A graph is used to illustrate the dependence of one variable on another. The
independent variable (the one you control) appears on the horizontal or x axis and the
dependent variable (the one whose dependence on the independent variable you wish
to determine) appears on the vertical or y axis. The graph is labeled as y vs. x. So for
example if you wish to determine the dependence of acceleration (a) on force (F), you
would set up an experiment that allows you to apply a series of different forces to an object and
measure the acceleration of the object for each force. Since you control the force, it is the
independent variable. Since the acceleration for each case depends on the force, the acceleration
is the dependent variable. To examine the relationship between the two, .make a graph of a vs. F,
i.e. F would appear on the horizontal axis and a would appear on the vertical axis. Or if you
wanted to determine the dependence of period (T) of a pendulum on its length (L), you would
measure the period (dependent variable) of the pendulum for each of a set of lengths which you
control (independent variable) and make a T vs. L graph, i.e. one with L on the horizontal axis
and T on the vertical axis.
1
Distance traveled (cm)
The graph shown on the left is a graph of
________________ vs.________________ .
The independent variable is ________________ .
The dependent variable is ________________ .
Time (s)
If you answered distance traveled, time, time, distance traveled, respectively in the blanks above,
go to
2
If you answered anything else, continue.
The independent variable is the quantity that the experimenter controls in an experiment. It
appears on the horizontal (x) axis of a graph. The dependent variable is the one that changes in
response to changes in the independent variable. It appears on the vertical (y) axis. For example,
if you wish to determine how the speed of an object just before hitting the floor depends on the
height from which it is dropped, you would drop an object from various heights and measure the
speed of the object just before hitting the ground for each height. The variable you controlled was
the height, so that is the independent variable. It goes on the horizontal axis of your graph. The
variable for which you measured the changes as a result of the change in height is the speed just
before hitting the ground, so that is the dependent variable. It would appear on the vertical axis
on your graph. This graph would be labeled as final velocity vs. height of drop, as shown below.
(Units on each axis should of course be included.)
Final Velocity (cm/s)
Final Velocity vs. Height of drop for
falling object
The graph on the right shows
________________ vs.________________ .
________________ is the dependent variable.
________________ is the independent variable.
Potential difference(volts)
Height of drop (cm)
Current (amps)
If you answered potential difference, current, potential difference, current, respectively continue
to
2
Otherwise, return to
1
If the graph of the dependent variable vs. the independent variable is a straight
line, we say that the dependence is linear. That is, the dependent variable depends
on the first power of the independent variable. If the independent variable is
labeled x and the dependent variable is labeled y and the relationship between y and
x is linear, then the relationship can be written in the form y = m x + b, where m
and b are constants representing the slope and the y-intercept respectively.
2
The graph at right shows such a linear relationship. Two
points on the graph have been labeled (x1, y1) and (x2, y2).
The slope of the line is given by
y2
(x2 ,y2 )
y1
(x1,y1)
b
The units of the slope will be the units of the variable on the
vertical (y) axis divided by the units of the variable on the
horizontal (x) axis.
27.0
24.0
Velocity in m/s
21.0
18.0
15.0
12.0
9.0
6.0
3.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Time in s
The slope of the graph shown above is __________.
If you answered 4.2 m/s2, go to
3
If not continue.
x1
x2
x
To find the slope of the graph, select two points near the ends of the line and identify the x and y
coordinates of each point.
(5.0 s, 27.0 m/s)
27.0
24.0
Velocity in m/s
21.0
18.0
15.0
12.0
9.0
6.0
(0 s, 6.0 m/s)
3.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Time in s
Next find the difference in the y-coordinates (Δy = y2 - y1) and the difference in the x-coordinates
(Δx = x2 - x1) between the two points. Don’t forget to include units.
27.0
24.0
21.0
18.0
Δy = y2 -y1 = 27.0 m/s-6.0 m/s = 21.0 m/s
15.0
12.0
9.0
6.0
3.0
Δx = x 2 -x 1 = 5.0s - 0 s = 5.0 s
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Time in s
Then divide Δy by Δx to obtain the slope.
If the line slopes downward, the slope is negative. For example, for the graph shown below,
Δy = 40 m - 180 m = -140 m and Δx = 6.0 s - 0 = 6.0 s.
180
Displacement in m
160
So
140
120
100
y
80
60
40
x
20
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
Time in s
Find the slope of the line in the graph below.
4.5
4.0
3.5
Force in N
3.0
2.5
2.0
1.5
1.0
0.5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
Distance in m
If your answer is 1.8 N/m, continue to
3
If not, return to
2
3
The y-intercept can be obtained from a linear graph simply by reading the vertical
scale at the point where the line crosses the y-axis. For the graph shown below, the
y-intercept is 1.5 N.
4.5
4.0
3.5
Force in N
3.0
2.5
2.0
1.5
1.0
0.5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
Distance in m
Once the slope and the y-intercept for a linear graph have been obtained, an equation can be
written which shows the specific functional dependence of y on x. The equation is written in
what is called “slope-intercept” form: y = mx + b, where m is the slope and b is the y-intercept.
The slope for the graph above was determined to be 1.8 N/m and the y-intercept is 1.5 N, so the
equation of the line (in slope-intercept form) is
y = (1.8 N/m) x + 1.5 N
Note: Sometimes the units may be left out of the equation if it’s understood that all quantities are
expressed in a particular set of units (usually SI units), however it should be remembered that for
a graph that represents any physical situation there are always units associated with both the slope
and the y-intercept.
Occasionally it is desired to obtain the x-intercept. This can be obtained directly from the
graph by reading the x-axis at the point where the line crosses it, or it can be obtained from the
equation of the line in slope-intercept form (or in any other form) by setting y=0 and solving for
x.
In the graph shown above, the line doesn’t cross the x-axis in the region shown so, in order to
obtain the x-intercept from the graph, it is necessary to extend the line backwards until it crosses
the x-axis.
4.5
4.0
3.5
Force in N
3.0
2.5
2.0
1.5
1.0
0.5
-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
Distance in m
We see now that the x-intercept is at -0.8 m.
We could have obtained the same result from the equation for the line:
y = (1.8 N/m) x + 1.5 N
Set y = 0
0 = (1.8 N/m) x + 1.5 N
and solve for x.
(1.8 N/m) x = -1.5 N
Find the slope and the x and y intercepts for the graph shown below, and write the equation for
the line shown in slope-intercept form.
9.0
8.0
7.0
Force in N
6.0
5.0
4.0
3.0
2.0
1.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
Distance in m
If you obtained the answers:
slope = -7.0 N/m
y-intercept = 8.0 N
x-intercept = 1.1 m
equation of line: y = (-7.0 N/m) x + 8.0 N
go to
4
If not, continue.
For the graph above,
The y-intercept (b) can be read from the graph as 8.0 N
So the equation of the line, in slope intercept form (y = m x + b) is
y = (-7.0 N/m) x + 8.0 N
The x-intercept can be read from the graph as 1.1 m, or it can be obtained by setting y = 0 in the
above equation.
0 = (-7.0 N/m) x + 8.0 N
(7.0 N/m) x = 8.0 N
Find the slope and the x and y intercepts for the graph shown below, and write the equation for
the line in slope-intercept form.
90
Displacement in m
80
70
60
50
40
30
20
10
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0
Time in s
If you obtained:
slope = 2.7 m/s
y-intercept = 10 m
x-intercept = -3.7 s
equation of line: y = (2.7m/s) x + 10 m
continue to
4
If not, go back to
3
4
If the equation expressing the relationship between two variables is
not in slope-intercept form, it can be rearranged to put it in that form
and the slope and y-intercept can then be obtained.
For example, if the equation 3T – 6k = 12 gives the relationship
between the variables T and k, the slope and y-intercept of the graph of T vs. k
can be found by rearranging the equation as follows:
3 T - 6 k = 12
3 T = 6 k + 12
T=2k+4
The equation is now in slope intercept form, y = m x + b with the slope, m, equal
to 2 and the y-intercept, b = 4. So the graph showing this relationship would
look like this: (Units have been omitted because the T and k used here
represent arbitrary variables and so don’t have any specific units attached to
them. In any real experiment, units would need to be included.)
T
9.0
8.0
7.0
6.0
5.0
4.0
3.0
2.0
1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
k