Graphing Linear Equations
Graphing Review
Objectives
Review graphing techniques in Microsoft Excel
Review the elements of a properly constructed scientific graph
Use linear and non-linear fits to determine the value of a quantity
Equipment
Computer with Microsoft Excel
Theory
The information presented in graphs can be used to determine unknown quantities. If a
single piece of data is used to calculate an unknown, what is the accuracy? Each data point is
affected by uncertainty; by using a best fit to the data set with some kind of regression, such as
the method of least squares, uncertainties are minimized and the fit will be a better representation
of the relationship. An R2 value can also be determined for the fit. This quantity indicates the
spread in the data about the best fit line. Values of this quantity range from 0, for a very poor fit,
to 1, for a fit that passes through every data point.
A well-prepared graph will include the following features:
Title: a title must indicate what the graph is being used to determine. Using the title “Y vs. X” is
redundant as the reader can already see this information by examining the axes. Titles should
answer the question “Why did I plot this graph?”
Labels: the axis labels should indicate the quantity plotted, including appropriate units.
Best fit equation: the equation shows the mathematical relationship between the plotted
variables.
R2 value: this value indicates the quality of the best fit
Legend: a legend is needed only when more than one set of data is plotted on the same set of
axes. If only one data set is plotted, the legend is redundant and should be deleted.
Scale: the scale of your graph should “frame” the data. Excess blank space makes a graph too
small and difficult to interpret.
For all future graphs, grades will be determined based on the inclusion of all these quantities. The
graph shown on the following page incorporates all the required elements for proper graphing.
1.
The data points are not connected. All graphs should be scatter plots because there is no
information at locations other than the specific data points.
2.
The title is appropriate. It tells the reader exactly why the data has been plotted and what
the experimenter is trying to determine. It does not incorporate the axis labels.
3.
There are labels, written as a full word, and units, written symbolically, on both axes.
4.
The best fit is present, with the equation and R2 value for the fit.
5.
There is no legend because there is only one set of data plotted.
6.
The scale is appropriate so that there is little blank space on the graph.
8/9/23 MAG
Graphing 1 of 4
Graphing Linear Equations
Determining the Permeability Constant
Magnetic Field (mT)
0.25
0.20
0.15
0.10
y = 0.114x - 0.01
R² = 0.9941
0.05
0.00
0.0
0.5
1.0
1.5
2.0
2.5
Current (A)
Figure 1: A properly formatted linear graph.
Procedure
1.
Partner with one member of the class.
2.
Decide which lab partner will work through Version A and which will do Version B of
the lab. Once you leave class, you will not need to work together on this lab if you don’t
want to.
3.
All work for this lab should be typed and printed in the order asked for in the lab. All
equations should be written using Microsoft Equation Editor or a similar program.
4.
All graphs must be included with their data tables.
5.
Turn in your personal version of the lab according to your instructor’s specifications.
8/9/23 MAG
Graphing 2 of 4
Graphing Linear Equations
Version A
Your new television contains a capacitive circuit that the manufacturer states will discharge with
a time constant (τ) of 2.25 seconds after you turn off the set. You collect the following data to
verify this claim.
Time (seconds)
Charge (mC)
0.5
160
1.0
122
1.5
110
2.0
93
2.5
70
3.0
64
3.5
48
4.0
39
4.5
35
5.0
25
5.5
20
6.0
17
6.5
15
7.0
10
7.5
10
8.0
7
1.
Graph the data as given and use an appropriate curve fit to fit the data. Note that the
𝑡
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
decay follows the theoretical relationship 𝑄(𝑡) = 𝑄0 𝑒 − ⁄𝜏 where the manufacturer’s
value for Q0 is 200mC.
Determine the experimental value of Q0. Explain how you arrived at this value.
Calculate the percent error between the given value of Q0 and the value you found using
your curve fit. Show your work.
Determine the time constant (τ) of the decaying circuit using the curve fit found in #1.
Show your work.
Determine the percent error between your value for τ and the manufacturer’s value of
2.25 s. Show your work.
Linearize the theoretical fit. Show your work.
Use this linearized fit to decide how to linearize the original data. Create a new data table
with the linearized data set.
Create a properly formatted linear graph.
Use the slope and y-intercept of the linear fit of your graph to determine the experimental
values of Q0 and τ. Show your work.
Calculate the percent error between the given value of Q0 and the experimental value
found in #9. Show your work.
Calculate the percent error between the given value of τ and the experimental value found
in #9. Show your work.
Compare the two experimental values of τ that you got from your graphs. Should you use
percent error or percent difference? Why? Show your work for whichever one you chose.
8/9/23 MAG
Graphing 3 of 4
Graphing Linear Equations
Version B
Your new television contains an inductive circuit that regulates the rate at which the current drops
when the set is turned off. The manufacturer states the decay has time constant (τ) of 4.15 seconds
after you turn off the set. You collect the following data to verify this claim.
Time (seconds)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
1.
Current (A)
2.8
2.3
2.2
1.9
1.6
1.4
1.3
1.1
1.0
0.8
0.8
0.7
0.6
0.5
0.4
0.3
Graph the data as given and use an appropriate curve fit to fit the data. Note that the
𝑡
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
decay follows the theoretical relationship 𝐼(𝑡) = 𝐼0 𝑒 − ⁄𝜏 where the manufacturer’s value
for I0 is 3.00A.
Determine the experimental value of I0. Explain how you arrived at this value. Calculate
the percent error between the given value of I0 and the value you found using your curve
fit. Show your work.
Determine the time constant (τ) of the decaying circuit using the curve fit found in #1.
Show your work.
Determine the percent error between your value for τ and the manufacturer’s value of
4.15 s. Show your work.
Linearize the theoretical fit. Show your work.
Use this linearized fit to decide how to linearize the original data. Create a new data table
with the linearized data set.
Create a properly formatted linear graph.
Use the slope and y-intercept of the linear fit of your graph to determine the experimental
values of I0 and τ. Show your work.
Calculate the percent error between the given value of I0 and the experimental value
found in #9. Show your work.
Calculate the percent error between the given value of τ and the experimental value found
in #9. Show your work.
Compare the two experimental values of τ that you got from your graphs. Should you use
percent error or percent difference? Why? Show your work for whichever one you chose.
8/9/23 MAG
Graphing 4 of 4