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Further Function Models
IB DP Mathematics: Applications and Interpretation | SL & HL | Topic 2.9 — Functions
📋 Learning Intentions
- Identify the key features of logistic, natural logarithmic, sinusoidal, and piecewise function models from equations and graphs.
- Use the GDC to fit logistic, logarithmic, and sinusoidal regression models to real-world data sets.
- Interpret the parameters of each model in context — including carrying capacity, period, amplitude, and phase shift.
- Construct piecewise functions from contextual descriptions and evaluate them for given domain values.
- Determine which function model best fits a given data set, justifying the choice using \( r^2 \) values and contextual reasoning.
- Solve equations involving these models using the GDC intersection or zero-finding tools.
Real-World Introduction
Not every real-world relationship is linear or quadratic. The spread of a virus through a population, the loudness of sound measured in decibels, the height of a tide over 24 hours, and the pricing structure of a mobile phone plan all follow fundamentally different mathematical shapes. Choosing the right model is one of the most important skills in applied mathematics.
Each of the four models in this topic captures a distinct pattern of change. The logistic model describes growth that accelerates then slows as it approaches a limit. The natural logarithm models rapid initial growth that gradually levels off. The sinusoidal model captures perfect cyclical repetition. The piecewise model handles situations where the rule itself changes at certain thresholds.
🌍 Why This Matters Epidemiologists used logistic models to forecast COVID-19 case peaks. Acoustical engineers use logarithmic functions to design concert halls. Oceanographers use sinusoidal models to predict tidal heights for safe shipping. Electricity providers use piecewise tariff functions to calculate bills. All four models appear directly in IB Math AI examinations.
Core Concepts & Definitions
When a population grows in an environment with limited resources, growth is initially exponential but eventually slows as the population approaches the maximum the environment can support. This S-shaped pattern cannot be captured by a simple exponential — it requires the logistic model.
📖 Key Definition — Logistic Function The logistic function has the form: \[ f(t) = \frac{L}{1 + Ce^{-kt}} \] where:
- \( L \) is the carrying capacity — the upper horizontal asymptote (maximum sustainable value)
- \( C \) is a constant that determines the initial value: \( f(0) = \frac{L}{1+C} \)
- \( k \) is the growth rate constant (\( k > 0 \) for growth)
- \( t \) is the independent variable (often time)
The logistic function has two horizontal asymptotes: \( f(t) \to 0 \) as \( t \to -\infty \), and \( f(t) \to L \) as \( t \to +\infty \). The graph is S-shaped (sigmoid). The point of inflection — where growth is fastest — occurs at \( f = \frac{L}{2} \), exactly halfway to the carrying capacity.
Logistic Model — Key Features
\[ \begin{aligned} &\text{Upper asymptote: } y = L \\ &\text{Lower asymptote: } y = 0 \\ &\text{Initial value: } f(0) = \frac{L}{1 + C} \\ &\text{Point of inflection: } f = \frac{L}{2} \end{aligned} \]
🖩 GDC