Modelling with Functions

students, have you ever wondered how scientists predict the spread of a virus, how a phone battery drains over time, or how a company estimates sales for the next month? 📈 These questions all involve modelling with functions. In this lesson, you will learn how functions can represent real situations, how to choose suitable function types, and how to interpret models using graphs, equations, and technology.

What it means to model with functions

A function model is a mathematical rule that connects an input to an output in a situation from the real world. In IB Mathematics: Applications and Interpretation HL, modelling is not just about writing an equation. It is about deciding whether the equation makes sense in context.

For example, if $t$ represents time and $h(t)$ represents the height of a ball, then a model such as $h(t)= -4.9t^2+12t+1.5$ may describe the ball’s motion. Here, the variables have meaning:

• $t$ is the input, often called the independent variable
• $h(t)$ is the output, often called the dependent variable

A model should match the situation well enough to make useful predictions. Real-life data are often messy, so a good model is usually an approximation rather than a perfect description.

Choosing a suitable function type

Different situations suggest different types of functions. students, one of the most important skills in modelling is recognizing patterns in data and choosing a function family that fits the pattern.

Linear models

A linear model has the form $f(x)=mx+b$. It is useful when the rate of change is constant.

Example: A taxi fare may have a starting cost plus a fixed cost per kilometer. If the fare is $f(x)=4+2.5x$, then $4$ is the base fee and $2.5$ is the cost per kilometer. If the trip is $8$ km, then $f(8)=4+2.5(8)=24$.

Linear models are common when something changes steadily, such as distance traveled at constant speed or total cost with a fixed fee.

Quadratic models

A quadratic model has the form $f(x)=ax^2+bx+c$. It often appears in motion, area, and projectile problems.

Example: The height of a thrown ball can often be modeled by a parabola because gravity causes a constant downward acceleration. If $h(t)= -5t^2+20t+2$, the $-5t^2$ term shows the height eventually decreases after rising.

The vertex of a quadratic model can give useful information such as maximum height or minimum cost.

Exponential models

An exponential model has the form $f(x)=ab^x$ or $f(x)=ae^{kx}$. It is used when change happens by a constant percentage rather than a constant amount.

Example: If a population grows by $3\%$ each year, a model could be $P(t)=P_0(1.03)^t$. If $P_0=500$, then after $4$ years, $P(4)=500(1.03)^4\approx 563.1$.

Exponential models are useful for population growth, inflation, compound interest, and radioactive decay.

Other models

Not every real situation is best described by a linear, quadratic, or exponential function. Sometimes a power model like $y=ax^b$, a logarithmic model like $y=a+b\ln x$, or a trigonometric model like $y=A\sin(Bx+C)+D$ is more appropriate.

For example, sound intensity levels and earthquake scales often involve logarithms, while seasonal data like temperature may follow a sinusoidal pattern.

Building a model from data

In IB AI HL, you often work with data tables, graphs, or scatter plots. The goal is to identify a pattern, fit a function, and judge whether the model is reasonable.

A typical modelling process is:

1. Understand the context and define variables
2. Plot or inspect the data
3. Choose a function type
4. Use technology to estimate parameters
5. Interpret the parameters in context
6. Test whether the model is reasonable
7. Use the model to make predictions

Suppose a student records the temperature of a drink as it cools. The data may decrease quickly at first and then level off toward room temperature. That shape suggests a model related to exponential decay. Technology can help find a best-fit equation such as $T(t)=22+58e^{-0.4t}$.

Here, $22$ represents the room temperature, and the exponential term shows how the drink approaches that temperature over time.

Regression and fitting

When data do not lie exactly on a curve, we use regression to find a model that fits the pattern as closely as possible.

Regression is especially important in IB because real measurements include uncertainty. A regression line, curve, or other function is chosen to minimize the overall error between the data and the model.

Common regression methods include:

• linear regression
• quadratic regression
• exponential regression
• power regression
• logarithmic regression

Technology, such as a graphing calculator or dynamic mathematics software, can calculate regression equations quickly. But students, you still need to understand what the equation means.

For example, if a regression equation is $y=2.1x+7.3$, then $2.1$ is the estimated rate of change and $7.3$ is the estimated value when $x=0$. If the data describe advertising cost versus sales, then the slope means how much sales are expected to change for each extra unit of advertising.

A strong model is not only one with a high correlation value or a good visual fit. It must also make sense in the context. A curved model may fit better than a line, but if the trend should logically be linear, the simpler model may be more appropriate.

Interpreting graphs, transformations, and parameters

Function models are not just equations. Their graphs communicate important information.

A shift, stretch, or reflection can change how a model behaves:

• $f(x)+k$ shifts the graph up by $k$
• $f(x-k)$ shifts it right by $k$
• $af(x)$ stretches vertically by factor $a$
• $f(ax)$ changes horizontal scale
• $-f(x)$ reflects in the $x$-axis

These transformations help build models from a parent function. For example, a sinusoidal model for tides might be written as $h(t)=3\sin\left(\frac{\pi}{6}(t-2)\right)+5$. Here:

• $3$ is the amplitude
• $\frac{\pi}{6}$ affects the period
• $2$ is the horizontal shift
• $5$ is the midline

Each parameter has a real-world meaning. In context, that is extremely important. A model is useful when its parameters describe real features of the situation, not just numbers on a screen.

Graph interpretation also includes domain and range. If $t$ represents time after an event, then negative values of $t$ may not make sense. So the mathematical domain must be restricted to the realistic domain.

Using a model responsibly

students, every model has limits. A function that fits data well in one interval may fail outside that interval. This is called extrapolation when you use the model beyond the observed data.

For example, suppose a model estimates a company’s monthly sales for the next $2$ months. Using it to predict sales for the next $10$ years may be unreliable, because conditions can change.

Good modelling asks:

• Is the function type reasonable for this context?
• Are the units correct?
• Does the domain match the situation?
• Are the predictions realistic?
• Does the model respect any restrictions, such as nonnegative values?

A population model that predicts negative people is clearly impossible, so the model must be used only where it makes sense.

Example: fitting a relationship with technology

Imagine a school records the number of hours studied, $x$, and the test score, $y$, for several students. The scatter plot shows a positive trend. A linear regression model gives $y=5.8x+42$.

Interpretation:

• $42$ is the estimated score when $x=0$
• $5.8$ means the score is expected to increase by about $5.8$ points for each extra hour studied

If a student studies $6$ hours, the model predicts $y=5.8(6)+42=76.8$.

But the model should not be treated as exact. A score depends on many factors, including sleep, prior knowledge, and test difficulty. The model gives a trend, not a guarantee.

This is the heart of modelling with functions: combining mathematics with context and judgment.

Conclusion

Modelling with functions is about turning real situations into useful mathematics. students, you should now understand that a function model describes a relationship between variables, that different function types suit different patterns, and that regression helps fit a model to data. You should also know that interpretation matters as much as calculation. In IB Mathematics: Applications and Interpretation HL, modelling connects algebra, graphs, technology, and real-world reasoning into one powerful skill. ✅

Study Notes

• A function model connects inputs and outputs in a real context.
• The independent variable is the input, and the dependent variable is the output.
• Common models include linear $f(x)=mx+b$, quadratic $f(x)=ax^2+bx+c$, exponential $f(x)=ab^x$, power $y=ax^b$, logarithmic $y=a+b\ln x$, and trigonometric models.
• Linear models suit constant rates of change.
• Quadratic models often describe motion and parabolic behavior.
• Exponential models describe constant percentage growth or decay.
• Regression finds a curve or line that fits data well.
• A good model must fit the data and make sense in context.
• Parameters in an equation often have real-world meanings.
• Graph transformations help describe changes in position, scale, and shape.
• Always check domain, range, and whether predictions are realistic.
• Extrapolation beyond the data can be unreliable.
• In IB AI HL, modelling combines technology, interpretation, and mathematical reasoning.