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Graphing
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Most people at one time or another
during their careers will have to interpret
data presented in graphical form.
This means of presenting data allows one
to discover trends, make predictions, etc.
To take seemingly unrelated sets of
numbers (data) and make sense out of
them is important to a host of disciplines.
An example of graphing techniques used
in physics follows.
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When weight is added to a spring hanging
from the ceiling, the spring stretches.
How much it stretches depends on how
much weight is added.
The following slide depicts this
experiment.
Add
another
weight
Stretch
islevel
now
here
Starting
a
weight
We control the weight that is added.
It is the independent variable.
The stretch is dependent on what weight is added.
It is the dependent variable.
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The following data were obtained by
adding several different amounts of
weight to a spring and measuring the
corresponding stretch.
Stretch
(meters)
0.1240
0.1475
0.1775
0.1950
0.2195
0.2300
0.2525
0.2675
0.2875
Weight
(Newtons)
6.0
14.0
22.0
30.0
38.0
40.0
47.0
54.0
58.0
The Newton
is a unit of
force or
weight
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There are two variables or parameters that can
change during the experiment, weight and
stretch.
As mentioned earlier the experimenter controls
the amount of weight to be added.
The weight is therefore called the independent
variable.
Again as mentioned before the amount that the
spring stretches depends on how much weight is
added. Hence the stretch is called the
dependent variable.
The dependent variable is the quantity that
depends on the independent variable.
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A graph of this experimental data is shown
on the next slide.
The independent variable is always plotted
on the horizontal axis.
The dependent variable is plotted on the
vertical axis.
Notice that each axis is not only labeled as
to what is plotted on it, but also, the units
in which the variable is displayed.
Units are important.
Stretch Versus Weight - Susie Que
Weight,
0.30the independent variable, will be plotted along the horizontal axis.
Each graph should be identified with a title and the experimenter's name.
Stretch, the dependent variable, will be plotted along the vertical axis.
Stretch (meters)
0.26
0.22
The graph should be made so that the data fills as much of the page
as possible. To do this, sometimes it is better not to start numbering
an axis at zero, but rather a value near the first data point.
0.18
0.14
0.10
0
10
20
30
Weight (Newtons)
40
50
60
Stretch Versus Weight - Susie Que
0.30
(58.0, 0.2875)
(54.0, 0.2675)
Stretch (meters)
0.26
(47.0, 0.2525)
(40.0, 0.2300)
(38.0, 0.2195)
Let’s plot the data.
0.22
(30.0, 0.1950)
0.18
This is not a connect-the-dot exercise.
(22.0, 0.1775)
(14.0, 0.1475)
0.14
(6.0, 0.1240)
The data appears to fit a straight line somewhat like this one.
0.10
0
10
20
30
Weight (Newtons)
40
50
60
Stretch Versus Weight - Susie Que
0.30
If there is a general trend to the data, then a best-fit
curve describing this trend can be drawn.
In this example the data points approximately fall
along a straight line.
This implies a linear relationship between the stretch
and the weight.
A wealth of information can be obtained if the
equation that describes the data is known.
With an equation one is able to predict what values
the variables will have well beyond the scope or
boundaries of the graph.
A timid mathematician should not be scared away,
since finding the equation is not hard and requires
very little knowledge of math.
Stretch (meters)
0.26
0.22
0.18
0.14
0.10
0
10
20
30
Weight (Newtons)
40
50
60
Stretch Versus Weight - Susie Que
0.30
If data points follow a linear relationship (straight line),
the equation describing this line is of the form
Stretch (meters)
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0.22
A very important
equation.
y = mx + b
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where y represents the dependent variable
(in this case, stretch), and
x represents the independent variable (weight).
0.14
0.10
0
10
20
30
Weight (Newtons)
40
50
60
Stretch Versus Weight - Susie Que
The value of the dependent variable when x = 0 is
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given by b and is known as the y-intercept.
0.26
The y-intercept is found graphically by finding the
Stretch (meters)
intersection of the y-axis (x = 0) and the smooth curve
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through the data points.
(From the equation y = mx + b, if we set x = 0
0.18
then y = b.)
y = mx + b
In this case b = 0.11 meters.
0.14
0.10
0
10
20
30
Weight (Newtons)
40
50
60
Stretch Versus Weight - Susie Que
The quantity m is the slope of the best-fit line.
0.30
Stretch (meters)
It is found by taking any two points,
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for instance (x2, y2) and (x1, y1), on the straight line and
subtracting their respective x and y values.
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Note
y = mx + b
y2 = mx2 + b
y1 = mx1 + b
0.18
We’ll call this
Subtracting one equation from the other yields
y2 - y1 = mx2 - mx1
0.14
y2 - y1 = m(x2 -x1) Therefore
y 2  y1 rise

m
run
x 2  x1
0.10
0
10
20
30
Weight (Newtons)
40
50
60
Stretch Versus Weight - Susie Que
0.30
0.129 meters
slope  rise 
 0.00293 meters / Newton
run 44.0 Newtons
Point 2
0.26Y2 = 0.263
Stretch (meters)
rise = (0.263 – 0.134) meters = 0.129 meters
0.22
To find the slope of this line pick a
couple of points on the line that are
somewhat separated from each other.
0.18
0.14
Y1 = 0.134
run = (52.0 – 8.0) Newtons = 44.0 Newtons
Point 1
y-intercept = 0.11 meters
X2 = 52.0
X1 = 8.0
0.10
0
10
20
30
Weight (Newtons)
40
50
60
Stretch Versus Weight - Susie Que
0.30
0.129 meters
slope  rise 
 0.00293 meters / Newton
run 44.0 Newtons
0.26
Stretch (meters)
At this point everything needed to write the
equation describing the data has been found.
0.22
Recall that this equation is of the form
0.18
y = mx + b
0.14
Weight +
Stretch
y-intercept = 0.11 meters
0.10
0
10
20
30
Weight (Newtons)
40
50
60
Stretch Versus Weight - Susie Que
0.30
Here are two ways we can gain useful information
from the graph and from the equation of the line.
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If we wanted to know how much weight would give
Stretch (meters)
us a 0.14 m stretch, we could read it from the plot
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thusly.
This would be about 10.2 Newtons.
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Solving the equation for x when y=0.14 m gives
x=10.2 Newtons.
0.14
y = mx + b
Stretch  (0.00293 meters / Newton)  Weight  (0.11 meters)
0.10
0
10
20
30
Weight (Newtons)
40
50
60
6
This relationship would be
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Suppose you have some x and y data
related to each other in the following way.
2
y  mx  b
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A curve through this data is not
straight and x and y are not linearly
related. Their relationship could be
complicated.
y2 3
2
ALet’s
replot
data
try of
y2 this
versus
x.might
straighten this line some to
give a linear relationship.
1
0
0
10
20
30
x
40
50
60
Graphing is a powerful analytical tool.
Graphing skills are important in many of today’s
professions.