Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Unit 2: Functions
📋 Learning Intentions
- Identify the key parameters of sinusoidal functions \( f(x) = a\sin(b(x-c)) + d \) and interpret each in a real-world context.
- Construct a sinusoidal model from given data by determining amplitude, period, principal axis, and phase shift.
- Fit a cubic regression model to data using the GDC and interpret the coefficients in context.
- Distinguish between direct variation \( y = kx^n \) and inverse variation \( y = \frac{k}{x^n} \) and determine the constant of variation \( k \).
- Apply each function type to solve real-world modelling problems, selecting the most appropriate model for a given data set.
- Evaluate the suitability of a model by interpreting parameters and commenting on the domain of validity.
Real-World Introduction
The height of a tide at a harbour, the temperature variation across a day, and the number of daylight hours across a year all share a common mathematical structure — they rise and fall in a repeating, wave-like pattern. No linear or quadratic model can capture this behaviour. A sinusoidal function, however, fits these patterns precisely and allows engineers, meteorologists, and navigators to make accurate predictions.
Other phenomena require different function families. The stopping distance of a car increases with the cube of speed under certain conditions, suggesting a cubic model. The intensity of light from a source decreases with the square of the distance — an inverse variation. Recognising which function family fits a given context is a core modelling skill in IB Math AI.
🌊 Why This Matters Coastal engineers use sinusoidal models to predict safe docking windows. Physicists use inverse square variation to model gravitational and electromagnetic fields. Cubic models appear in fluid dynamics and economics. Each function family is a tool — and choosing the right tool is what mathematical modelling is about.
Core Concepts & Definitions
When a quantity oscillates — rising and falling repeatedly between a maximum and minimum value — a sinusoidal function provides the natural mathematical description. The key insight is that such functions are completely determined by four measurable features of the oscillation.
📐 Key Definition — Sinusoidal Function A sinusoidal function has the form: \[ f(x) = a\sin(b(x - c)) + d \] where:
- \( a \) = amplitude — half the vertical distance between the maximum and minimum values; \( |a| = \frac{\text{max} - \text{min}}{2} \)
- \( b \) = angular frequency — determines the period; \( \text{period} = \frac{2\pi}{|b|} \)
- \( c \) = phase shift (horizontal translation) — shifts the graph left or right
- \( d \) = vertical translation (principal axis) — the midline of the oscillation; \( d = \frac{\text{max} + \text{min}}{2} \)
The cosine form \( f(x) = a\cos(b(x-c)) + d \) is equally valid and may be more convenient when the data begins at a maximum.
The amplitude \( a \) is always taken as a