture the opening of a cave. If a curve looks like this or part of this, it is concave (from below).

70. Convex
If a curve looks like the letter U or part of a U, it is convex (from below).