Techniques of Graphing
Student Handout
Determining Coordinates and Axes
Most experiments involve two or more variables that change concurrently. One
variable, the independent variable, is the variable that you control. By convention
it is plotted on the horizontal axis (x-axis). Time is usually, but not always, plotted
on the x-axis. The second variable, the dependent variable, is the variable that
changes as a result of the first variable. It is plotted on the vertical (y-axis). The
axes need to be labeled with both the variable name and the unit (example:
weight, newtons or N.) Every graph needs a title stating in words the dependent
variable as a function of the independent variable such as “Velocity vs Time”. If
there is more than one graph per report, they should be numbered.
Graphing Points and Drawing the Best Curve to Fit the Data
Once the points have been plotted, the type of curve drawn depends on the nature
of the data. In some cases it may be clear that the relation is linear. A straight edge
should be used and a “best fit” line that is drawn to best represent the average
value. This is done by having some points fall above and some points fall below
the straight line to balance the uncertainties in the data. A transparent ruler is
useful when drawing a straight best fit line. If the graph represents a nonlinear
curve, the points should be connected with a smooth curve so that the points
average around the line as they do in the graphs below. For accuracy, when
drawing a graph by hand the data points should be as small as possible with a
small circle called a point protector drawn around the data point. The circle
protectors will help in determining the position of the best fit line. A point that is
obviously off might be given a larger point protector.
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nonlinear
linear
Depending upon the data, the graph paper can be turned so that the long edge of
the paper is horizontal or vertical. The scale should be chosen so that all the points
fit on the graph and so that positions between the divisions on the graph are easily
determined. If necessary, the data can be written in scientific notation before
plotting to make it easier to determine the desired division size.
Jan Mader and Mary Winn
Graphs were drawn using Graphical Analysis by Vernier Software.
Interpretation of Data
Direct Proportionality
There are several relationships that occur frequently in physical processes. The
following graphs will show examples of some of these.
The slope of a direct relationship is
the rate of change of the dependent
variable with respect to the
independent variable. By definition
the slope is
m
y d

x t
In the diagram on the right, the
change in distance Δd divided by
the change in time Δt is 30
kilometers per hour which is an
average speed. The slope of a
distance versus time graph is an
average velocity or speed
distance in kilometers
Best fit line drawn
through data points
which represent a
linear relationship.
If the dependent variable varies
directly with the independent
variable, the graph will be a
straight line. The relationship is
described as direct or linear. If the
line slants upward to the right, the
slope of the line is positive. If the
line slants downward to the right,
the slope of the line is negative.
Jan Mader and Mary Winn
Graphs were drawn using Graphical Analysis by Vernier Software.
Inverse (or indirect) proportionality
The curve represents an
inverse (or indirect)
proportionality.
If the dependent variable varies
inversely with the independent
variable the graph will be a
hyperbola. An inverse
relationship is one in which one
variable increases as the second
variable decreases. Another
name for an inverse relationship
is an indirect relationship.
For this type of relationship,
plotting the dependent variable
vs. the reciprocal of the
independent variable will
produce a straight line.
Directly proportional to the square
An example of a direct
square relationship is
distance vs. time for
accelerated motion. If the
object is starting from rest
which makes the initial
velocity zero, the formula
becomes d = ½ at2. From
the formula it can be seen
that distance is directly
proportional to the square
of time. The graph of d vs. t
is a parabola.
If distance is graphed vs.
time squared, a straight line
will be produced.
Inverse square Relationship
Jan Mader and Mary Winn
Graphs were drawn using Graphical Analysis by Vernier Software.
Direct
square
relationship
There are many inverse square relationships in physics. Gravitational force,
electrical force, magnetic force and the intensity of light versus distance are all
inverse square relationships. This means that as the distance increases the values
for the dependent variable will decrease as the inverse square of the independent
variable.
In the graph to the left, the force
of gravitational attraction is
plotted vs the distance between
the centers of mass.
F
For the gravitational
attraction between two
masses the force between
An inverse
the
massessquare
will vary with
relationship.
the
square of the distance
between the centers of
the masses. This is called
an inverse square
relationship.
Gm1m2
r2
If the force were plotted against
the inverse square of the distance
between the masses, the result
would be a straight line.
Direct Square Root Proportionality
Sometimes one variable is
directly proportional to the
square root of the other
variable. This is the
situation in the graph on
the right where the length
of a pendulum is plotted
against the period of the
pendulum according to the
formula:
T  2
l
g
Jan Mader and Mary Winn
Graphs were drawn using Graphical Analysis by Vernier Software.
A direct square
root relationship