Modelling with Functions

In this lesson, students, you will learn how functions are used to describe real situations and make predictions 📈. A function model links inputs to outputs in a clear rule, so it can represent things like population growth, taxi fares, cooling coffee, or the spread of a virus. The key idea is that the shape and formula of a function should match the pattern in the data or situation. By the end of this lesson, you should be able to explain the main language of modelling, choose suitable function types, and interpret what a model tells you in context.

What does it mean to model with a function?

A mathematical model is a simplified representation of a real situation. When we model with functions, we use a formula, table, graph, or verbal description to connect one quantity to another. In IB Mathematics Analysis and Approaches SL, this means recognizing which function family fits a pattern and using that function to describe the relationship.

For example, if a phone plan charges a fixed fee plus a cost per minute, the total cost can be written as $C(m)=a+bm$, where $a$ is the fixed fee and $b$ is the cost per minute. Here, $m$ is the input, $C(m)$ is the output, and the model is linear because the rate of change is constant.

A good model should be:

sensible in context,
based on patterns in data or information,
able to make reasonable predictions,
and interpreted with awareness of its limits.

This last point is important, students. A model is not reality itself. It is a useful approximation.

Function language and key terminology

To model well, you need precise function language. The input is the independent variable, and the output is the dependent variable. A function can be written as $y=f(x)$, where $x$ is the input and $f(x)$ is the output. The domain is the set of allowed inputs, and the range is the set of possible outputs.

In modelling, the domain often comes from the real situation. For instance, if $t$ represents time after the start of an experiment, then $t\ge 0$. If a quantity cannot be negative, that also restricts the domain or range.

Important terms include:

variable: a quantity that can change,
parameter: a constant that shapes the model,
rate of change: how fast the output changes with the input,
intercept: a value where the graph crosses an axis,
asymptote: a line the graph approaches but may never reach.

Suppose a population is modelled by $P(t)=1200(1.03)^t$. The initial value is $1200$, and the growth factor is $1.03$. Since the factor is greater than $1$, this is exponential growth.

When interpreting a model, ask:

What does each number mean?
What are the units?
Is the result realistic?

For example, if $h(t)$ gives height in metres, then any negative value may be impossible in context.

Choosing the right function type

A major skill in modelling is deciding whether the relationship is linear, quadratic, rational, exponential, or logarithmic.

Linear models

A linear model has the form $f(x)=mx+c$. It is used when the quantity changes by a constant amount. For example, if a water tank loses $5$ litres per hour, the amount remaining might be $V(t)=200-5t$. The slope $-5$ means the volume decreases steadily.

Linear models are useful for simple, short-term situations, but many real patterns are not perfectly linear forever.

Quadratic models

A quadratic model has the form $f(x)=ax^2+bx+c$. It is useful when there is a turning point, such as the path of a thrown ball or the profit from a business with increasing then decreasing returns.

For example, the height of a ball might be modelled by $h(t)=-4.9t^2+12t+1.5$. The negative coefficient of $t^2$ means the graph opens downward, so the ball rises first and then falls.

The vertex is important because it gives a maximum or minimum value. In real life, that may represent the highest point of a ball or the best profit.

Rational models

A rational model is a ratio of polynomials, such as $f(x)=\frac{1}{x-2}$. These models are often used when a quantity becomes very large near a restricted value or when two variables are inversely related.

For example, if $T(n)=\frac{100}{n}$ represents time per person for $n$ workers sharing a job, then increasing the number of workers reduces the time per person. Rational functions often have asymptotes, which help describe limits in the real situation.

Exponential models

An exponential model has the form $f(x)=ab^x$, where $b>0$ and $b\ne 1$. It is used when the change is proportional to the current amount. This is common in population growth, compound interest, and radioactive decay.

For compound interest, the amount may be written as $A(t)=P\left(1+\frac{r}{n}\right)^{nt}$, where $P$ is the principal, $r$ is the annual interest rate, $n$ is the number of compounding periods per year, and $t$ is time in years.

If a quantity decays, the base satisfies $0<b<1$. For example, $N(t)=500(0.92)^t$ means the amount decreases by $8\%$ each time period.

Logarithmic models

A logarithmic model is the inverse of an exponential model. It has the form $f(x)=a\log_b(x)+c$ or $f(x)=a\ln(x)+c$. These are used when growth is very rapid at first and then slows down, such as sound intensity or some learning curves.

Because logarithms are only defined for positive inputs, the domain restriction matters. For example, $y=\ln(x-3)$ requires $x>3$.

Transformations in modelling

Real data rarely matches a parent function exactly, so transformations are often used. A transformed function might look like $y=a f(b(x-h))+k$.

Each parameter has a meaning:

$a$ gives vertical stretch or compression and possible reflection in the $x$-axis,
$b$ gives horizontal stretch or compression and possible reflection in the $y$-axis,
$h$ shifts the graph horizontally,
$k$ shifts the graph vertically.

These changes help fit a model to data. For example, if the basic exponential $f(x)=2^x$ starts too low, then $g(x)=3\cdot 2^{x-1}+4$ moves the graph right, stretches it vertically, and shifts it up.

In context, transformations can represent:

delayed start times,
different starting values,
steeper or gentler growth,
or changes in scale.

When fitting a model, students, always check whether the transformed graph makes sense in the situation. A model can look mathematically correct but still be unrealistic if the context is ignored.

Composite and inverse functions in modelling

Composite functions combine two processes. If $f$ and $g$ are functions, then $(f\circ g)(x)=f(g(x))$.

This is useful when a real situation happens in stages. For example, suppose $g(x)$ converts Celsius temperature to Fahrenheit, and $f(x)$ estimates a material property based on Fahrenheit. Then $f(g(x))$ gives the property directly from Celsius.

Inverse functions reverse a process. If $f(x)$ turns an input into an output, then $f^{-1}(x)$ returns the original input when possible. This is especially helpful when solving a model for the input variable.

For example, if $A(t)=200(1.05)^t$ models savings, then to find the time needed to reach a certain amount, you may solve using logarithms or an inverse relationship. Rewriting gives:

$$t=\frac{\log\left(\frac{A}{200}\right)}{\log(1.05)}$$

This shows how inverse reasoning helps in modelling: you are not only predicting outputs, but also finding the input needed to achieve a target.

Making and checking a model with data

A common IB task is to compare data to a suitable function and decide which model is best. The process usually involves:

identifying the variables,
plotting or reading the pattern,
choosing a likely function family,
estimating parameters,
checking the fit,
and interpreting the model in context.

Suppose a company records sales over time. If the sales increase by about the same amount each month, a linear model may fit. If sales grow by a percentage, an exponential model may be better. If sales rise quickly and then level off, a logarithmic model may be more suitable.

You should also evaluate the model using residuals or error. A model that follows the general trend but misses some points may still be acceptable if it is simple and useful.

For example, if data for a bacterial culture is close to $B(t)=150(1.12)^t$, the base $1.12$ suggests $12\%$ growth per time unit. If the real values begin to level off later, the model may stop being accurate because resources become limited.

Conclusion

Modelling with functions connects algebra, graphs, and real situations. It is not just about writing equations; it is about choosing a function that fits evidence and interpreting what the formula means. In IB Mathematics Analysis and Approaches SL, students, you should be able to identify the right function family, use transformations, combine functions, apply inverses, and explain the meaning of your model clearly. The most important habit is to check whether the model matches the context and whether its predictions are reasonable. When you do this well, functions become a powerful tool for understanding the world 🌍.

Study Notes

A function model links an input to an output using a rule such as $y=f(x)$.
The domain and range must make sense in context.
Linear models have a constant rate of change and the form $f(x)=mx+c$.
Quadratic models often describe motion or optimization and have the form $f(x)=ax^2+bx+c$.
Rational models use ratios of polynomials and often include asymptotes.
Exponential models have the form $f(x)=ab^x$ and describe repeated percentage change.
Logarithmic models are inverses of exponential functions and require positive inputs.
Transformations can shift, stretch, compress, and reflect graphs using $y=a f(b(x-h))+k$.
Composite functions are written as $(f\circ g)(x)=f(g(x))$ and model processes in stages.
Inverse functions reverse a process and are useful for solving for the input.
A good model is mathematically correct, contextually sensible, and supported by data.