GRAPHS

1
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
• -------- ------
• 1000 140
• 1100 150
• 1200 160
• 1300 170
• 1400 180

4
Graph of Weight and Calories
weight 180 170 160 150 140
1000 1100 1200 1300 1400 calories
5
Graph of Weight and Calories
weight 180 170 160 150 140
calories weight 1000 140 1100
150 1200 160 1300
170 1400 180
1000 1100 1200 1300 1400 calories
6
Graph of Weight and Calories
weight 180 170 160 150 140
calories weight 1000 140 1100
150 1200 160 1300
170 1400 180
1000 1100 1200 1300 1400 calories
7
Graph of Weight and Calories
weight 180 170 160 150 140
calories weight 1000 140 1100
150 1200 160 1300
170 1400 180
1000 1100 1200 1300 1400 calories
8
Graph of Weight and Calories
weight 180 170 160 150 140
calories weight 1000 140 1100
150 1200 160 1300
170 1400 180
1000 1100 1200 1300 1400 calories
9
Graph of Weight and Calories
weight 180 170 160 150 140
calories weight 1000 140 1100
150 1200 160 1300
170 1400 180
1000 1100 1200 1300 1400 calories
10
Graph of Weight and Calories
weight 180 170 160 150 140
calories weight 1000 140 1100
150 1200 160 1300
170 1400 180
1000 1100 1200 1300 1400 calories
11
Graph of Weight and Calories
weight 180 170 160 150 140
calories weight 1000 140 1100
150 1200 160 1300
170 1400 180
1000 1100 1200 1300 1400 calories
12
Graph of Weight and Calories
weight 180 170 160 150 140
calories weight 1000 140 1100
150 1200 160 1300
170 1400 180
1000 1100 1200 1300 1400 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.

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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 100 weight
10

• calories 100/100 weight
  10/100calories 1
  weight 1/10 .1

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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

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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.

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weight 180 170 160 150 140
slope rise/run 10/100 1/10
rise 10
run 100
1000 1100 1200 1300 1400 calories
21

calories weight 1000
140 900 130 800
120 700 110 600
100 500 90 400
80 300 70 200
60 100 50
0 40
Theoretically, we can determine what you would
weigh if your calories were zero. According to
the pattern, your weight would be 40 pounds.

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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 180 170 160 150 140
40
y-intercept
0 1000 1100 1200 1300
1400 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, andthe vertical intercept
  (b) is 40.
• So the equation of this line isweight 40 0.1
  calories .

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Example 2 STUDYING

Consider your course grade and the number of
hours studied per week. Let A 4, B 3, C 2,
D 1, and F 0.
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Suppose the graph of the relation looks like this
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
27
If you studied for two hours per week, what would
your grade be?

• 1 (D)

grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
28
If you studied for eight hours per week, what
would your grade be?

• 4 (A)

grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
29
If you studied for zero hours per week, what
would your grade be?

• 0 (F)

grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
30
At what number does the line intersect the
vertical axis?

• 0

grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
31
What is the Y-intercept?

• 0

grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
32
study time grade 8 4 6
3 4 2 2
1 0 0

• Filling in the other points, we have this
  table

grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
33
study time grade 8 4
6 3 4 2 2
1 0 0
You used to study 2 hours per week. You decide to
study an additional 2 hours per week. By how
much does your grade increase? 1 You used to
study 6 hours per week. You decide to study an
additional 2 hours per week. By how much does
your grade increase? 1

What is the change in the Y variable (grade)
divided by the change in the X variable (study
time)? 1/2 What is the slope of the relation?
1/2
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Given that for this example, the independent
variable is hrs studied, the dependent variable
is grade, and we found that the slope is 0.5 and
the intercept is 0, what is the equation of the
relation?

• grade 0 0.5 hrs studied or grade
  0.5 hrs studied

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Horizontal LinesExample 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.
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calories weight
1000 180
1100 180 1200
180 1300 180
1400 180

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weight 180 170 160 150 140
calories weight 1000 180 1100
180 1200 180 1300 180
1400 180
1000 1100 1200 1300 1400 calories
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Notice that Y never changes.
Y 180
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.

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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 180 0 calories or weight 180

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Example 4 STUDYING

Consider the following graph of the relation
between study time and grades.
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Suppose the graph of the relation looks like this
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
42
If you studied for 2 hours per week, what would
your grade be?

• 2 (C)

grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
43
If you studied for 8 hours per week, what would
your grade be?

• 2 (C)

grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
44
If you studied for zero hours per week, what
would your grade be?

• 2 (C)

grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
45
At what number does the line intersect the
vertical axis?

• 2

grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
46
What is the Y-intercept?

• 2

grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
47
study time grade 8 2
6 2 4 2
2 2 0 2
Filling in the other points we have this table
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
48
study time grade 8 2
6 2 4 2 2
2 0 2
You used to study 2 hours per week. You decide to
study an additional 2 hours per week. By how
much does your grade increase? 0 You used to
study six hours per week. You decide to study an
additional 2 hours per week. By how much does
your grade increase? 0

What is the change in the Y variable (grade)
divided by the change in the X variable (study
time)? 0/2 0 What is the slope of the
relation? 0
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Given that for this example, the independent
variable is hrs studied, the dependent variable
is grade, and we found that the slope is 0 and
the intercept is 2, what is the equation of the
relation?

• grade 2 0 hrs studied or grade 2

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Vertical LinesExample 5 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.
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calories weight
1100 140
1100 150 1100
160 1100 170
1100 180

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weight 180 170 160 150 140
40
calories weight 1100 140 1100
150 1100 160 1100 170 1100
180
0 1000 1100 1200 1300
1400 calories
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Even though we don't change calories (the X
variable), weight (the Y variable) does change.
wgt
1100 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.

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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
  1100 .

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Example 6 STUDYING

Consider the following graph of the relation
between study time and grade.
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How many hours did you study to get a grade of
2 (C)?

• 6

grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
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How many hours did you study to get a grade of
3 (B)?

• 6

grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
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How many hours did you study to get a grade of
4 (A)?

• 6

grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
59

• study time grade
• 6 4
• 6 3
• 6 2
• 6 1
• 6 0

Filling in the other points, we have this table
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
60

You always studied the same amount. Your grade
varied with other factors, such as the amount of
sleep you had and your diet.
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study time grade 6 4
6 3 6 2
6 1 6 0
What is the change in the Y variable (grade)
divided by the change in the X variable (study
time)?

• 1/0 undefined or infinity
• What is the slope of the relation?
• undefined or infinity

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What is the equation of our relation in which the
value of the independent variable, hours studied,
is always 6?

• hrs studied 6 .

63
Downward Sloping LinesExample 7 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.

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• hours slept minutes per mile
• 6 8
• 7 7
• 8 6
• 9 5
• 10 4

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min./mile
hrs min/mi 6 8 7 7
8 6 9 5 10 4
8 7 6
5 4
hrs. slept/day
0 1 2 3 4 5 6 7 8 9 10
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What is the slope of the relation?
hrs min/mi 6 8 7 7
8 6 9 5 10 4

• slope D Y/ D X
• D min/ D hrs
• -1/1 -1
• A positive change denotes an increase. A
  negative change denotes a decrease.

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• 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.

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• 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.

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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.

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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 6
8 5 9 4
10 3 11 2
12 1 13
0 14

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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
0 1 2 3 4 5 6 7 8 9 10
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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 14 - 1 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
  .

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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.
74
beats/min.
med. beats/min 0 75 100
70 200 65 300 60 400 55
500 50
75 70 65
60 55 50
medicine (mg.)
0 100 200 300 400 500
75
beats/min.
If you took no medication, what would your heart
rate be?
75 70 65
60 55 50

• 75
• At what number does the line intersect the
  vertical axis?
• 75
• What is the Y-intercept?
• 75

medicine (mg.)
0 100 200 300 400 500
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If you increase your medication from 200 to 300
milligrams, by how much does your heart rate
change?
med. beats/min 0 75 100 70
200 65 300 60 400 55 500
50

• - 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
• What is the change in the Y variable
  (beats/min) divided by the change in the X
  variable (medication)?
• - 5/100 or - .05
• What is the slope of the relation?
• - .05

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• The slope is negative, because the variables
  are inversely related.
• When the amount of medication increases, the
  heart rate decreases.
• When the amount of medication decreases, the
  heart rate increases.

med. beats/min 0 75 100 70
200 65 300 60 400 55
500 50
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• The negative slope is evident in the graph by
  the fact that the line slopes downward toward the
  right.

beats
mg.
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Given that for this example, the independent
variable is mgs of medication, the dependent
variable is beats per min, and we found that the
slope is -0.05 and the intercept is 75, what is
the equation of the relation?

• beats 75 (-0.05) med or beats 75
  0.05 med
• Again, remember that multiplication and
  division take precedence over addition and
  subtraction. So the right side of this equation
  is not 74.95 med .

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We will next considerNonlinear 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.

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Example 9 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.

82

• calories weight
• 1000 162 1100 163
• 1200 165
• 1300 170
• 1400 180

If you reduce your intake from 1400 to 1300
calories, your weight drops 10 pounds.
83

• calories weight
• 1000 162 1100 163
• 1200 165
• 1300 170
• 1400 180

When you reduce your intake from 1300 to 1200
calories, your weight only drops 5 pounds.
84

• calories weight
• 1000 162 1100 163
• 1200 165
• 1300 170
• 1400 180

When your reduce your intake from 1200 to 1100
calories, your weight drops just 2 pounds.
85
weight 180 175 170 165 160
1000 1100 1200 1300 1400 calories
86
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).

87

• calories wgt D wgt
• 1000 162
• 1
  
• 1100 163
• 2
  
• 1200 165
• 5
  
• 1300 170
• 10
  
• 1400 180

88

• calories wgt D wgt
  slopeDwgt/Dcal
• 1000 162
• 1
  .01
• 1100 163
• 2
  .02
• 1200 165
• 5
  .05
• 1300 170
• 10
  .10
• 1400 180

89

• calories wgt D wgt
  slopeDwgt/Dcal
• 1000 162
• 1
  .01
• 1100 163
• 2
  .02
• 1200 165
• 5
  .05
• 1300 170
• 10
  .10
• 1400 180

As calories increase, the slope increases the
curve gets steeper.
90

• This curve is upward sloping and convex from
  below.
• 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
91
Example 10 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.

92

• calories weight
• 1000 100 1100 110
• 1200 115
• 1300 118
• 1400 119

If you increase your intake from 1000 to 1100
calories, your weight increases 10 pounds.
93

• calories weight
• 1000 100 1100 110
• 1200 115
• 1300 118
• 1400 119

When you increase your intake from 1100 to 1200
calories, your weight only increases 5 pounds.
94

• calories weight
• 1000 100 1100 110
• 1200 115
• 1300 118
• 1400 119

When your increase your intake from 1200 to 1300
calories, your weight increases just 3 pounds.
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weight 120 115 110 105 100
1000 1100 1200 1300 1400 calories
96

• calories weight D wgt
• 1000 100
• 10
  
• 1100 110
• 5
  
• 1200 115
• 3
  
• 1300 118
• 1
  
• 1400 119

97

• calories weight D wgt slopeDwgt/Dcal
• 1000 100
• 10
  .10
• 1100 110
• 5
  .05
• 1200 115
• 3
  .03
• 1300 118
• 1
  .01
• 1400 119

98

• calories weight D wgt slopeDwgt/Dcal
• 1000 100
• 10
  .10
• 1100 110
• 5
  .05
• 1200 115
• 3
  .03
• 1300 118
• 1
  .01
• 1400 119

As calories increase, the slope decreases the
curve gets flatter.
99

• This curve is upward sloping and concave from
  below.

wgt
calories
100
Example 11 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.

101
hours slept minutes per mile
6 8.0
7 7.0
8 6.4
9 6.1
10 6.0

102
min./mile
hrs min/mi 6 8.0 7
7.0 8 6.4 9 6.1 10
6.0
8.0 7.8 7.6 7.4 7.2
7.0 6.8 6.6 6.4 6.2 6.0
hrs. slept/day
0 1 2 3 4 5 6 7 8 9 10
103
hrs. slept min. D min. 6
8.0 -1.0
7 7.0
- .6 8
6.4 -
.3 9 6.1
- .1
10 6.0

104
hrs. slept min. D min. slopeD min/D
hrs 6 8.0
-1.0 -1.0 7
7.0 - .6
- .6 8 6.4
- .3 -
.3 9 6.1
- .1 - .1 10
6.0

105
hrs. slept min. slope 6
8.0 -1.0 7
7.0
- .6 8 6.4
- .3 9 6.1
- .1 10
6.0

As sleep increases, the absolute value of the
slope decreases the curve gets flatter.
106
This curve is downward sloping and convex from
below.
min. per mile
hrs. slept per day
107
Example 12 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.
108
beats/min.
med. beats/min 0 75 100
74 200 72 300 69 400 64
500 56
75 74 72 69 64
56
medicine (mg.)
0 100 200 300 400 500
109
med. beats D beats 0 75
-1
100 74 -2
200 72
- 3 300 69
- 5
400 64
-8 500 56

110
med. beats D beats slope Dbeats/D
med. 0 75
-1 -.01 100 74
-2 -
.02 200 72 -
3 - .03 300 69
- 5 -
.05 400 64
-8 - .08 500 56

111
med. beats slope 0 75
-.01 100 74
- .02 200 72
- .03 300 69
- .05 400 64
- .08 500 56

As medication increases the absolute value of
the slope rises the curve gets steeper. This
pattern indicates that the effects of the
medicine increase as you take more of it.

112
This curve is downward sloping and
concave from below.
beats/min.
medicine (mg.)
113
Concave

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

114
Convex

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