AHL 2.9 — Modelling with Advanced Functions

📘 1. Exponential Models and Half-Life

  • General form: f(x) = a·bx where a is the initial value and b > 0 is the base (growth if b > 1, decay if 0 < b < 1).
  • Half-life concept: The time it takes for a quantity to reduce to half its original amount.
  • Formula: N(t) = N0·(½)t / h where h is half-life.

Example: A 100 g radioactive sample decays with a half-life of 5 hours.

After 15 hours: N = 100(½)15 / 5 = 100(½)3 = 12.5 g

The exponential model accurately describes exponential decay in physics, chemistry, and finance (e.g., depreciation).

Applications: Radioactive decay, bacterial growth, inflation, cooling of objects, and charging/discharging of capacitors.

📗 2. Natural Logarithmic Models

  • General form: f(x) = a + b·ln(x)
  • Meaning: Models processes that increase rapidly and then level off, or where growth slows as x increases.
  • Properties:
    • Domain: x > 0
    • Range: all real numbers
    • Vertical asymptote at x = 0

Example: Sound intensity in decibels (dB) can be expressed as I = 10·log10(P / P0), which is logarithmic in nature.

At x = 1 → f(1) = a (since ln(1) = 0).

🌍 Real-World Connections

  • Exponential models: Half-life, depreciation, and radioactive decay.
  • Logarithmic models: Richter scale, pH, or sound intensity (decibel scale).
  • Sinusoidal models: Tides, seasonal changes, alternating current circuits.
  • Logistic models: Population growth with limited resources.
  • Piecewise models: Income tax brackets, postage rates, and cell phone billing.

📐 IA Ideas

  • Investigate how logistic models predict disease spread or urban growth.
  • Use sinusoidal models to analyze daylight hours over a year for a city.
  • Model depreciation or population change using exponential and logarithmic functions.
  • Explore a piecewise cost model in an everyday scenario (e.g., cab fare structure)

📙 3. Sinusoidal Models

  • General form: f(x) = a·sin[b(x − c)] + d
  • Parameters:
    • a: amplitude (vertical stretch, height of wave)
    • b: affects the period, which is 2π / b (in radians)
    • c: horizontal shift, called the phase shift
    • d: vertical shift (moves graph up/down)

Example: f(x) = 3·sin(2x − π/2) + 1

  • Amplitude = 3
  • Period = 2π / 2 = π
  • Phase shift = π/4 to the right
  • Vertical shift = +1

Interpretation: This could represent temperature variation with time, where amplitude is the seasonal temperature range, period is one year, and vertical shift represents the average temperature.

📕 4. Logistic Models

  • General form: f(x) = L / (1 + Ce−kx), with L, C, k > 0
  • Meaning: Used when growth begins exponentially but slows as it nears a maximum limit (carrying capacity L).
  • Key features:
    • Horizontal asymptote: y = L (maximum population or capacity)
    • Inflection point: where growth rate is maximum

Example: A bacterial population grows according to f(t) = 100 / (1 + 9e−0.5t).

  • At t = 0 → f(0) = 100 / (1 + 9) = 10
  • As t → ∞ → f(t) → 100 (the limiting population)

❤️ CAS Ideas

  • Create visual posters showing how different models (exponential, logistic, sinusoidal) describe natural or social processes.
  • Conduct an awareness project explaining exponential spread in public health (e.g., virus transmission).

📝 Paper 1 & 2 Exam Tips

  • Always state domain and range for all models.
  • Clearly identify asymptotes and key features (period, amplitude, half-life, carrying capacity).
  • Label graphs with parameter changes (a, b, c, d, k, L, etc.).
  • Show working when fitting or verifying model equations with data.

📒 5. Piecewise Models

  • Definition: A function defined by different expressions over separate intervals of x.
  • Purpose: Used when a situation behaves differently in different conditions.

Example:

f(x) = {
  1 + x ,   0 ≤ x < 2
  a·x² + x ,   x ≥ 2
}

To make f(x) continuous at x = 2:

1 + 2 = a(2)² + 2 → 3 = 4a + 2 → a = ¼

🌐 EE Focus

  • Study how logistic models describe population equilibrium and sustainability.
  • Examine the role of logarithmic models in measuring inequality (Gini index, Lorenz curve).
  • Explore transformations and continuity in piecewise-defined models.

🔍 TOK Discussion

Mathematics models reality, but no model perfectly represents it. To what extent can a mathematical function be said to “describe” a phenomenon, rather than merely approximate it? Is one type of model more useful than another depending on the area of knowledge?

🧠 Examiner Tip

Examiners look for reasoning, not memorization. Always interpret the context: What does amplitude, half-life, or carrying capacity represent in the problem? Models without contextual meaning score fewer marks in reasoning criteria.