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

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.

Standard transformations

• Exponential form \( y = Ab^x \): take logs to obtain \( \log y = \log A + x \log b \).
• Power form \( y = Ax^n \): take logs to produce \( \log y = \log A + n \log x \).
• Reciprocal form \( y = \dfrac{1}{ax + b} \): rearrange to \( \dfrac{1}{y} = ax + b \).
• Mixed models (e.g. \( y = \dfrac{A}{x^n} \)) combine the above manipulations.

Worked example 1 — Power model

Given \( y = kx^{1.5} \), rewrite in linear-law form and estimate \( k \) from the data \( (x, y) = (4, 32) \), \( (25, 395) \).

1. Take base-10 logs: \( \log_{10} y = \log_{10} k + 1.5 \log_{10} x \).
2. Let \( Y = \log_{10} y \), \( X = \log_{10} x \), \( C = \log_{10} k \).
3. Compute: \( X_1 = \log_{10} 4 = 0.6021 \), \( Y_1 = \log_{10} 32 = 1.5051 \).
4. For the second point: \( X_2 = \log_{10} 25 = 1.3979 \), \( Y_2 = \log_{10} 395 = 2.5960 \).
5. Gradient of straight line: \( m = \dfrac{2.5960 - 1.5051}{1.3979 - 0.6021} = 1.373 \), close to the theoretical \( 1.5 \) (allowing rounding errors).
6. Intercept: \( C = Y_1 - m X_1 = 1.5051 - 1.373 \times 0.6021 = 0.678 \).
7. Hence \( k = 10^{0.678} = 4.77 \) (3 s.f.).

Worked example 2 — Reciprocal model

A dataset follows \( y = \dfrac{1}{a + bx} \). Show how to plot a straight line and determine \( a \), \( b \) using the points \( (1, 0.42) \), \( (3, 0.21) \).

1. Rearrange: \( \dfrac{1}{y} = a + bx \).
2. Define \( Y = \dfrac{1}{y} \); plotting \( Y \) against \( x \) gives intercept \( a \) and gradient \( b \).
3. Compute values: for \( x = 1 \), \( Y_1 = \dfrac{1}{0.42} = 2.381 \); for \( x = 3 \), \( Y_2 = \dfrac{1}{0.21} = 4.762 \).
4. Gradient: \( b = \dfrac{4.762 - 2.381}{3 - 1} = 1.1905 \).
5. Intercept: \( a = Y_1 - b = 2.381 - 1.1905 = 1.1905 \).

Try this

Data believed to satisfy \( y = A \mathrm{e}^{-kx} \). Outline how you would estimate \( A \) and \( k \) from a semi-log plot, then test with two sample points of your choice.