# Graphing

```Jerry Crawford – June 23, 2011
1
Thinking Ahead about Graphing – You should understand these questions fully before the
with this work from the following sources:
•
•
•
Visit your instructor during office hours
Go to the MAC (Math Assistance Center) 700 BH
Go to http://www.themathpage.com/aPreCalc/graphs-of-functions.htm
Experimental data is often graphed in order to see if any
type of relationship exists between the independent
variable and the dependent variable. If the plot shows
random behavior, perhaps there is no useful relationship.
Even then, it is often possible to see some range of
expected behavior for a particular variable.
For example, Graph #1 shows that the average height of
adults is not particularly related to their month of birth, but
a trend shown by the graph is that the average adult height
falls mostly in the range between 5 feet and 6.5 feet.
Graph #1
Three particular relationships between independent and dependent variables are fairly common
in the physical sciences. The first is a linear relationship,=
y mx + b , shown in Graph #2,
which is a plot of the data from Table 1,
the speed of fall (dependent variable)
versus time of fall (independent
variable). This graph shows a
representation of the data as if handplotted on linear graph paper.
Table 1: Speed of Fall vs. Time of Fall.
Time (s) Speed (m/s)
0
0
1
9.8
2
19.6
3
29.4
4
39.2
5
49.0
6
58.8
Graph #2: Linear plot of Speed of Fall vs. Time
The next common relationship is a power law relationship, y = Ax , where the dependent
variable is related to some power of the independent variable. If data related in this way are
plotted on linear paper, a curved line results unless the power, m = 1, in which case the data are
linearly related as in the first example. It is possible to “straighten out” power law data by
m
Jerry Crawford – June 23, 2011
2
plotting on log-log paper. This has the same effect as plotting the logarithm of y against the
logarithm of x on linear paper. This is demonstrated in the log-log plot of Graph #3.
If the logarithm is taken of both sides of the power law relation, the result is:
log(
=
y ) log( A) + m log( x) .
This is equivalent to a linear relationship with log(y) being the dependent variable, log(x) the
independent variable, m the slope of the straight line, and log(A) the intercept.
Table 2: Speed of Fall vs. Distance of Fall
Distance
(m)
Speed
(m/s)
1
2
3
4
5
6
7
8
9
10
20
30
40
50
60
70
80
90
4.4
6.3
7.7
8.9
9.9
10.8
11.7
12.5
13.3
14.0
19.8
24.2
28.0
31.3
34.3
37.0
39.6
42.0
Graph #3: Log-Log Plot of Speed of Fall vs. Distance of Fall
Note the both the horizontal and vertical scales are non-linear, and that after reaching 10
(1,2,3,…8,9,10), the scale intervals increment by 10’s (10,20,30,….80,90,100) instead by 1’s.
Exercise: Plot the speed versus distance data on a graph with linear scales, either by hand, or
with a calculator or computer (if you are familiar with using those to graph), to see that the data
relationship is not linear. The speed actually varies as the square root of the distance.
The third common relationship between measured variables is exponential, y = Ae , where x
is the independent variable and y the dependent. An example of this is radioactive decay. If the
natural logarithm of both side of this equation is taken, the result is:
mx
ln( y ) =
ln( A) + mx ln(e) =
ln( A) + mx .
Jerry Crawford – June 23, 2011
3
This is equivalent to a linear relationship with ln(y) being the dependent variable, x the
independent variable, m the slope of the straight line, and ln(A) the intercept. If the natural
logarithm of y is plotted versus x on linear paper, a straight line should result. Alternatively, y
may be plotted against x on semi-logarithmic paper, and a straight line will result. The vertical
axis is logarithmic and the horizontal axis linear for semi-log plots.
Table 3: Nuclei Remaining vs. Time
12000
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
10000
9512
8607
7408
6065
4724
3499
2466
1653
1054
639
369
202
106
52
25
11
5
2
1
0
Graph #4 shows the curve resulting
from plotting the data from Table 3
on a linear graph. Graph #5 shows
the same data plotted on a semi-log
graph, resulting in a straight line.
Note that the vertical scale is
logarithmic, with powers of ten
sharing equal increments of height.
10000
Number of Nuclei
Remaining
Nuclei
8000
6000
4000
2000
0
0
20
40
60
80
100
120
Time (days)
Graph #4: Linear plot of Number of Nuclei vs. Time
10000
1000
Number of Nuclei
Time
(days)
100
10
1
0
20
40
60
80
100
120
Time (days)
Graphs #4 and #5 were produced
using Microsoft Excel, taking
Graph #5: Semi-log plot of Number of Nuclei vs. Time
advantage of the feature allowing the
axis scales to be changed from linear to logarithmic. The first graph was plotted, and then
copied, and the vertical axis was re-scaled to be logarithmic.
Jerry Crawford – June 23, 2011
The ability to quickly re-scale is common in most graphing software, and is a very useful timesaving feature.
Exercise: Plot the data in the following table on linear, log-log, and semi-log graphs either by
hand, or using calculator or computer graphing software, to determine if the relationship is
linear, a power law, exponential, or none of these. Graph the thickness of the absorber as the xaxis variable (independent) and the relative intensity as the y-axis variable (dependent).
Thickness (cm)
0.1
0.5
1
1.5
2
3
5
7.5
10
Relative Intensity
0.917
0.65
0.422
0.274
0.178
0.075
0.013
0.0016
0.00018
4
```