Modelling Using Linear and Non-linear Functions
Core concepts
You must recognise, fit and interpret linear, quadratic, exponential and logarithmic models. For linear: gradient (m) and intercept (c) in (y=mx+c). For quadratic: vertex form (y=a(x-h)^2+k), axis of symmetry, and roots. For exponential: (y=A,b^x) where (b>1) grows and (0<b<1) decays; doubling time and half-life. For logarithmic: (y=a\ln x + b) to linearise exponential data. Know residuals, coefficient of determination (R^2), and how to interpolate vs extrapolate.
Traps to watch
- Mixing up growth vs decay bases - a base of 1.05 means 5% growth per period, not 5% of the total.
- Extrapolating beyond the data range - models fitted to a small window often fail outside it.
- Confusing (R^2) with causation - high (R^2) means good fit, not cause.
- Log scale misreads - on a log-linear plot, a straight line indicates exponential growth.
- Forgetting domain restrictions - e.g. length cannot be negative.
- Rounding the model coefficients too early - keep full precision until the final answer.
Worked rule of thumb
To fit an exponential to data ((x,y)), take logs: if (y=A,b^x), then (\ln y=\ln A + x\ln b), a straight line with gradient (\ln b). Example: a population grows from 500 to 800 in 6 years. Then (b=(800/500)^{1/6}=1.6^{1/6}\approx1.081), meaning roughly 8.1% growth per year. Predict year 10: (500(1.081)^{10}\approx1095). Always state units and check plausibility.