GRAPHS
Professor Karen Leppel
Economics 202
Upward-sloping lines
Example 1: DIETING
Consider your weight and the number of
calories you consume per day. Suppose the
following relation holds.
calories
-------1000
1100
1200
1300
1400
weight
-----140
150
160
170
180
weight
Graph of Weight and Calories
180
170
160
150
140
1000 1100 1200 1300 1400 calories
weight
Graph of Weight and Calories
180
170
160
150
140
calories
1000
1100
1200
1300
1400
weight
140
150
160
170
180
1000 1100 1200 1300 1400 calories
weight
Graph of Weight and Calories
180
170
160
150
140
calories
1000
1100
1200
1300
1400
weight
140
150
160
170
180
1000 1100 1200 1300 1400 calories
weight
Graph of Weight and Calories
180
170
160
150
140
calories
1000
1100
1200
1300
1400
weight
140
150
160
170
180
1000 1100 1200 1300 1400 calories
weight
Graph of Weight and Calories
180
170
160
150
140
calories
1000
1100
1200
1300
1400
weight
140
150
160
170
180
1000 1100 1200 1300 1400 calories
weight
Graph of Weight and Calories
180
170
160
150
140
calories
1000
1100
1200
1300
1400
weight
140
150
160
170
180
1000 1100 1200 1300 1400 calories
weight
Graph of Weight and Calories
180
170
160
150
140
calories
1000
1100
1200
1300
1400
weight
140
150
160
170
180
1000 1100 1200 1300 1400 calories
weight
Graph of Weight and Calories
180
170
160
150
140
calories
1000
1100
1200
1300
1400
weight
140
150
160
170
180
1000 1100 1200 1300 1400 calories
weight
Graph of Weight and Calories
180
170
160
150
140
calories
1000
1100
1200
1300
1400
weight
140
150
160
170
180
1000 1100 1200 1300 1400 calories
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.
your calories
your weight
your calories
your weight
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.
calories
calories
calories
100
100/100
1
weight
10
weight
10/100
weight
1/10 = .1
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
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.
weight
180
slope = rise/run
= 10/100 = 1/10
170
160
150
rise = 10
run = 100
140
1000 1100 1200 1300 1400 calories
Theoretically, we can
determine what you
would weigh if your
calories were zero.
According to the
pattern, your weight
would be 40 pounds.
calories
1000
900
800
700
600
500
400
300
200
100
0
weight
140
130
120
110
100
90
80
70
60
50
40
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.
weight
180
170
160
150
140
y-intercept
40
0
1000 1100 1200 1300 1400 calories
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 = 40 + 0.1 * calories .
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.
hours slept minutes per mile
6
8
7
7
8
6
9
5
10
4
min./mile
hrs
6
7
8
9
10
8
7
6
5
min/mi
8
7
6
5
4
4
0
1 2 3 4
5 6
7 8
9
10
hrs. slept/day
hrs
6
7
8
9
10
min/mi
8
7
6
5
4
What is the slope of the
relation?
slope = D Y/ D X
= D min/ D hrs
= -1/+1 = -1
A positive change denotes an
increase.
A negative change denotes a
decrease.
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.
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
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.
We know it takes 8
hours slept min/mile
minutes to run a mile
6
8
when you have had 6
5
9
hours of sleep. We can
work down from there.
4
10
So when the number of
3
11
hours slept is zero, you
2
12
need 14 minutes to run
the mile.
1
13
The number 14 is the Y0
14
intercept.
min./mile
15
12
9
You can also find the intercept by extending
the line in the graph to the vertical axis.
The Y-intercept tells the value of the
Y variable (minutes needed to
run a mile) when the value
of the X variable (hours
y-intercept
slept) is zero.
6
3
hrs. slept/day
0
1 2 3 4
5 6
7 8
9
10
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 .
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.
calories
weight
1000
180
1100
180
1200
180
1300
180
1400
180
weight
180
170
160
150
140
calories
1000
1100
1200
1300
1400
weight
180
180
180
180
180
1000 1100 1200 1300 1400 calories
Y
Notice that
Y never changes.
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.
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
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.
calories
weight
1100
140
1100
150
1100
160
1100
170
1100
180
weight
180
170
calories weight
1100
140
1100
150
1100
160
1100
170
1100
180
160
150
140
40
0
1000 1100 1200 1300 1400 calories
wgt
1100
calories
Even though we don't
change calories (the X
variable), weight (the Y
variable) does change.
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.
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 .
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.
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.
If you reduce your intake
from 1400 to 1300
calories, your weight
drops 10 pounds.
calories
1000
1100
1200
1300
1400
weight
162
163
165
170
180
When you reduce your
intake from 1300 to 1200
calories, your weight only
drops 5 pounds.
calories
1000
1100
1200
1300
1400
weight
162
163
165
170
180
calories
1000
When your reduce your
1100
intake from 1200 to 1100
1200
calories, your weight drops
1300
just 2 pounds.
1400
weight
162
163
165
170
180
weight
180
175
170
165
160
1000 1100 1200 1300 1400 calories
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).
calories
1000
wgt
162
D wgt
1
1100
163
2
1200
165
5
1300
170
10
1400
180
calories
1000
1100
1200
1300
1400
wgt
162
D wgt slope=Dwgt/Dcal
1
.01
2
.02
5
.05
10
.10
163
165
170
180
calories
1000
1100
1200
1300
1400
wgt
162
D wgt slope=Dwgt/Dcal
1
.01
2
.02
5
.05
10
.10
163
165
170
180
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. Here your weight is increasing at an increasing rate.
wgt
calories
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.
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.
If you increase your intake
from 1000 to 1100
calories, your weight
increases 10 pounds.
calories
1000
1100
1200
1300
1400
weight
100
110
115
118
119
When you increase your
intake from 1100 to 1200
calories, your weight only
increases 5 pounds.
calories
1000
1100
1200
1300
1400
weight
100
110
115
118
119
When your increase your
intake from 1200 to 1300
calories, your weight
increases just 3 pounds.
calories
1000
1100
1200
1300
1400
weight
100
110
115
118
119
weight
120
115
110
105
100
1000 1100 1200 1300 1400 calories
calories weight
1000
100
D wgt
10
1100
110
5
1200
115
3
1300
118
1
1400
119
calories weight
1000
100
1100
1200
1300
1400
D wgt slope=Dwgt/Dcal
10
.10
5
.05
3
.03
1
.01
110
115
118
119
calories weight
1000
100
1100
1200
1300
1400
D wgt slope=Dwgt/Dcal
10
.10
5
.05
3
.03
1
.01
110
115
118
119
As calories increase, the slope decreases; the curve gets flatter.
Here your weight is increasing at a decreasing rate.
This curve is upward sloping and concave
from below.
wgt
calories
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.
hours slept minutes per mile
6
8.0
7
7.0
8
6.4
9
6.1
10
6.0
min./mile
hrs
6
7
8
9
10
8.0
7.8
7.6
7.4
7.2
7.0
6.8
6.6
6.4
6.2
6.0
0
1 2 3 4
5 6
7 8
9
10
min/mi
8.0
7.0
6.4
6.1
6.0
hrs. slept/day
hrs. slept
6
7
8
9
10
min. D min.
8.0
-1.0
7.0
- .6
6.4
- .3
6.1
- .1
6.0
hrs. slept
6
7
8
9
10
min. D min. slope=D min/D hrs
8.0
-1.0
-1.0
7.0
- .6
- .6
6.4
- .3
- .3
6.1
- .1
- .1
6.0
hrs. slept
6
min.
8.0
slope
-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.
Here your number of
minutes needed is
decreasing at a
decreasing rate.
This curve is downward sloping
and convex from below.
min.
per
mile
hrs. slept per day
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.
beats/min.
med. beats/min
0
75
75
74
72
69
64
56
0
100
200
300
400
500
100
74
200
72
300
69
400
64
500
56
medicine (mg.)
med.
0
beats
75
D beats
-1
100
74
-2
200
72
-3
300
69
-5
400
64
-8
500
56
med.
0
100
200
300
400
500
beats
75
D beats slope = Dbeats/D med.
-1
-.01
-2
- .02
-3
- .03
-5
- .05
-8
- .08
74
72
69
64
56
med.
0
beats
75
slope
-.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. Here your
heart rate is decreasing
at an increasing rate.
This pattern indicates
that the effects of the
medicine increase as you
take more of it.
This curve is downward sloping
and concave from below.
beats/min.
medicine (mg.)
Concave
Picture the opening of
a cave. If a curve
looks like this or part
of this, it is concave
(from below).
Convex
If a curve looks like
the letter U or part
of a U, it is convex
(from below).