Interpreting Graphs
In the laboratory investigations, you generally control one variable and measure the effect it has on another
variable while you hold all other forces constant. After the data is collected, you then make a graph to give you
a better understanding of the relationship between the variables.
In this set of notes, I will use “x” and “y”
for the variables. Despite the bias that mathematicians have towards these two letters, we in physics, use a great
deal of other symbols. Be careful when graphing to use the proper symbols and notation.
Sometimes you need information about a value that you have determined experimentally. Reading from the
graph between data points is called interpolation. Reading from the graph beyond the limits of your
experimentally determined data points is called extrapolation. Extrapolation must be used with caution because
you cannot be sure that the relationship between the variable remains the same beyond the limits of your
investigation.
Once you have graphed a set of data, you must determine the shape. This is part of your job as the
experimenter. Therefore, when you ask me what I think, my response will probably be, “What do you think?”.
You can use the theory to help you through this important step. When this cannot be done, you will have to rely
on your data and your instinct (which will be developed as the course moves along). Draw a smooth curve
(using a ruler for linear relationships or a bendable ruler for non-linear relationships).
In general, you find the graph yields one of two shapes: straight or not straight. This may seem trivial to
identify these two groups, but there is a difference in how you treat these to shapes. Once you have identified
the shape, the next problem is what to do with the graph in order to determine the relationship.
Straight line relationships:
There are two types of straight line relationships: linear and direct. Direct relationships are lines that go
through the origin (i. e. (0,0) not just the bottom left corner of the graph) and therefore have an y-intercept of
zero. Linear relationships are straight lines that do not go through the origin and therefore have a non-zero yintercept.
What to do with the graph: Once you have identified a straight line relationship and have drawn the line,
determine the slope and y-intercept. The equation for this shape is y = m x + b where m is the slope and b is
the y-intercept. Be sure to include the units for the slope and intercept. Using the units, identify the quantities
that the slope and intercept represent.
Non-straight line relationships:
When the curve of the graph is not a straight line, you need to identify the type of curve. The general shape
will be one of the following:
y = k xn
Power curve
y = k √x
Root curve
y=k/x
Inverse curve
To verify which of these curves you have, the data are systematically manipulated until the graph yields a
straight line. The equation for the line can be written and you will have obtained the relationship between the
variables. If your graph was a power curve, you would expect the equation to be y = k x n. In order to arrive at
the correct relationship, you must “straighten” the graph. Suppose an experiment yield the following data:
y
0.0
1.0
4.0
9.0
16
x
0.0
1.0
2.0
3.0
4.0
When graphed, this would yield a parabola. The parabola does not tell us what power is in the relationship
between x and y. So now we start to manipulate the data. To do this we square all of the x data (or square root,
or cube, or take the inverse, etc.) and then re-graph the new data. In this case, the new data would be:
y
x
x2
0.0
0.0
0.0
1.0
1.0
1.0
4.0
2.0
4.0
9.0
3.0
9.0
16
4.0
16
By plotting y versus x2 we would get a straight line. We would now go to the straight line relationship
guidelines. When writing the equation for the line in this second graph we would have y = m x 2 + b.