Cambridge O Level Additional Mathematics · Syllabus 4037 · Straight-Line Graphs
Constructed Variables
What is Constructed Variables?
Constructed variables are the new quantities X and Y formed from the original x and y so that a non-linear relationship plots as a straight line. They need not involve logarithms: a relationship of the form y squared equals A x cubed plus B becomes a straight line when y squared is plotted against x cubed, with gradient A and intercept B. Reading such a graph backwards means substituting the constructed quantities into the straight-line equation Y equals m X plus c and then replacing X and Y by what they stand for. The horizontal and vertical assignments must be identified explicitly and cannot be interchanged, because plotting the pair the other way round gives a line whose gradient is the reciprocal and whose intercept is different.
This definition is part of the Straight-Line Graphs chapter in Cambridge O Level Additional Mathematics.
Common mistakes with Constructed Variables
- 18. Trying to linearise a sum with logarithms The error Given \(y^{2}=Ax^{3}+B\), taking logarithms to get \(2\ln y=\ln A+3\ln x\). Why it fails That step silently discards \(B\) and treats the model as \(y^{2}=Ax^{3}\). There is no law for \(\ln(P+Q)\), so the logarithm of the right-hand side cannot be split at all. Logarithms linearise products and powers, never sums. Fix Look at the model first. If the unknowns already sit outside as a coefficient and a constant, plot the constructed variables directly — here \(y^{2}\) against \(x^{3}\) — and leave logarithms out of it.