Modelling with Functions

students, have you ever used your phone to predict how long a battery will last, or checked a graph to see how a population changes over time? 📱📈 That is the heart of modelling with functions: using a function to describe real-life data and make predictions. In IB Mathematics: Applications and Interpretation SL, this topic connects algebra, graphs, technology, and real-world interpretation. By the end of this lesson, you should be able to explain what a model is, choose a suitable function, interpret key features of a graph, and judge whether a model is reasonable.

Learning objectives:

Explain the main ideas and terminology behind modelling with functions.
Apply IB Mathematics: Applications and Interpretation SL reasoning to build and interpret models.
Connect modelling with functions to graphs, transformations, and regression.
Summarize how modelling with functions fits into the wider topic of functions.
Use examples to support understanding of models in context.

What it means to model with a function

A function is a rule that links each input to exactly one output. In modelling, the inputs and outputs represent real quantities, such as time, distance, temperature, height, cost, or population. For example, if $t$ represents time in hours and $B(t)$ represents battery percentage, then $B(t)$ is a function that describes how the battery changes over time.

A model is a simplified mathematical description of a real situation. It is not meant to be perfect. Instead, it should be useful, accurate enough, and based on the context. This is an important IB idea: a model must fit the purpose. A model that is slightly imperfect may still be very useful if it predicts well within a realistic range.

When making a model, you should think about:

the variables and their units,
the shape of the relationship,
the domain and range,
whether the model is linear, quadratic, exponential, or another type,
whether the result makes sense in context.

For example, the number of bacteria in a culture often grows quickly at first. That kind of pattern may be modelled with an exponential function such as $N(t)=N_0e^{kt}$, where $N_0$ is the initial amount and $k$ is the growth rate. If $k>0$, the function increases; if $k<0$, it decreases. In a real lab, growth cannot continue forever, so the model may only be valid for a limited time. 🌱

Choosing a suitable function type

Different situations often suggest different kinds of functions. A big part of modelling is recognizing the pattern in the data.

A linear model has the form $y=mx+b$. It works well when the rate of change is constant. For example, if a taxi fare is a fixed start fee plus a charge per kilometre, the total cost can often be modelled linearly.

A quadratic model has the form $y=ax^2+bx+c$. It is useful when the data shows a curved shape with a turning point. For instance, the height of a ball thrown upward can often be modelled by a quadratic function because gravity causes the motion to rise and then fall.

An exponential model has the form $y=ab^x$ or $y=ae^{kx}$. It is useful for rapid growth or decay, such as money in compound interest or radioactive decay.

A power model has the form $y=ax^b$. It may fit relationships like area and side length, or some biological measurements. For example, the surface area of a square is $A=s^2$, which is a power relationship.

A logarithmic model often appears when growth is fast at first but slows down over time. One example is perceived loudness or some scales in science.

In IB, you are not expected to guess randomly. You should look at the data and use evidence. If a scatter plot looks roughly like a straight line, a linear model may be sensible. If it curves upward more steeply as $x$ increases, an exponential model may be better. Technology can help you compare models by fitting several possibilities and checking which one is most appropriate. 💡

Reading graphs and interpreting features

Graphs are one of the most important tools in modelling. A graph can show the overall trend, local changes, and whether a model fits the data well.

When interpreting a graph, pay attention to:

the intercepts,
the gradient or rate of change,
maximum and minimum points,
asymptotes,
intervals where the function increases or decreases,
the meaning of the domain and range.

Suppose a model for the temperature of a cooling drink is $T(t)$. If the graph decreases quickly at first and then levels off, that suggests the drink is approaching room temperature. A horizontal asymptote can represent a value that the function gets closer to but does not reach in the model.

If a model has an $x$-intercept, that may represent when a quantity becomes zero. For example, if $h(t)$ is the height of a dropped object, then $h(t)=0$ might show the moment it hits the ground. In context, negative height would not make sense, so the domain may need to be restricted to values where the function is meaningful.

A key IB skill is interpreting the graph in words, not just in algebra. For example, if $P(t)$ is a population model and $P(0)=1200$, then the initial population is 1200. If $P(t)$ increases more and more quickly, that indicates accelerating growth. If the slope becomes smaller over time, the growth is slowing.

Regression and fitting data

When data is collected from experiments, surveys, or real-life measurements, the points rarely lie exactly on a perfect curve. Regression is the process of finding a function that best fits the data. This is often done using technology such as a graphing calculator or software.

A scatter plot shows the data points. Then you can test a model such as linear, quadratic, exponential, or power regression. The software usually gives an equation and a goodness-of-fit measure. One common measure is the coefficient of determination, $R^2$. Values of $R^2$ closer to $1$ usually indicate a better fit, though you must still check whether the model makes sense in context.

For example, suppose a student records how the number of visitors to a museum changes with time after opening. If the data rises quickly and then slows down, a logarithmic or exponential model with a leveling effect may fit better than a linear model.

It is important to remember that a high $R^2$ does not automatically mean the model is correct. A model can fit the given data well but still behave unrealistically outside the data range. This is called extrapolation. If a model predicts negative values for a quantity that cannot be negative, then the model should not be used there.

Interpolation means using the model within the range of observed data. Extrapolation means using it beyond the data range. Interpolation is usually safer. For example, if a model for shoe size versus age is built from teenagers aged 13 to 17, it is not reliable to predict the same relationship for toddlers or adults. 👟

Transformations and function behaviour in context

Many models are based on parent functions that are shifted, stretched, or reflected. These transformations help match a function to data.

For example, the graph of $y=x^2$ can be transformed into $y=a(x-h)^2+k$. Here, $h$ and $k$ shift the graph horizontally and vertically, while $a$ controls vertical stretch or reflection. This is useful when a data set has a parabolic shape but is not centered at the origin.

Similarly, an exponential model can take the form $y=ae^{k(x-h)}+c$. This model can shift the graph so it fits a real situation better. The value $c$ may represent a long-term level, such as a background temperature.

In context, transformations are not just about algebra. They help explain real behaviour. For example, if $f(t)$ is the number of people using a service, then $f(t-h)$ may show the same pattern beginning later in time. The horizontal shift $h$ can represent a delayed start.

One important habit is to interpret parameters in words. If a model is $C(x)=3x+15$, then $15$ is the initial cost and $3$ is the cost per unit. If a model is $A(t)=500(1.08)^t$, then $500$ is the initial amount and $1.08$ means an $8\%$ increase each time period. Always connect the mathematics to the context. ✅

Checking whether a model is reasonable

A good model must be mathematically correct and contextually sensible. IB expects you to evaluate whether a result is realistic.

Ask these questions:

Do the units match?
Are the outputs possible in real life?
Is the model valid only for certain values of $x$ or $t$?
Does the graph match the observed trend?
Does the model make sense beyond the data?

For example, if $d(t)$ represents distance travelled, then $d(t)$ should usually not decrease unless the object is returning. If a model predicts a child’s height shrinking over time, that would likely be unreasonable. If a model for mass gives a negative value, it is physically impossible.

Sometimes two models both seem possible. In that case, use the context and the evidence from the data. For example, both linear and quadratic curves may pass through some points, but the situation may indicate constant rate of change, which supports the linear model. In other cases, the shape of the scatter plot and the behaviour of residuals may show that a curved model is better.

A strong modeller is careful, not just fast. They use the function to understand the situation, make a prediction, and then question whether the prediction is believable. That critical thinking is a major part of IB Mathematics: Applications and Interpretation SL.

Conclusion

Modelling with functions means using mathematics to describe real-world patterns in a clear and useful way. students, you should now understand that a model is a simplified representation of reality, that different function types suit different situations, and that graphs and regression help you test and improve a model. The most important habit is to interpret results in context. A model is valuable when it helps explain data, supports prediction, and stays reasonable within its domain. In the broader study of functions, modelling brings together algebra, graphs, transformations, and technology into one practical skill. 🌍

Study Notes

A function assigns each input exactly one output.
A model is a simplified mathematical description of a real situation.
Common models include linear, quadratic, exponential, power, and logarithmic functions.
Use the shape of data to choose a suitable function type.
Interpret intercepts, gradients, turning points, and asymptotes in context.
Regression uses technology to find a function that fits data.
$R^2$ helps judge fit, but context is still essential.
Interpolation is safer than extrapolation.
Parameters in an equation often have real-world meanings, such as starting value or rate.
A good model must be mathematically correct, realistic, and useful within its domain.