What a model does

A mathematical model is a simplified mathematical representation of selected features from a real situation. It isn’t the situation itself. Instead, assumptions are chosen deliberately so that a useful relationship can be expressed and analysed.

For a function model,

.

In the formulas below, coefficients such as $a$, $b$, $c$, $d$, $k$, $m$ and $r$ are constants; their meanings and units depend on the context. The symbol $n$ is an integer exponent (dimensionless), while $e$ is the natural exponential base (dimensionless).

Choosing a model means choosing a pattern of change. Look for what remains approximately constant: perhaps an additive change, a multiplicative factor, a repeated cycle or another structural feature.

Comparison of common function families and key graph features.

Linear and piecewise linear models

A linear model is a function model with a constant rate of change:

Linear models work well for conversions, charges involving a fixed fee plus a constant usage rate, and quantities that change steadily with time. Repeated equal differences also connect them directly with arithmetic sequences.

A piecewise linear model has a domain divided into intervals, with a linear rule on each interval. Use one when the rate changes at a threshold—for example, a delivery charge with different rates for different mass bands. Pay attention to endpoints. A filled point includes the boundary value; an open point excludes it.

Quadratic models

A quadratic model is a polynomial function of degree two:

Its graph is a parabola, and the sign of $a$ decides whether it opens upwards or downwards. The vertex is the turning point. The axis of symmetry is the vertical line passing through it. Input values that satisfy $f(x)=0$ are called the zeros or roots; on the graph, they are the $x$-intercepts. The $y$-intercept is $f(0)=c$.

Quadratic models can describe a quantity that rises and then falls, a minimum operating cost or a relationship governed by a parabolic shape. Technology can dynamically generate a family of parabolas and solve $f(x)=0$. Roots outside the contextual domain must be rejected—for example, a negative time when time begins at release.

Quadratic relationships have a long history. Algebraic procedures developed in ancient Mesopotamian and South Asian traditions show that mathematical knowledge is not the product of one culture alone. One historical multiplication identity is

There is also a geometric view of a parabola: it is a conic section formed when an appropriate plane cuts a cone. This geometric representation and the algebraic graph describe the same mathematical object.

Exponential growth and decay

An exponential model is a function model where equal increases in the input multiply the changing part of the output by a constant factor. Useful forms include

and

The line $y=c$ is a horizontal asymptote. It is a horizontal line approached by the graph as the relevant input becomes very large or very negative. For $ka^x+c$, growth occurs when $a>1$ and decay when $0<a<1$, subject to the sign of $k$. For $ke^{rx}+c$, positive $r$ gives growth and negative $r$ gives decay when $k>0$.

Exponential models connect to geometric sequences, compound interest, loan repayment and depreciation because every equal interval applies a constant multiplier. They also model population change, transmission processes, cooling and radioactive decay. Use the terminology carefully: rapid growth isn’t necessarily exponential. In mathematics, “exponential growth” requires an approximately constant multiplicative factor over equal intervals, rather than simply a dramatic increase.

Direct and inverse variation

A power variation model is a function model in which the output is proportional to a fixed power of the input:

When $n>0$, the model describes direct power variation. When $n<0$, it describes inverse variation because $x^n=1/x^{-n}$. In that case, the $y$-axis, with equation $x=0$, is a vertical asymptote. This is a vertical line approached by the graph while the function becomes unbounded or remains undefined.

These models represent relationships such as pressure-volume effects, demand responses and inverse-square behaviour in physical systems. The exponent shows how strongly the output responds: doubling $x$ multiplies the output by $2^n$.

Cubic models

A cubic model is a polynomial function of degree three:

A cubic can change direction twice and may have one, two or three distinct $x$-intercepts. That flexibility makes cubic models useful for volume relationships, efficiency curves and quantities moving through several phases of increase and decrease. A graph by itself is not enough. The model still needs a meaningful domain and must follow physical restrictions, such as non-negative lengths or volumes.

Sinusoidal models

A sinusoidal model is a periodic function model based on a sine or cosine curve:

or

Using degree measure, the amplitude is $|a|$ and the period is

while the principal axis is the midline $y=d$. The maximum output is $d+|a|$, and the minimum is $d-|a|$. Sinusoidal models suit repeating phenomena, including seasonal measurements, rotating mechanisms and wave-like motion.

At this level, treat sine and cosine forms as given instead of translating one into the other. You need to interpret the amplitude, period and principal axis.

Models as representations

Models connect mathematics with experimental science, economics and business. A single family of curves can describe structurally similar patterns in quite different settings. Models work by removing detail while keeping the relationship that matters. Simplicity helps when it makes assumptions and parameters understandable, but too much simplicity may remove the feature needed for prediction.

This leads to the central knowledge question: mathematical models can produce exact consequences from chosen assumptions, whereas models in other areas are also judged against observation. Their conclusions remain conditional—if the assumptions are unsuitable, elegant mathematics will not rescue the result.

The modelling cycle

Mathematical modelling is an iterative process. It translates a real situation into mathematics, analyses the resulting representation, checks the output against reality and, when needed, revises the model. Think of this as a cycle—finding an equation isn’t the end of the process.

A practical modelling process is:

1. identify the variables, units, purpose and assumptions;
2. inspect the context or data, then choose a plausible function family;
3. establish a reasonable domain;
4. determine the parameters;
5. use the model to calculate, interpret or predict;
6. test the output against the data and the context;
7. refine or reject the model if necessary.

Choosing the function and domain

Use both shape and context when choosing a model. Roughly constant first differences point towards a linear model, while constant second differences suggest a quadratic. Approximately constant ratios indicate an exponential, and repeated peaks suggest a sinusoidal model. Shape alone isn’t conclusive. For example, a short section of an exponential curve can look almost linear.

A parameter is a constant in a model whose value controls a feature of the relationship, such as slope, initial value, curvature, amplitude or growth factor. Before calculating a parameter, identify what it would represent.

The domain of a model is the set of input values over which the model is intended to represent the situation. This can be narrower than the algebraic domain. Time might begin at zero, dimensions may need to be positive, and a pricing rule may apply only during a stated contract. A population model may also be reliable only over the period covered by the observations.

Finding parameters

Initial conditions, known points or simultaneous equations can be used to find parameters. Substitute every known point into the proposed function. A linear model usually requires two independent conditions. A quadratic needs three, and a cubic needs four unless extra structural information is given.

When a quadratic is fitted through three known points, substitution gives three linear equations in the three unknown coefficients. Use technology to solve systems of up to three linear equations in three variables. Keep full calculator precision during the working, then round only the final reported parameters as the context requires.

An initial condition is a known value of a model at a specified starting input, commonly $f(0)$. It may immediately identify an intercept or starting amount. When the functional form has already been supplied, direct substitution is often quicker and clearer than regression.

Regression offers another way to fit models to data, although formal regression methods belong with statistics. Non-linear regression is not required here at this level. The focus is on choosing a theoretical model and finding its parameters from conditions or simultaneous equations.

Testing and reflecting

Fitting a curve doesn’t automatically produce a good model. Test it by asking:

• Does its shape agree with the observed pattern?
• Do its intercepts, turning points, asymptotes and parameter signs make contextual sense?
• Are predicted values possible and reasonably close to observations?
• Does the model respect known bounds, thresholds or cycles?
• Are the assumptions plausible over the stated domain?

A model may be mathematically accurate at the supplied points but still behave unreasonably between or beyond them. Scientific modelling calls for judgment as well as calculation. When two models explain the data nearly equally well, the simpler one is often preferable because its assumptions are clearer. Simplicity is not helpful, though, if it hides an important mechanism.

Reading and using a model

A model can be used in several ways: substitute an input to predict an output, solve an equation to find the input that produces a target output, or interpret a feature of its graph in context. Give the meaning and units of the result instead of reporting an isolated calculator value.

Interpolation is prediction for an input within the range of observed data. Extrapolation is prediction beyond that observed range. Extrapolation is far less secure because the established pattern may not continue. Growth could meet a resource limit, a tariff might change, or a periodic pattern may shift. Treat a prediction more cautiously as it moves further outside the fitted range.

Models work well because they convert assumptions into consequences that can be tested. They play a particularly important role in experimental work: observations shape the model, and its predictions guide further observations. Mathematics supplies the internal deductions, but evidence determines whether the representation remains useful.

Half-life from an exponential model

The half-life is the time it takes for the changing quantity in an exponential decay model to fall to half its current value. For

the half-life is

When the model has a non-zero asymptote, as in $f(x)=ke^{rx}+c$, the difference $f(x)-c$ halves—not necessarily the complete output.

Half-life models apply to decaying substances and attenuation processes in chemistry and physics. After each half-life, the amount above the asymptote is multiplied by $1/2$ rather than reduced by a fixed amount.

Natural logarithmic models

A natural logarithmic model is a function model in which the output changes linearly with the natural logarithm of the input:

The restriction $x>0$ is essential. Multiplying $x$ by a constant produces a constant additive change in $f(x)$. Its graph has vertical asymptote $x=0$. It changes rapidly near zero, then progressively more slowly as $x$ increases. Logarithmic models are useful when equal ratios in the input matter more than equal differences.

Shifted sinusoidal models

A shifted sinusoidal model can be written as

Angles are measured in radians unless a degree symbol is shown. The amplitude is $|a|$, and the principal axis is $y=d$. The period is

while the horizontal translation $c$ is called the phase shift. This is the displacement of the cycle along the input axis. If the model is explicitly written with a degree symbol, use degree-based periods instead.

This form works well for periodic measurements such as oscillations, cyclic environmental records and alternating electrical signals. Check the starting position carefully: amplitude and period alone don’t determine the phase.

Logistic models

A logistic model is a bounded growth function. Growth begins approximately exponentially, then slows as a limiting resource becomes important and approaches an upper level:

The horizontal asymptote $y=L$ is the carrying capacity, the limiting level that the environment or system can sustain under the model's assumptions. Also,

The curve is S-shaped. Growth is slow at first, reaches its fastest rate in the middle region, then slows as the output approaches $L$. This model is appropriate only when growth has a plausible restriction, such as finite space, nutrients or market size.

One theoretical origin comes from the differential equation

If $P=P_0$ at $t=0$, where $P_0$ is the initial quantity (the same units as $P$), then

This shows why growth slows: as $P$ approaches the capacity, the factor $1-P/L$ approaches zero.

Piecewise models and continuity

A piecewise model is a function defined by different rules on different subdomains. It fits systems whose behaviour changes at a threshold, including progressive charges, staged motion and materials that follow different rules under different conditions.

A piecewise function is continuous at a boundary if the adjoining pieces meet at the same included point, leaving no jump or hole in the graph. You don’t need a formal definition of continuity; simply match the output values at the boundary.

For example, consider

As $x$ approaches $3$, the left-hand rule approaches $5$. Continuity requires $9a-3=5$, so $a=8/9$. Although the first interval excludes $3$, its value as $x$ approaches $3$ must still match the value given by the second rule.

You may also meet an unfamiliar model in context. Before using it, identify its domain, parameters, intercepts, asymptotes, turning behaviour and long-term pattern. No hierarchy of subjects makes one model universally “best”; its usefulness depends on the question, assumptions and evidence involved.

Why use logarithmic scales?

A logarithmic scale is an axis scale where equal distances show equal multiplicative factors, not equal additive differences. It keeps very large ranges manageable while drawing attention to relative change. Repeated multiplication by ten, for instance, appears as equal spacing even though the absolute increases become much larger.

Logarithmic axes work well when one or both variables cover several orders of magnitude, or when growth rate matters more than raw size. Long-term biological growth, online activity and physical decay can then be compared more easily. There is a catch: an alarming rise may look visually moderate. Always check the axis labels and scale before interpreting the graph.

Semi-log graphs and exponential relationships

A semi-log graph is a coordinate graph with one logarithmic axis and one linear axis. If

then taking natural logarithms gives

A plot of $\ln y$ against $x$ should therefore be approximately linear. The gradient is $r$, while the vertical intercept is $\ln k$. Hence,

Likewise, for $y=ka^x$,

Here, the gradient is $\ln a$, so $a=e^{\text{gradient}}$. A straight-line pattern on a semi-log representation supports an exponential relationship.

Log-log graphs and power relationships

A log-log graph is a coordinate graph where both axes use logarithmic scales. A general power relationship can be written as

Taking logarithms gives

The plot of $\ln y$ against $\ln x$ should be approximately linear. Its gradient is $p$ and its intercept is $\ln K$, giving $K=e^{\text{intercept}}$.

Changing the logarithm base does not affect whether the transformed relationship is linear, as long as the same base is used consistently. If base-ten logarithms are used, recover an intercept parameter with powers of ten rather than $e$.

Log-scale linearization for exponential and power laws

Linearizing and interpreting data

Linearization is a transformation of variables that turns a proposed non-linear relationship into an approximately straight-line relationship. Use this procedure:

1. identify the proposed relationship;
2. take logarithms and rearrange into straight-line form;
3. transform the appropriate variable or variables;
4. fit a best-fit straight line using technology;
5. use the gradient and intercept to recover the original parameters;
6. interpret those parameters in the original context.

A strong linear association after transformation supports the proposed model. Pearson's product moment correlation coefficient can help assess this linear association. It does not prove that the model is correct. Transformed data may hide clusters, unusual points or systematic curvature, and correlation alone does not establish a causal mechanism.

For a supplied graph, remember:

• straight on linear-linear axes suggests a linear relationship;
• straight on semi-log axes suggests an exponential relationship;
• straight on log-log axes suggests a power relationship;
• equal distances on a logarithmic axis represent equal ratios;
• zero and negative values cannot appear on an ordinary logarithmic axis.

You are expected to interpret semi-log and log-log graphs, not draw or sketch them by hand. Technology should carry out the transformations and produce the best-fit line, but you still need to understand what the slope and intercept mean.

These representations appear across science. Logarithmic response scales compress wide ranges, exponential decay becomes linear on semi-log axes, and transformed laboratory data can reveal parameters such as a reaction's activation behaviour. Their applicability is powerful, but it is not the sole measure of mathematical value. A method may offer theoretical insight before it has an obvious practical use. Equally, a useful graph can mislead when its scale or assumptions are ignored.