Chapter 2: Graphing and Relationships of Variables
A graph is a diagram that illustrates by means of points
and lines a relationship between different variables whose
values are plotted on the graph. Graphical analysis also
helps us determine the type of mathematical equation that
relates the variables being graphed.
6.0
5.0
Part 1: Graphing procedures
weight on
spring (N)
distance spring
stretched (cm)
weight
4.0
3.0
2.0
1.0
0
5
0
10
15
distance
20
25
Graph A
3.0
weight on spring versus distance spring is stretched
2.5
weight on spring (N)
1. The data is most commonly plotted on Cartesian
paper in which the lines are set up in a linear
fashion. The horizontal line with the value of zero is
called the x-axis and the vertical line with the value
of zero is called the y-axis. You may be told to plot
one variable versus another. If so, the variable
given before the "versus" goes on the y-axis and
the variable after versus goes on the x-axis. In
general when plotting data, the independent
variable (the one the experimenter controls) goes
on the x-axis and the dependent variable goes on
the y-axis. This is not a hard and fast rule, and may
be changed by the instructor during some labs.
2. When plotting data, choose axis scales that are
easy to plot and read. If possible, choose scales so
that over half of each axis is used to plot data
points. (At times, for the graphs to start at zero, this
rule may need to be ignored.) Graph A illustrates
an example of scales that are too small. Graph B
shows the data plotted with more appropriate
scales. The scales must be uniform, i.e. every
small box is worth the same amount.
3. When the data points are plotted, draw a smooth
best-fit line for the points. It is appropriate to ignore
a data point that seems far off. Never go dot to dot
as shown in Graph A. If the best fit is a straight
line, always use a ruler to draw the line. If the best
fit for the data is a curved line, use a French curve if
you are unable to draw a smooth curve.
4. In cases where several sets of data are taken for
each trial, the average value is plotted. If more than
one line is plotted, be sure to label the lines or
provide a key.
5. Make sure all graphs have the following:
a) Each axis labeled with quantity plotted.
b) The unit of the quantity plotted.
c) The title of the graph on the paper (commonly
listed as the y variable versus the x variable).
Graphs A and B were made with the following data
obtained by hanging weights on a spring and
measuring how far the spring stretched.
2.0
1.5
1.0
0.5
0
2
0
4
6
distance spring stretched (cm)
.25
.50
.75
1.00
1.25
1.50
1.75
2.00
.80
2.41
2.74
3.69
5.20
6.28
6.72
8.85
8
10
Graph B
*Notice how the data is completely labeled. You are expected to write down
more than weight and distance when taking data. Indicate what weight and
what distance.
9
Part 2: Processing straight line graphs
3.0
weight on spring versus distance spring is stretched
2.5
x2
,y
2
2.0
1.5
1.0
,y
1
y
0.5
x1
weight on spring (N)
A graph in which the data points yield a straight line is called a
linear graph. When the intercept is zero, we say that the
variable on the y-axis is directly proportional to the variable
on the x-axis. That phrase “directly proportional” tells you that
if the variables doubles, so does the other. When the intercept
is not zero, we say that the change in the variable on the y-axis
is directly proportional to the change in the variable on the xaxis. For the Graph B, the relationship of the data is that the
change in the weight hanging on the spring is directly
proportional to the change in the distance the spring is
stretched. This means by whatever factor the force is
changed, the displacement will change by the same factor. To
understand what this means, on graph B, the value for the line
at zero stretch is .05 N and at 4.0 cm of stretch the weight is
.95 N. Let’s double the stretch to 8.0 cm and we see the weight
is now 1.85 N. We doubled the stretch and the weight did not
quite double. Now let’s look at the change in the variables.
When the stretch is changed from zero to 4.0 cm, the weight
changes from .03 N to .94 N or a change or .91 N. When the
stretch is doubled to 8.0 cm, the weight changes from .03 N to
1.85 N, for a change of 1.82 N. This shows that when the
change in stretch is doubled (from 4.0 cm to 8.0 cm) the
change in weight also doubled (from .91 N to 1.82 N).
0
0
x
2
6
4
distance spring is stretched (cm)
8
Graph B
When a graph is linear, the x and y variables have an algebraic relationship of the form y = mx+b. The b in the
equation is called the intercept and is equal to the value of y when the line crosses the y-axis. The m in the
equation is called the slope and is equal to change in y divided by change in x (y/x). Any two points on the line
should yield the same value, however it is best to pick points that are far apart. It is best if data points are not
used, and never use a data point that is not exactly on the line. On graph B, two points have been indicated by
arrows. They will be used to calculate the slope. One point has the values (x1 = 1.20 cm, y1 = .30N) and the
other (x2 = 6.40 cm, y2 = 1.50 N).
To find the slope we use the equation for slope, as shown below.
m
y y2  y1
1.50N-.30N
1.20N
N



.23 cm
x x2  x1 6.4cm -1.2cm 5.2cm
Notice that the units are included when the slope is calculated.
N
The complete equation for the data is weight = (.23 cm
)distance + .03 N
In physics, we usually let symbols or letters represent the variables. In this case, the final equation becomes
N
W = (.23 cm
) d + .03 N
Notice in this equation the slope and the intercept have units, and the variables do not. When any value for the
distance stretched is put into the equation with its unit, the similar unit in the slope should cancel out.
Horizontal straight lines
Occasionally data will produce a graph with an appearance similar to the diagram
to the right. This graph indicates that as the variable plotted on the x-axis
changes, the variable on the y-axis stays the same. There is no relationship
between the two variables. When asked to state what this type of graph
indicates, one says, “y is independent of x”.
The line has a slope of zero and the intercept is the same value as every other
point on the line. Therefore the algebraic relationship is y = b.
y
x
10
10
Part 3: Curved graphs
In the work you will be doing, when the data produces a curved graph, the graph will need to be linearized before
you can write the equation for the relationship. The shape of the graph will give you a clue on how to linearize it.
Here are the three basic shapes you will run into for curved graphs.
A. Top opening parabola
The shape the graph to the right is a top opening parabola. The line on the
graph becomes steeper as the line goes to the right. In the work we will do,
in the equation y = kxn, the power “n” is either 2 or 3. In other words, the
equation relating the two variables will be either y=kx2 or y=kx3 .
To straighten out a graph with this shape, all the data graphed on the x-axis
(horizontal) must be squared or cubed. The data graphed on the y-axis
(vertical) is not changed.
If asked to state the relationship of two variables when y = kx2, one states
that “y is proportional to x squared”. For example, when an object is dropped,
the distance it falls is proportional to the time squared. This means if the time
the object falls is doubled, the distance it falls is quadrupled.
B. Hyperbola
The shape of the graph to the right is a hyperbola. The curved line on the
graph goes down as the line goes to the right. This type of graph indicates
illustrates what is called an inverse relationship. If one quantity
increases, the other decreases. In the work we will do, in the equation y =
kxn, the power “n” is either -1 or -2. In other words, the equation relating
the two variables will be either y  kx-1 or y  kx -2 or more commonly,
k
k
y= or y= 2 .
x
x
To straighten out a graph with this shape, all the data graphed on the xaxis must be divided into 1, or it must be squared and that number
divided into 1. The data graphed on the y-axis is not changed.
k
y
x
top opening parabola
y
x
hyperbola
If asked to state the relationship of two variables when y = x , one states
that “y is inversely proportional to x”. For example, in an electric circuit,
when the resistance is doubled, the amount of current is cut in half.
Therefore the current is inversely proportional to the resistance. The power carried by a sound is inversely
proportional to the distance squared from the source of the sound. This means if you double the distance
from the sound source, the power carried by the sound is one fourth as much.
C. Side opening parabola
The shape of the graph to the right is a side opening parabola. The
curved line on the graph goes up and becomes less steep as the line goes
to the right. In the work we will do, in the equation yn = kx, the power “n” is
either 2 or 3. In other words, the equation relating the two variables will be
either y2 = kx or y3 = kx.
y
This means the square or the cube of the data graphed on the y-axis must be
plotted to straighten out the graph. The data graphed on the x-axis is not
changed. (Notice this is the opposite of what we did on the other graphs.)
If asked to state the relationship of two variables when y2 = kx, one
states, “y squared is proportional to x”. For example, when the mass
x
hanging on a spring is quadrupled, the time it takes for one trip up and
side opening parabola
down for the mass is twice as much. Therefore, the time squared for a
trip is proportional to the mass on the spring.
**On the inside of the back cover is a table that summarizes the graphs and the relationships.
11
Part 4: Straightening curved graphs
time for one oscillation versus mass on spring
The data below was taken during an experiment in which
masses were hung on a vertical spring. The mass was
pulled down and released. The time for one complete
oscillation was recorded. This data will be used to show
the process of straightening a curved graph and writing
the equation relating the mass on the spring and the time
for one oscillation. The data is plotted on the graph to
the right.
mass hung on
spring (kg)
time for one
oscillation (s)
.10
.20
.30
.40
.50
.60
.68
.92
1.16
1.29
1.50
1.66
2.0
1.6
time for one oscillation (s)
For a curved graph, before the exact relationship can be
determined and the equation for it written, the graph
needs to be straightened out. This is done by modifying
the data plotted on x-axis and replotting the points.
1.2
.8
.4
0
0
.1
.2
.3
.4
.5
4
Graph C
mass on spring (kg)
1. Plot the data. (Shown on graph C.)
2. Identify the shape of the curve. This curve is a side opening
parabola. That shape tells us that the time squared or cubed is
proportional to the mass.
mass hung on
spring (kg)
.10
.20
.30
.40
.50
.60
time2 for one
oscillation (s2)
.46
.85
1.35
1.66
2.25
2.76
4. Plot the modified data. (Shown on graph D.) Since the
points are linear (ignoring the one bad point), a straight
line is drawn. We can now definitely say the time
squared is proportional to the mass. (For example, if
the mass is increased by a factor of 4, the time will
increase by a factor of 2.)
5. Take the slope of the line and write the equation for
the relationship.
time2 versus mass on spring
3.0
2.5
time2 for one oscillation (s2)
3. The data on the y-axis must be taken to one of the
powers that go with the type of curve. To make a test
graph, take the square of the times in the original data
table. Do not change the data on the x-axis.
.6
2.0
1.5
1.0
.5
The arrows indicate points used for the slope. Point 1 is
x = .20 kg, y = .90 s2, point 2 is x = .48 kg, y = 2.2 s2.
m=
2.2s2 -.90s2 1.3s2
s2
. So the equation
=
=4.6
.48kg-.20kg .28kg
kg
0
.2
0
.6
.4
mass on spring (kg )
.8
Graph D
)
2
2
for the relationship is time2 = 4.6 s mass + 0 , or 2 = 4.6 s m .
kg
kg
Part 5: Sample problems involving variable relationships
1
1) Answer the following questions about the following equation: distance = 2 acceleration x time2, d  2 at .
a) State the relationship of distance and acceleration. Distance is directly proportional to the acceleration.
Quantities that are both to the same power are directly proportional.
b) If time is kept constant and the acceleration is doubled, by what factor does the distance traveled change?
Quantities that are directly proportional change by the same factor. If the acceleration is doubled, the
distance traveled will be doubled.
12
1.0
1
2
c) State the relationship of distance and time. Distance is directly proportional to time squared or distance is
directly proportional to the square of time.
d) If the acceleration is kept constant and the time is tripled, by what factor does the distance traveled
change?
2
Looking at the equation d1  21 at , if a 3 is placed in front of “t”, d 2  21 a(3t) 2  d 2  21 a(9t 2 ) , the 3
needs to be squared along with the “t”. This makes the value for “d” nine (32) times larger.
e) State the relationship for acceleration and time.
Since the variables are on the same side of the equation, first manipulate the equation to get
acceleration alone on the left side.
2d
d  21 at 2  a  2 Acceleration is inversely proportional to time squared.
t
f) If the distance is kept constant and the time to cover that distance is cut in half, by what factor does the
acceleration change?
A trick that can be used is to replace everything that does not change with a 1. Since the distance and
the 2 don’t change, lets give “2d” a value of 1. If we do this we must replace the equals with a
proportional symbol “”. This makes the original value, a1 1 . When “t” is half as much,
t2
1
1
4
a2 1 2  a2 1 2  a2 2 . This is 4 times larger than the original, so the acceleration would be 4
( 2 t)
t
t
4
times greater.
g) If the acceleration is constant, and the distance is quadrupled, what happens to the time?
This type of question is easiest to do if the variable you are asked about is to the first power. So the
equation is rearranged so time is alone on the left side. Then the change for d is placed in the equation.
(Since the ½a is constant it can be replaced with a 1.)
t1
d
4d
2 d
 if d is quadrupled, t 2 
 t 2
 t 2  2t1 Therefore the time is doubled.
1
1
1
Problems:
60
1.2
50
1.0
40
.8
force (N)
distance (m)
1) Title the two graphs below, calculate the slopes and write the equation for the line.
30
20
.6
6
.4
4
10
0
0
.2
5
15
10
time (s)
20
25
0
0
1.0
2.0
distance2
3.0
4.0
5.0
(cm2)
13
2) In the data in the table below, the first column is the
independent variable, and the second column is the
dependent variable. Graph the following set of data
on the grid provided. When done, calculate the slope
of the line and write the equation for the line.
Potential difference (volts)
Current (amps)
0.25
0.30
0.40
0.50
0.75
1.00
15.0
20.0
24.2
31.0
45.0
59.8
Slope:
Equation:
3) Plot the data below on the grid.
What type of relationship is shown?
Modify the data and show the new data table in the
area below.
volume (in3)
pressure (bars)
12.0
11.0
10.0
9.0
7.5
6.0
5.0
29.1
31.0
35.3
39.3
47.1
58.8
70.7
Modified data table:
Plot the modified data on the grid on the next page.
14
Write the complete equation for the modified graph.
paycheck vs. time worked
paycheck ($)
250
200
150
100
50
0
0
5
10
15
20
25
30
35
40
time worked (hr)
4) This is a graph that Lena Genster made for her first two weeks at Burger World.
a) Which is the dependent variable and which is the independent? How do you know?
b) What is the slope for this graph? What is the slope the value of?
c) What is the mathematical model (equation) which describes the graph?
d) State the relationship of the variables?
15
5) Dan Druff performed an experiment with a metal sphere. He shot the sphere from a slingshot and measured
its maximum height. Six different trials were performed with the sphere being shot at a different angle from
the horizontal for each trial.
a) What is the relationship being studied?
b) What is the independent variable in this experiment?
c) What is the dependent variable in this experiment?
d) What other variables must be held constant throughout this experiment?
current vs. resistance
current (amps)
1.2
1
0.8
0.6
0.4
0.2
0
0
100
200
300
400
500
600
700
resistance (ohms)
6) Sally Veight did an electricity lab that produced the graph above.
a) What is the shape of this graph? What type of relationship does this graph suggest?
b) What variables should Sally plot to linearize the data?
c) If what Sally did in (b) linearized the graph, what are the units of the line’s slope?
d) If what Sally did in (b) produced a graph with shape of the sketch to the right,
what variables should she plot to try and linearize the data?
16
7) a) Bob N. Furapels did a careful study of the relationship between wirbles and didacs. His data produced the
linearized graph below. Title the graph and determine the equation for the graph below? (Show setup for
the slope.)
350
wirble (b)
300
250
200
150
100
50
0
0
5
10
15
2
20
25
30
2
didacs (n )
b) Sketch the shape of the graph before it was linearized? What is the shape of that graph called? What was
its independent variable?
c) State the relationship of the variables?
Answer the questions following each of the equations below.
potential difference = current x resistance, V = IR
8a) What is the relationship between potential difference and current?
b) If the potential difference is doubled, by what factor will the current change?
c) What is the relationship between current and resistance?
d) If the potential difference is unchanged and the resistance is doubled, by what factor will the current
change?
2
kinetic energy = 21 mass x velocity
, KE =
1
2
mv2
9a) What is the relationship between kinetic energy and velocity?
17
KE =
1
2
mv2
b) If the velocity increases by a factor of 4, by what factor will the kinetic energy change?
c) If the velocity changes to one half of the original, by what factor will the kinetic energy change?
d) If the velocity remains the same and the mass is tripled, by what factor will the KE change?
force=(Coulomb's constant)
qq
charge one  charge two
, F  k 1 22
distance apart 2
d
10a) What is the relationship between force and distance apart?
b) What is the relationship between force and charge two?
c) If the charges are unchanged and the distance apart is tripled, by what factor will the force change?
d) If the charges are unchanged and the distance apart is 1/4th as much, by what factor will the force change?
period2  4 2
length of pendulum
,
gravitational field strength
Τ2 =(4 2 )
g
11a) What is the relationship between the period and the length?
b) If the period is to be tripled, by what factor must the length change?
c) If the length is doubled, by what factor will the period change?
d) If the length is decreased to one sixteenth as much, by what factor will the period change?
e) If the pendulum is taken to a planet where gravitational field strength is 4 times more, by what factor will the
period change?
18