IP AMaths Notes (Upper Sec, Year 3-4): 03) Linear Law

Study guide08 Nov 2025, 00:00 ZUpdated 17 Jul 2026

Transform non-linear relationships into straight lines for graph paper work and regression checks in IP AMaths.

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For Integrated Programme students: Your current school materials, teacher instructions, and assessment scope take precedence because IP topic sequence and depth vary by school. This is an Eclat IP guide, not the O-Level / SEC G3 exam-track guide.

How this chapter applies

Eclat core: transforming power and exponential relationships into straight-line form, reading gradient and intercept, and recovering unknown constants form the main route.
School-sensitive extension: reciprocal, mixed, or log-linear models beyond the named K341 families may be school-sensitive depth.
2027 national comparison: K341 Topic G2.5 explicitly requires transforming relationships including y = ax^n and y = kb^x to linear form to determine unknown constants from a straight-line graph.
Check your school: confirm permitted axis transformations, plotting precision, and whether regression tools are allowed.
Exam-track route: use the separate O-Level and SEC G3 Additional Mathematics notes for K341 topic ownership.

Q: What does IP AMaths Notes (Upper Sec, Year 3-4): 03) Linear Law cover?
A: Transform non-linear relationships into straight lines for graph paper work and regression checks in IP AMaths.

Linear-law questions ask you to recast a model so plotting gives a straight line. Identify the transformation, compute plotting coordinates, and interpret the resulting gradient and intercept.

Keep the full topic roadmap handy via our IP Maths tuition hub so you can jump into related drills, quizzes, or diagnostics as you move through these notes.

New to the Integrated Programme? Start with What is IP? | Browse all free IP notes.

The core idea is simple: Linear Law turns a curved model into a straight-line graph.

Use it as a working check: Decide what to plot on each axis before calculating values. The gradient and intercept must translate back to the original constants.

Then go one layer deeper: Example: for a power model y = A x^n, plot log(y) against log(x); the gradient gives n and the intercept gives log(A).

Choosing the plotting axes

Before making a table of values, decide what quantity should become the straight-line vertical axis and what should become the horizontal axis. This prevents the common mistake of plotting the original $y$

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Model form	Plot vertical axis	Plot horizontal axis	What gradient and intercept mean
$y = Ab^x$	$\log y$	$x$	Gradient is $\log b$ ; intercept is $\log A$ .
$y = Ax^n$	$\log y$	$\log x$	Gradient is $n$
$y = \dfrac{1}{a + bx}$	$\dfrac{1}{y}$
$y = a + b\ln x$	$y$	$\ln x$	Gradient is $b$
$y = A\mathrm{e}^{-kx}$	$\ln y$	$x$	Gradient is $-k$

Original model	Straight-line form	What the graph gives	What to report
$y = Ab^x$	$\log y = x\log b + \log A$	gradient $m = \log b$ , intercept $c = \log A$	$b = 10^m$ , $A = 10^c$ if base-10 logs are used.
$y = Ax^n$	$\log y = n\log x + \log A$
$y = A\mathrm{e}^{-kx}$	$\ln y = -kx + \ln A$

If the graph uses	Intercept $c$ means	Convert back with	Common trap
$\log_{10} y$ against $x$ for $y=Ab^x$	$c=\log_{10} A$	$A=10^c$	Using $\mathrm{e}^c$ because the model looks exponential.
$\ln y$ against $x$ for $y=A\mathrm{e}^{-kx}$	$c=\ln A$
$\log_{10} y$ against $\log_{10} x$ for $y=Ax^n$

IP AMaths Notes (Upper Sec, Year 3-4): 03) Linear Law

How this chapter applies

Choosing the plotting axes

Sources

Gradient-intercept decoding checkpoint

Log-base consistency checkpoint

1 Standard transformations

2 Worked example - Power model

3 Worked example - Reciprocal model

4 Worked example - Log-linear model

5 Practice Quiz

6 Try this

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