A table of measurements can hide a simple relationship in a crowd of numbers. A well-chosen graph lets the pattern speak—but only if the axes, units and fitted line are asking the right question.
A useful graph is more than a neat picture. Its axes carry quantities and units, its slope has a physical meaning, and its deviations can reveal an incomplete model. This chapter follows the full route from a table of observations to an interpreted parameter, including the limitations of a fitted line.
What you will learn to do
- Choose axes that match a physical model and retain their units.
- Extract a slope and intercept from a line without forcing it through the origin.
- Use a change of variables to test a nonlinear relationship.
- Distinguish area, slope and reading height on a graph.
- Interpret residuals, bounds and extrapolation without claiming unsupported certainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Make an experiment speak through a graph Learning goals: - Choose axes that match a physical model and retain their units. - Extract a slope and intercept from a line without forcing it through the origin. - Use a change of variables to test a nonlinear relationship. - Distinguish area, slope and reading height on a graph. - Interpret residuals, bounds and extrapolation without claiming unsupported certainty. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
On a distance–time plot, why does the slope have speed units?
Compare your reasoning
Slope is a change in the vertical quantity divided by a change in the horizontal quantity. Metres divided by seconds gives m/s. That unit check also helps interpret less familiar plots.
Revisit the explanation →01 / Make an experiment speak through a graph
Choose the question before drawing the axes
Suppose you control an input x and record a response y. Plotting x horizontally and y vertically is a common choice, but the real reason for the choice is the model you want to test. If the proposed relation is y = mx + b, a straight-line plot exposes both its scale m and its offset b.
Name the physical quantity and unit on each axis. “Time (s)” is a complete axis description; “t” without its unit is incomplete unless the unit is explicitly supplied elsewhere. A plotted coordinate such as (2, 5) means nothing physical until the axis scales tell you whether it represents seconds and metres, amperes and volts, or something else.
Choose a scale that makes the data legible and uses the available space. A graph need not start at zero, but a truncated axis must be labelled honestly. Do not judge a slope from the apparent steepness of a line on the screen: stretching the horizontal or vertical scale changes its visual angle without changing the physical relation.
| Graph | Slope | Area under the curve |
|---|---|---|
| Position against time | Velocity | Not displacement; units are m·s |
| Velocity against time | Acceleration | Signed displacement |
| Force against displacement | Force gradient | Work when the plotted component is along displacement |
| Force against time | Rate of force change | Impulse along the chosen axis |
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Make an experiment speak through a graph Focus: Choose the question before drawing the axes Learning goals: - Choose axes that match a physical model and retain their units. - Extract a slope and intercept from a line without forcing it through the origin. - Use a change of variables to test a nonlinear relationship. - Distinguish area, slope and reading height on a graph. - Interpret residuals, bounds and extrapolation without claiming unsupported certainty. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
02 / Make an experiment speak through a graph
A slope is a ratio of changes, with units
For a straight line, any two distinct points on that line give the same ratio Δy/Δx. Choose points far apart to reduce the relative effect of coordinate-reading uncertainty. If you have drawn or fitted a best-fit line through scattered data, read suitable points on the line; they need not be original measured points.
The intercept b is the predicted y value when x is zero. It might represent an initial position, a baseline signal or an unwanted offset. A nonzero intercept is not automatically an error: its interpretation comes from the model. If x = 0 is outside the measured range, the intercept is an extrapolated parameter rather than a directly observed reading.
Changing the order of subtraction in both numerator and denominator preserves the slope. Reversing only one changes its sign incorrectly.
WORKED EXAMPLE
Find a scale factor and an offset separately
An idealised sensor follows V = RI + b. Measurements are (I, V) = (2.0 mA, 0.90 V), (5.0 mA, 1.50 V) and (8.0 mA, 2.10 V). Find R and b.
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Separate the slope from the baseline
- Sensor output
- 0.5
STOP & THINK
A straight voltage–current line rises by 0.60 V over a current increase of 3.0 mA. What is its slope in ohms?
Answer in Ω.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Make an experiment speak through a graph
Focus: A slope is a ratio of changes, with units
Learning goals:
- Choose axes that match a physical model and retain their units.
- Extract a slope and intercept from a line without forcing it through the origin.
- Use a change of variables to test a nonlinear relationship.
- Distinguish area, slope and reading height on a graph.
- Interpret residuals, bounds and extrapolation without claiming unsupported certainty.
Equations in this section (LaTeX):
m=\frac{y_2-y_1}{x_2-x_1},\qquad b=y_1-mx_1
Meaning: Changing the order of subtraction in both numerator and denominator preserves the slope. Reversing only one changes its sign incorrectly.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.03 / Make an experiment speak through a graph
A fitted trend is not a dot-to-dot path
Measured points usually do not fall exactly on one line. Joining them in acquisition order would imply that every small fluctuation is a true feature of the response. If the physical hypothesis is a linear trend plus reading scatter, estimate a line representing that trend instead.
A simple least-squares line minimises the sum of squared vertical residuals. This method fits a particular error model: commonly x is treated as known much more accurately than y, and observations have comparable independent vertical scatter. If x uncertainties are important or y uncertainties differ greatly, the fitting method must account for them. A fitted line is not automatically the correct statistical analysis.
Residual means observed y minus the fitted y at the same x. Plot or inspect residuals after fitting. Random-looking small residuals are compatible with the model, but do not prove it. A curved pattern, a changing spread or a common displacement can signal model inadequacy, changing noise or a calibration issue.
Residuals compare data with a chosen model. They are not known errors relative to an unknown true value.
WORKED EXAMPLE
Use the unused pattern to question the model
A proposed position model is x(t) = 1.0 + 2.0t metres, with t in seconds. At t = 0, 1, 2, 3 s, observations are 1.0, 3.1, 5.4, 7.9 m. Find the residuals and interpret the pattern.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Make an experiment speak through a graph Focus: A fitted trend is not a dot-to-dot path Learning goals: - Choose axes that match a physical model and retain their units. - Extract a slope and intercept from a line without forcing it through the origin. - Use a change of variables to test a nonlinear relationship. - Distinguish area, slope and reading height on a graph. - Interpret residuals, bounds and extrapolation without claiming unsupported certainty. Equations in this section (LaTeX): r_i=y_i-(mx_i+b) Meaning: Residuals compare data with a chosen model. They are not known errors relative to an unknown true value. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
04 / Make an experiment speak through a graph
Change the plotted variable to test the model
A relationship can be nonlinear in the directly measured variables and linear in a transformed variable. For an ideal small-angle simple pendulum, T = 2π√(L/g). Squaring both sides gives T² = (4π²/g)L. A plot of T² against L should therefore be linear under that model.
Do the algebra before choosing axes. Plotting T against L and forcing a straight line through it does not test the square-root prediction. Conversely, a straight plot after transformation is evidence about that specific relationship, not a universal sign that the experiment has no systematic errors.
A transformation also changes uncertainty. For a small period uncertainty, the uncertainty in T² is approximately 2T times the uncertainty in T. Equal absolute uncertainty in T does not imply equal absolute uncertainty in T² at all periods. Weighting and error bars should reflect the transformed quantity.
WORKED EXAMPLE
Extract a physical constant from a transformed slope
For an ideal small-angle pendulum experiment, a T²-versus-L line has slope 4.00 s²/m. Use π² = 9.87 for this calculation. Estimate g.
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STOP & THINK
If a model predicts y = C/x² with positive x, which plot should be straight through the origin?
- y against x
- y against 1/x²
- y² against x
MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Make an experiment speak through a graph Focus: Change the plotted variable to test the model Learning goals: - Choose axes that match a physical model and retain their units. - Extract a slope and intercept from a line without forcing it through the origin. - Use a change of variables to test a nonlinear relationship. - Distinguish area, slope and reading height on a graph. - Interpret residuals, bounds and extrapolation without claiming unsupported certainty. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
05 / Make an experiment speak through a graph
A slope estimate needs an uncertainty interpretation
If a problem gives admissible reading bounds, you can ask which slopes are compatible with those bounds and with the proposed model. Lines that touch different extreme corners can illustrate a range, but a rigorous fit to many bounded observations must satisfy all of their allowed regions, not just two convenient points.
A statistical standard uncertainty in a fitted slope is a different object. It depends on a noise model, the number and spread of measurements, and the fitting procedure. Do not label a worst-case two-point calculation as a one-standard-deviation confidence result.
Spread measurements over a useful horizontal range when the model remains valid. A larger Δx makes a fixed absolute Δy reading uncertainty less important to the slope. However, extending the range into a new physical regime can invalidate the model; more separation is helpful only when you are still measuring the same relationship.
WORKED EXAMPLE
Calculate a bounded two-point slope
Two positions measured at exact times 1 s and 5 s are (3.0 ± 0.1) m and (11.0 ± 0.1) m. The ± terms are independently admissible bounds. Find the secant velocity and its worst-case bound.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Make an experiment speak through a graph Focus: A slope estimate needs an uncertainty interpretation Learning goals: - Choose axes that match a physical model and retain their units. - Extract a slope and intercept from a line without forcing it through the origin. - Use a change of variables to test a nonlinear relationship. - Distinguish area, slope and reading height on a graph. - Interpret residuals, bounds and extrapolation without claiming unsupported certainty. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
06 / Make an experiment speak through a graph
Read accumulated quantities without confusing them with slopes
A narrow vertical strip beneath a graph has approximate area y Δx. Adding strips accumulates a quantity with units equal to vertical-unit times horizontal-unit. On a force–time graph this is impulse; on a velocity–time graph it is displacement. The units identify the physical candidate before you calculate.
For signed quantities, area below the horizontal zero line contributes negatively. Total distance from a velocity graph requires adding the magnitudes of areas instead of allowing cancellation. A force–displacement graph gives work only for the appropriate force component along the displacement.
Extrapolation deserves particular care. A line that describes the measured interval may fail when a spring leaves its elastic regime, a resistor heats, or a sensor saturates. Report the range over which the data support the model, and distinguish an interpolation inside that range from a prediction outside it.
STOP & THINK
A force–time pulse is a triangle with base 0.40 s and height 30 N, entirely above zero. What is its impulse?
- 75 N/s
- 12 N·s
- 6 N·s
MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Make an experiment speak through a graph Focus: Read accumulated quantities without confusing them with slopes Learning goals: - Choose axes that match a physical model and retain their units. - Extract a slope and intercept from a line without forcing it through the origin. - Use a change of variables to test a nonlinear relationship. - Distinguish area, slope and reading height on a graph. - Interpret residuals, bounds and extrapolation without claiming unsupported certainty. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BRING THE IDEAS TOGETHER
What should stay with you
- Choose axes from the physical question and label units explicitly.
- A slope measures a ratio of changes; an intercept records a baseline or extrapolated starting value.
- Fit the proposed model and inspect residuals rather than joining every fluctuation.
- Linearisation changes both the plotted relation and its uncertainty.
- Check the dimensions of slope and area, and state the range over which a model is supported.
RETRIEVE, THEN EXPLAIN
Try it without the examples
Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.
SELF CHECK 1
A straight line passes through (x, y) = (1 s, 5 m) and (4 s, 14 m). What is its y intercept?
Answer in m.
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SELF CHECK 2
A force component increases linearly from 2 N to 8 N as displacement increases from 0 to 3 m. What work does it do?
Answer in J.
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SELF CHECK 3
An experiment predicts a zero intercept but a fit gives a nonzero intercept. What is the appropriate next step?
- Erase the intercept and force the fit through zero
- Investigate calibration, model assumptions and uncertainty
- Conclude the governing law has certainly failed
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APPLY THE CHAPTER
Take the ideas into a problem
Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.
Use the slope when the zero is uncertainAdvanced · Hints and guided reasoningFind the hidden open-circuit interceptIntermediate · Hints and guided reasoningExtract gravity from a straight-line graphIntermediate · Hints and guided reasoningRead the slope, not a single ratioIntermediate · Hints and guided reasoningRead capacitance from a logarithmic graphIntermediate · Hints and guided reasoningUse the slope to find a halving timeIntermediate · Hints and guided reasoningCOME BACK TO THE IDEA
What can you explain later?
Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.
Next study session
For T² proportional to pendulum length L, which plot should be straight? What would its slope represent in the small-angle model?
Compare your reasoning
Plot T² vertically against L horizontally. The slope is 4π²/g with units s²/m. Plotting T against L instead produces a curved relationship.
Revisit the explanation →A few days later: change the situation
Your measured points nearly follow a straight line but the intercept is nonzero. Should you force the fit through the origin?
Compare your reasoning
Only if the model and calibration justify that constraint. An offset may be evidence about the apparatus or model. Forcing it away can bias the slope and the quantity inferred from it.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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