Modelling with Functions

Introduction

In mathematics, a function is a rule that connects one quantity to another. When we use functions to describe real situations, we are modelling. students, this means turning a situation like population growth, cost, temperature, or sound into a mathematical form that can be studied, predicted, and compared 📈.

By the end of this lesson, you should be able to:

explain what it means to model with functions,
choose suitable function types for different situations,
interpret parameters in context,
use transformations, inverses, and composites in modelling,
solve equations and inequalities involving models.

A strong model does not have to be perfect. It should fit the situation well enough to make useful predictions. In IB Mathematics: Analysis and Approaches HL, modelling is important because it links function language, algebra, graphs, and real-world reasoning into one skill set.

What it means to model with functions

A model is a simplified description of reality. In function modelling, we assume that one variable depends on another. For example, the amount of money in a savings account may depend on time. The height of a ball may depend on the time since it was thrown. The cost of a taxi ride may depend on distance travelled.

The key idea is that the function is not just a formula. It is a representation of a relationship. A good model should:

match the trend in the data,
use variables clearly,
have realistic domain and range,
allow interpretation of parameters.

Suppose a bacteria population grows over time. If the population increases by about the same percentage every hour, an exponential model may fit:

$$P(t)=P_0e^{kt}$$

Here, $P(t)$ is the population after time $t$, $P_0$ is the initial population, and $k$ is the growth constant. If $k>0$, the population grows; if $k<0$, it decays.

If a taxi charges a fixed starting fee plus a cost per kilometre, a linear model may fit:

$$C(d)=ad+b$$

Here, $d$ is the distance, $a$ is the price per kilometre, and $b$ is the starting fee. This is useful because the meaning of each parameter is clear in context 🚕.

Choosing a function type

A major part of modelling is deciding which family of functions is appropriate. Different patterns in real life often match different function shapes.

Linear models

Linear functions are useful when the rate of change is constant. If a quantity increases by the same amount each time, a linear model is often suitable.

Example: A gym charges $20$ to join and $8$ per visit. If $v$ is the number of visits, the cost is

$$C(v)=8v+20$$

This model is easy to interpret. The gradient $8$ means each visit adds $8$ dollars, and the $y$-intercept $20$ means the fixed joining fee.

Polynomial models

Polynomials are useful when a relationship changes direction or when smooth curved behaviour is needed. A quadratic model is especially common for motion under constant acceleration.

For example, the height of a ball thrown upwards can be modelled by

$$h(t)=-4.9t^2+ut+h_0$$

where $u$ is the initial velocity and $h_0$ is the initial height. The negative coefficient on $t^2$ means the graph opens downward, so the ball eventually falls back down.

Higher-degree polynomials can fit more complex data, but students should be careful: a polynomial may fit a set of points well while still making unrealistic predictions far outside the data range.

Exponential models

Exponential models are ideal when the rate of change is proportional to the current amount. This happens in compound interest, population growth, radioactive decay, and cooling processes in simplified settings.

Example: If a phone battery loses $15\%$ per hour, a decay model may be

$$B(t)=100(0.85)^t$$

where $B(t)$ is the percentage battery remaining after $t$ hours.

A key feature is that exponential graphs have constant multipliers over equal time intervals, not constant differences.

Logarithmic models

Logarithmic models are often used when a quantity increases quickly at first and then slows down. This can describe sound intensity scales, perceived brightness, and some scientific measurements.

A logarithmic function may look like

$$y=a\ln(x)+b$$

Since the logarithm is only defined for positive inputs, the domain matters. In modelling, this means students must check that the input values make sense in context.

Rational models

Rational functions are helpful when one quantity is inversely related to another or when there are asymptotes.

Example: The time $t$ needed to complete a job may depend on the number of workers $n$ by

$$t(n)=\frac{k}{n}$$

If more workers are added, the time decreases. This is a simplified model, but it is useful for reasoning about inverse variation.

Transformations, inverses, and composites in modelling

Many models are based on a basic function that is transformed. This helps students understand how the graph changes without starting from scratch.

A transformation may involve:

vertical shifts, like $f(x)+c$,
horizontal shifts, like $f(x-c)$,
stretches, like $af(x)$ or $f(bx)$,
reflections, like $-f(x)$ or $f(-x)$.

For example, if $f(x)=e^x$, then

$$g(x)=3e^{x-2}+5$$

is a transformed exponential model. The graph is stretched vertically by a factor of $3$, shifted right by $2$, and shifted up by $5$.

Inverse functions are useful when you want to reverse a process. If a function takes an input and gives an output, its inverse finds the original input.

Example: If a model gives temperature in degrees Celsius from Fahrenheit,

$$C(F)=\frac{5}{9}(F-32)$$

then the inverse function converts Celsius back to Fahrenheit.

Composite functions model processes in stages. If one function describes one step and another describes a second step, the composite combines them.

For instance, if $p(x)$ converts time into production output and $c(x)$ converts output into cost, then

$$c(p(t))$$

gives the cost after time $t$. This is useful whenever one real-world quantity depends on another through more than one rule.

Equations and inequalities involving models

Modelling is not only about building functions. It is also about using them to answer questions.

If two models are equal, that may represent a point where two situations match. For example, if income is

$$I(x)=15x+200$$

and expenses are

$$E(x)=8x+500$$

then the break-even point is found by solving

$$15x+200=8x+500$$

This gives

$$7x=300$$

$$x=\frac{300}{7}$$

This means the business breaks even after about $42.9$ units, which may then be interpreted in context.

Inequalities are used when a model must meet a condition. If a company wants profit greater than zero, then

$$I(x)-E(x)>0$$

must be solved. If the model gives a height, and a safety rule requires the height to be at least $2$ metres, then solve

$$h(t)\ge 2$$

The graph often helps: the solution is the time interval where the graph lies above the line $y=2$.

When solving model-based equations or inequalities, always check whether your answers make sense in the domain. A negative time, for example, may not be meaningful. Likewise, if the model is built from a square root, the expression under the root must satisfy the required conditions.

How to judge whether a model is good

A model is judged by how well it fits the context and data. students should ask:

Does the graph follow the observed trend?
Are the parameters meaningful?
Is the domain realistic?
Does the model make sense outside the data range?
Is the model simple enough to use?

Sometimes different models can fit the same data reasonably well. Then the choice depends on the context. For example, short-term growth may appear linear, but longer-term growth may be better described by an exponential model. Data from a graphing calculator or technology can help compare models, but the final choice should still be justified mathematically and in context.

In IB Mathematics: Analysis and Approaches HL, it is important to connect algebraic manipulation with interpretation. If you solve for a parameter, you should explain what it means. If you find an intersection point, you should interpret it in words. If you use a graph, you should explain what the axes represent.

Conclusion

Modelling with functions is about turning real-life situations into mathematical relationships that can be studied and used. The main function families—linear, polynomial, exponential, logarithmic, and rational—each describe different patterns. Transformations help adapt standard functions to specific contexts, inverses reverse processes, and composites combine stages of a model. Equations and inequalities then allow students to answer practical questions from the model.

This topic connects directly to the rest of Functions because it uses function notation, graphs, transformations, inverse functions, and solving equations in a meaningful way. In IB Mathematics: Analysis and Approaches HL, modelling is not only about finding a formula. It is about building a mathematically sound representation of a real situation and using it to reason clearly and accurately.

Study Notes

A function model describes how one quantity depends on another.
Good models should fit the data, have a realistic domain, and make sense in context.
Linear models are useful when the rate of change is constant.
Polynomial models are useful for curved behaviour and motion problems.
Exponential models describe growth or decay by constant multiplication.
Logarithmic models are useful when growth is fast at first and then slows.
Rational models often describe inverse relationships or asymptotes.
Transformations change a basic graph by shifting, stretching, or reflecting it.
Inverse functions reverse a process.
Composite functions model multi-step situations.
Equations help find where two models are equal, such as break-even points.
Inequalities help find when a condition is satisfied, such as when a quantity is above a threshold.
Always check whether an answer is meaningful in the real-world domain.
In IB Mathematics: Analysis and Approaches HL, clear interpretation is as important as calculation.