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IB DP Mathematics: Applications and Interpretation | SL & HL | Topic 2 — Functions
📋 Learning Intentions
- Identify whether a data set is best modelled by a linear, quadratic, or exponential function by examining the structure of the data and the context.
- Write the equation of a linear model in the form \( f(x) = mx + c \), a quadratic model in the form \( f(x) = ax^2 + bx + c \), and an exponential model in the form \( f(x) = ka^x \) or \( f(x) = ke^{rx} \).
- Interpret the parameters \( m \), \( c \), \( a \), \( k \), and \( r \) in the context of a real-world problem.
- Use the GDC to find regression equations and evaluate the fit of a model using the coefficient of determination \( r^2 \).
- Calculate values using a model and comment on the validity of extrapolation beyond the data range.
- Sketch graphs of linear, quadratic, and exponential functions, labelling key features including intercepts, vertex, and asymptotes.
Real-World Introduction
Every time a scientist measures the cooling of a hot drink, an economist tracks the price of a commodity, or an engineer models the path of a projectile, they are choosing a mathematical function to represent a real pattern. The choice of model is not arbitrary — it is driven by the shape of the data and the underlying physical or economic process. Choosing the wrong model leads to predictions that diverge wildly from reality.
A taxi fare that increases at a fixed rate per kilometre is linear. The height of a ball thrown upward follows a quadratic path. The number of bacteria doubling every hour grows exponentially. Each of these patterns has a distinct algebraic structure, and recognising that structure is the first skill of mathematical modelling.
🌍 Why This Matters Exponential models are used by epidemiologists to predict the early spread of infectious diseases — the COVID-19 pandemic saw exponential growth models in daily use by governments worldwide. Linear models underpin economic forecasting. Quadratic models describe the trajectory of every projectile from a basketball to a satellite re-entry vehicle.
Core Concepts & Definitions
Before selecting a model, it is essential to understand what each function type looks like algebraically and graphically. The three models covered here — linear, quadratic, and exponential — each have a characteristic shape, a set of parameters with real-world meaning, and a specific domain of applicability.
1 — Linear Functions: \( f(x) = mx + c \)
A linear function produces a straight-line graph. The rate of change between any two points is constant. If a quantity increases by the same amount for every unit increase in the independent variable, a linear model is appropriate.
📖 Key Definition — Linear Function A linear function has the form \( f(x) = mx + c \), where:
- \( m \) is the gradient (slope) — the constant rate of change of \( f \) with respect to \( x \).
- \( c \) is the \( y \)-intercept — the value of \( f(x) \) when \( x = 0 \).
- The graph is a straight line with no turning points and no asymptotes.