Learning outcomes
- Distinguish linearity from direct proportionality.
- Determine and interpret an intercept.
- Use y = mx + c in experimental analysis.
- Choose transformed variables to obtain a straight line.
- Use graph evidence to test a prediction.
11.1 Linear relationship
A straight-line graph shows a linear relationship described by y = mx + c. The gradient m gives the rate of change of y with x, while c is the y-intercept when x = 0.
A non-zero intercept may represent a real offset, an unaccounted quantity or a systematic error. Interpretation depends on the experiment.
11.2 Direct proportionality
For y to be directly proportional to x, the graph of y against x must be a straight line through the origin within experimental accuracy. Constant gradient alone is not sufficient if the intercept is significant.
Use the phrase ‘within experimental accuracy’ because scatter and finite resolution may prevent the line from passing through the exact mathematical origin.
11.3 Intercepts
Extend the best-fit line graphically to the axis when necessary, but avoid long extrapolation. Read the intercept with units consistent with the axis.
A positive extension intercept in a spring graph may indicate that the reference length was measured incorrectly. A temperature intercept in a cooling model may have a physical interpretation supplied by the question.
11.4 Transforming variables
Some relationships are curved in their original variables but linear after transformation. If T is proportional to the square root of L, plotting T² against L should produce a straight line through the origin.
Calculate transformed values carefully and include the correct unit, such as s². A straightened graph can make comparison, gradient and proportionality judgments easier.
11.5 Evidence-based wording
Write conclusions that match the graph: ‘R increases linearly with L’ for a straight line with non-zero intercept; ‘R is directly proportional to L’ only when the origin condition is met.
If scatter is large, qualify the conclusion. The trend may support a prediction but not prove an exact law.
Worked examples
Interpreting c
A graph of measured length against true length has gradient near 1 but y-intercept +0.4 cm. This suggests a constant positive zero offset.
Linearising a pendulum relation
If T² plotted against L is straight through the origin, the evidence supports T² ∝ L and therefore T ∝ √L.
Practical focus
Investigation or training activity
Take three equations from the physics syllabus and identify a graph whose gradient or intercept would reveal a physical quantity. State axes and gradient units.
Examination guidance
- Do not call every straight line direct proportion.
- Check the origin or intercept.
- Give intercept units.
- Label transformed axes correctly.
- Use cautious evidence language when scatter is substantial.
Check your understanding
- What does c represent in y = mx + c?
- What graph condition indicates direct proportionality?
- Why plot T² against L for a pendulum?
- Can a straight line with non-zero intercept show direct proportion?
Answers
- The y-intercept.
- A straight line through the origin within experimental accuracy.
- To test whether T² is proportional to L using a straight line.
- No.
