Title: GRAPHS
1GRAPHS
- Professor Karen Leppel
- Economics 202
2Upward-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
4Graph of Weight and Calories
weight 180 170 160 150 140
1000 1100 1200 1300 1400 calories
5Graph 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
6Graph 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
7Graph 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
8Graph 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
9Graph 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
10Graph 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
11Graph 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
12Graph 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.
14 your calories your weight
15 your calories your weight
16The 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.
17calories 100 weight
10
- calories 100/100 weight
10/100calories 1
weight 1/10 .1
18The 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
19The 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.
20weight 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.
22The 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.
23weight 180 170 160 150 140
40
y-intercept
0 1000 1100 1200 1300
1400 calories
24Equation 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 .
25Example 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.
26Suppose the graph of the relation looks like this
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
27If you studied for two hours per week, what would
your grade be?
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
28If you studied for eight hours per week, what
would your grade be?
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
29If you studied for zero hours per week, what
would your grade be?
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
30At what number does the line intersect the
vertical axis?
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
31What is the Y-intercept?
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
32study 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
33study 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
34Given 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
35Horizontal 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.
36 calories weight
1000 180
1100 180 1200
180 1300 180
1400 180
37weight 180 170 160 150 140
calories weight 1000 180 1100
180 1200 180 1300 180
1400 180
1000 1100 1200 1300 1400 calories
38Notice 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.
39Given 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
40Example 4 STUDYING
Consider the following graph of the relation
between study time and grades.
41Suppose the graph of the relation looks like this
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
42If you studied for 2 hours per week, what would
your grade be?
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
43If you studied for 8 hours per week, what would
your grade be?
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
44If you studied for zero hours per week, what
would your grade be?
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
45At what number does the line intersect the
vertical axis?
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
46What is the Y-intercept?
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
47study 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
48study 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
49Given 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
50Vertical 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.
51 calories weight
1100 140
1100 150 1100
160 1100 170
1100 180
52weight 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
53Even 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.
54Since 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 .
55Example 6 STUDYING
Consider the following graph of the relation
between study time and grade.
56 How many hours did you study to get a grade of
2 (C)?
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
57 How many hours did you study to get a grade of
3 (B)?
grade
4 3 2 1
hrs. studied per week
0 2 4 6
8
58 How many hours did you study to get a grade of
4 (A)?
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
60You always studied the same amount. Your grade
varied with other factors, such as the amount of
sleep you had and your diet.
61study 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
62What is the equation of our relation in which the
value of the independent variable, hours studied,
is always 6?
63Downward 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.
64- hours slept minutes per mile
- 6 8
- 7 7
- 8 6
- 9 5
- 10 4
65min./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
66What 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.
67-
- 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.
68- 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.
69 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.
70We 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
71min./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
72Given 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
.
73Example 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.
74beats/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
75beats/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
76If 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
77- 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
78- The negative slope is evident in the graph by
the fact that the line slopes downward toward the
right.
beats
mg.
79Given 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 .
80 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.
81Example 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.
85weight 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
91Example 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.
95weight 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
100Example 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
102min./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
103hrs. slept min. D min. 6
8.0 -1.0
7 7.0
- .6 8
6.4 -
.3 9 6.1
- .1
10 6.0
104hrs. 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
105hrs. 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.
106This curve is downward sloping and convex from
below.
min. per mile
hrs. slept per day
107Example 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.
108beats/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
109med. beats D beats 0 75
-1
100 74 -2
200 72
- 3 300 69
- 5
400 64
-8 500 56
110med. 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
111med. 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.)
113Concave
- Picture the opening of a cave. If a curve
looks like this or part of this, it is concave
(from below).
114Convex
- If a curve looks like the letter U or part of
a U, it is convex (from below).