Graphs and mathematical relationships in Physics

Overview

Graphs help Physics learners see patterns that may be hard to notice in a table. A graph can show whether two quantities increase together, whether one decreases as another increases, whether a relationship is linear, and whether a result is unusual.

In Form II, this topic connects experimental data from light, magnetism, static electricity, and current electricity with mathematical relationships. A learner may plot current against voltage, image size against object size, deflection against number of turns, or another safe classroom relationship.

Key idea: a graph is evidence. Its axes, scale, plotted points, best-fit line, gradient, and conclusion must all match the quantities being investigated.

Prerequisites

• Physical quantities and SI units - Graph axes need quantities and units.
• Graphical presentation of experimental results - Form I graphing foundations: axes, scale, plotting, and gradient.
• Spreadsheet processing of experimental data - Processed tables can prepare graph-ready data.
• Mathematics for light and current electricity - Formula relationships support graph interpretation.
• Light and Current electricity - Form II contexts for graphs and relationships.

Learning Scope

This page covers:

• Preparing graph-ready data from experiments.
• Choosing axes, units, and scales.
• Plotting points and drawing a best-fit line or curve.
• Interpreting direct, inverse, and nonlinear relationships at an introductory level.
• Calculating and interpreting gradient for straight-line graphs.
• Connecting graph shape to Physics formulas.

This page does not teach advanced statistics, calculus, digital curve fitting, or professional laboratory graphing. It keeps graph work at learner-ready Form II level.

Subtopics

Variables And Axes

The independent variable is the quantity deliberately changed or selected. It usually goes on the horizontal axis. The dependent variable is the quantity measured in response. It usually goes on the vertical axis.

Example: in a circuit experiment, voltage may be changed while current is measured. Plot voltage on the horizontal axis and current on the vertical axis.

Scale And Units

Each axis must show a quantity and a unit.

Good axis labels:

• Voltage $V$ (V)
• Current $I$ (A)
• Image distance (cm)
• Number of turns

Choose a scale that uses most of the graph paper and is easy to read. Avoid awkward scales that make points difficult to plot.

A suitable scale should:

• cover the smallest and largest values in the data
• use simple intervals such as $1$, $2$, $5$, or $10$
• make the plotted points spread across the graph area
• allow values to be read without guessing too much

Example: if current values run from $0.10\ \text{A}$ to $0.90\ \text{A}$, a vertical scale of $0.10\ \text{A}$ per large square may be clearer than $0.07\ \text{A}$ per large square.

Plotting Points

Each point represents one pair of values. Plot points carefully and check that each point matches the table.

Example:

| Voltage $V$ (V) | Current $I$ (A) | |---:|---:| | 1 | 0.2 | | 2 | 0.4 | | 3 | 0.6 |

The points are $(1,0.2)$, $(2,0.4)$, and $(3,0.6)$.

Plotting checklist:

• read the horizontal value first
• read the vertical value second
• mark the point with a small cross or dot
• check the point against the table before moving on
• do not thicken points until they hide the exact position

Best-Fit Line Or Curve

A best-fit line shows the overall trend. It does not have to pass through every point. If the points form a smooth curve, draw a smooth curve instead of forcing a straight line.

Key insight: forcing a straight line through curved data gives a false relationship.

For learner work, a best-fit line should usually have a balanced number of points on both sides. If one point is far away from the pattern, do not bend the line to reach it. Mark it as a point to check.

Direct Relationship

A direct relationship means both variables increase together in a constant ratio. A graph may be a straight line through the origin.

For an ohmic conductor:

$$ V = IR $$

If $R$ is constant, $V$ and $I$ are directly related.

Another direct relationship can appear when image size changes in the same ratio as object size under a fixed setup. The learner should always connect the pattern back to the experiment, not only name the graph shape.

Inverse Relationship

An inverse relationship means one variable decreases when the other increases. At this level, learners should recognise the pattern and describe it carefully.

Example: if the same energy is spread over a larger area, the effect per unit area may decrease.

In a Form II practical context, inverse patterns should be described carefully unless the learner has enough data to prove the exact formula. It is acceptable to write, "as one quantity increases, the other decreases" when the exact relationship has not been established.

Nonlinear Relationships

A nonlinear relationship gives a curved graph. Curved graphs can still be meaningful.

Examples:

• brightness may not appear to increase evenly with distance because the eye is not a measuring instrument
• heating effect in a component may change as resistance changes with temperature
• some optical measurements may form a curve if distances are not chosen carefully

Key insight: a curved graph is not automatically wrong. It may show that the relationship is not a simple direct proportion.

Gradient

For a straight-line graph:

$$ \text{gradient} = \frac{\text{change in vertical quantity}}{\text{change in horizontal quantity}} $$

On a voltage-current graph with voltage on the vertical axis and current on the horizontal axis:

$$ \text{gradient} = \frac{\Delta V}{\Delta I} = R $$

The unit is ohm, because $\frac{\text{V}}{\text{A}} = \Omega$.

When calculating gradient:

1. choose two points on the best-fit line, not necessarily two original data points
2. make the points far apart
3. include units
4. interpret the answer in the Physics context

Intercepts

An intercept is where a graph crosses an axis. In some Physics graphs, an intercept can have meaning. In other cases, it only shows a systematic error or an incomplete range of data.

Example: if a current-voltage graph does not pass near the origin when it should, possible reasons include zero error, connection resistance, a wrong reading, or a non-ohmic component.

Key insight: do not force a graph through the origin unless the experiment and theory justify it.

Graph Interpretation

A good graph interpretation states:

• what increases or decreases
• whether the relationship is straight, curved, direct, or inverse
• what the gradient or intercept means when appropriate
• whether any points look unusual
• what conclusion is supported by the evidence

Comparing Table, Formula, And Graph

A table shows individual values. A formula states a relationship. A graph shows the pattern visually.

For current electricity:

| Representation | Example | What it helps with | |---|---|---| | Table | voltage and current readings | checking individual data | | Formula | $V = IR$ | calculating one quantity from others | | Graph | straight line for $V$ against $I$ | seeing proportionality and finding gradient |

A strong learner can move between all three.

Key Terms

• Axis: a line on a graph used to show a quantity.
• Scale: the numerical spacing on an axis.
• Independent variable: the variable changed or selected.
• Dependent variable: the variable measured in response.
• Best-fit line: a line showing the overall trend of plotted points.
• Gradient: the rate of change shown by a straight-line graph.
• Direct relationship: a relationship where two quantities increase together in a steady ratio.
• Inverse relationship: a relationship where one quantity decreases as another increases.
• Outlier: a plotted point that does not fit the general pattern.
• Intercept: the point where a graph crosses an axis.
• Nonlinear relationship: a relationship shown by a curve rather than a straight line.
• Proportional: changing in a constant ratio.
• Gradient triangle: a right triangle drawn on a straight line to calculate gradient.

Worked Examples

Example 1: Choose Axes

A learner changes voltage and measures current. Which axes should be used?

Voltage is the independent variable, so it goes on the horizontal axis. Current is the dependent variable, so it goes on the vertical axis.

Conclusion: label the axes as Voltage $V$ (V) and Current $I$ (A).

Example 2: Find A Gradient

A straight graph has points $(1.0, 0.20)$ and $(4.0, 0.80)$, where horizontal values are voltage in volts and vertical values are current in amperes.

$$ \text{gradient} = \frac{0.80 - 0.20}{4.0 - 1.0} $$

$$ \text{gradient} = \frac{0.60}{3.0} = 0.20\ \text{A/V} $$

The graph shows that current increases by $0.20\ \text{A}$ for every $1\ \text{V}$ increase.

If the graph were voltage on the vertical axis and current on the horizontal axis, the gradient would represent resistance instead. Always check which quantity is on which axis.

Example 3: Interpret A Relationship

Current readings double when voltage doubles:

| Voltage (V) | Current (A) | |---:|---:| | 2 | 0.4 | | 4 | 0.8 | | 6 | 1.2 |

The ratio $\frac{I}{V}$ is constant:

$$ \frac{0.4}{2} = \frac{0.8}{4} = \frac{1.2}{6} = 0.2 $$

The data suggest a direct relationship between voltage and current.

Example 4: Use Gradient To Find Resistance

A graph of voltage $V$ against current $I$ is a straight line. Two points on the best-fit line are $(0.20\ \text{A}, 1.0\ \text{V})$ and $(0.80\ \text{A}, 4.0\ \text{V})$.

$$ \text{gradient} = \frac{\Delta V}{\Delta I} $$

$$ \text{gradient} = \frac{4.0 - 1.0}{0.80 - 0.20} $$

$$ \text{gradient} = \frac{3.0}{0.60} = 5.0\ \Omega $$

The resistance is $5.0\ \Omega$ because the graph is $V$ against $I$.

Example 5: Spot A Possible Outlier

| Voltage (V) | Current (A) | |---:|---:| | 1.0 | 0.20 | | 2.0 | 0.40 | | 3.0 | 0.90 | | 4.0 | 0.80 |

The current at $3.0\ \text{V}$ does not fit the pattern. If the other points suggest about $0.60\ \text{A}$ at $3.0\ \text{V}$, the learner should check that reading before drawing a conclusion.

Example 6: Interpret A Curved Graph

A graph of shadow size against distance from a light source forms a curve. This does not mean the experiment failed. It means shadow size does not change by the same amount for every equal change in distance.

Conclusion: describe the trend from the graph rather than forcing a direct relationship.

Common Mistakes

• Mistake: forgetting units on axes. Correction: include both quantity and unit.
• Mistake: using the wrong variable on the horizontal axis. Correction: put the independent variable on the horizontal axis.
• Mistake: joining dot to dot when a best-fit line is better. Correction: show the overall trend.
• Mistake: treating one odd point as the main conclusion. Correction: check whether it is an outlier.
• Mistake: calculating gradient from tiny graph squares. Correction: use two well-spaced points on the best-fit line.
• Mistake: forgetting that gradient meaning depends on axis choice. Correction: read the vertical and horizontal quantities before interpreting gradient.
• Mistake: forcing every graph through the origin. Correction: use the data and theory to decide whether the origin is justified.
• Mistake: calling every increasing graph "directly proportional". Correction: direct proportionality needs a constant ratio and usually a straight line through the origin.

Practice Tasks

1. Choose axes for a graph where current is measured as voltage is changed.
2. Explain why a graph must include units.
3. Plot the points $(1,2)$, $(2,4)$, $(3,6)$, and describe the relationship.
4. Calculate the gradient between $(2,5)$ and $(6,13)$.
5. A graph is curved upward. Explain why forcing a straight line may be misleading.
6. A voltage-current graph has voltage on the vertical axis. Explain what the gradient represents.
7. Choose a suitable scale for current values from $0.0\ \text{A}$ to $1.2\ \text{A}$.
8. Identify a possible outlier in this set: $(1,2)$, $(2,4)$, $(3,11)$, $(4,8)$.
9. Explain why a graph may not pass through the origin even when theory suggests it should.
10. Write a conclusion for a graph where voltage and current form a straight line through the origin.

Generated Question Layer

• Axis-choice questions for light and circuit experiments.
• Scale-choice questions using realistic learner tables.
• Gradient calculation questions with units.
• Relationship-identification questions: direct, inverse, nonlinear, or no clear trend.
• Evidence-writing questions that turn graph patterns into conclusions.
• Axis-switch questions where gradient has different meaning depending on graph orientation.
• Outlier-checking questions using realistic practical data.
• Intercept interpretation questions with zero error and systematic error possibilities.
• Formula-table-graph translation questions.

Learner Aid Opportunities

• graph: interactive plotting of voltage-current and light-distance data.
• diagram: labelled graph showing axes, scale, plotted point, best-fit line, and gradient triangle.
• interactive: learner chooses scale and receives feedback.
• LLM tutor: checks graph interpretations for evidence and unit use.

Exam-Derived Signals

• No reviewed Physics exam mappings are attached to this page yet.
• CSEE_FORMATS_2022 may later provide assessment-only signals for graphing and data interpretation.
• The 2023 Physics syllabus remains the curriculum authority.

Source And Review Notes

• Official syllabus status: extracted from the 2023 Physics syllabus.
• Learner expansion status: original unreviewed chapter expansion from the official syllabus topic and existing wiki context.
• External enrichment status: not used.
• Textbook status: not used.
• Review risk: graph conventions and accepted wording should be checked by a Physics teacher before reviewed status.