Functions as Models
Functions as Models asks you to choose a family of functions that fits a real situation, estimate its parameters, and interpret them with units - linear, exponential, logarithmic, logistic, sinusoidal, or power. Many items are set in context: population, medication levels, temperature, revenue, or radioactive decay.
Core rules
- Linear: constant rate of change, y = mx + b, where m is the rate per unit and b the starting value.
- Exponential: constant percent change, y = a · b^t, with b = 1 + r for growth and b = 1 - r for decay; a is the initial amount.
- Doubling and half-life: y = a · 2^(t/d) or y = a · (1/2)^(t/h). Continuous compounding is A = P e^(rt); convert bases using b = e^k with k = ln b.
- Logarithmic models invert exponential growth and compress wide ranges; logistic models approach a carrying capacity L in y = L/(1 + a e^(-kt)).
- Sinusoidal: y = A sin(B(t - C)) + D has amplitude |A|, period 2π/B, midline D, and phase shift C.
- Interpretation counts: state each parameter with its unit and note the domain over which the model is reasonable.
Worked example
A culture of 4,000 bacteria doubles every 6 hours. Build a model and find when the population reaches 20,000. The doubling time goes in the denominator of the exponent, so P(t) = 4000 · 2^(t/6) with t in hours. Setting P = 20,000 gives 2^(t/6) = 5, so t/6 = log base 2 of 5 = ln 5 / ln 2, which is about 2.322. Therefore t is about 13.9 hours. The exponent divides by 6 because doubling happens once per 6 hours, not 6 times per hour.
Traps that cost marks
- Writing 2^(6t) instead of 2^(t/6) when a doubling time is given.
- Confusing a nominal annual rate with continuous compounding - e^r and 1 + r are not the same effective rate.
- Forcing a linear fit onto data that grow multiplicatively.
- Confusing the amplitude of a sinusoidal model with its midline or maximum value.
- Extrapolating far beyond the data, for instance pushing a logistic curve past its carrying capacity.