Maths › Algebra and functions › Functions in modelling
Functions in modelling
By now the course has a shelf of function families, and modelling is the art of choosing the right one, fitting its constants to the situation, and saying honestly where it stops working. Tides call for trig, cooling calls for exponentials, and the choice itself is the examined skill.
Builds on Log graphs and exponential models and Trigonometric graphs and equations.
IN THIS TOPIC
- Match a situation's behaviour to the right function family.
- Fit and interpret a model's constants in context, with units.
- State limitations honestly and suggest refinements.
WHAT YOU PROBABLY THINK
The best model is the one that fits the data most closely.
Choosing the family
Each function family in the course has a behavioural signature, and diagnosis comes before fitting. Behaviour that repeats at a fixed interval points to trig; a rate proportional to the current amount points to an exponential; two quantities whose product stays constant point to a reciprocal. The families can also combine, a constant plus a trig term for tides, a constant plus a decaying exponential for cooling.
| Behaviour observed | Family to reach for |
|---|---|
| repeats every fixed interval | trigonometric: a + b sin(kt) |
| rate proportional to amount | exponential: Ae to the kt |
| product of the two quantities constant | reciprocal: k/x |
| steady rate of change | linear: mx + c |
WORKED EXAMPLE
Reading a tide model
The depth of water in a harbour is modelled by h = 5 + 2.4 sin (30t)°, with t in hours after midnight. Find the period of the tide, the greatest and least depths, and the first time of high tide.
The sine completes a cycle when 30t reaches 360, so the period is 12 hours.
Sine runs between ±1, so h runs from 5 − 2.4 = 2.6 m to 5 + 2.4 = 7.4 m.
High tide needs sin (30t)° = 1, first at 30t = 90, so t = 3, three in the morning.
The 5 is the mean depth, the 2.4 the tidal amplitude, and the 30 sets the clock. Every constant answers to a physical question, which is what separates a model from a formula.
Fitting, and knowing the limits
A fitted model earns trust only inside the conditions it was built for, which is where the opening lie falls down. A curve through every data point can still be the wrong family, and the wrong family extrapolates into nonsense. The examined skills are naming the assumption a model makes, spotting where reality breaks it, and proposing the refinement.
YOUR TURN
A cooling drink, read in full
A drink's temperature is modelled by T = 20 + 60e−0.05t, t in minutes. State the initial temperature, explain the physical meaning of the 20, and find when T reaches 50°, before opening the working.
Show the working
At t = 0 the exponential is 1, so T starts at 20 + 60 = 80°.
As t grows the exponential dies away and T flattens onto 20°, the room's temperature; the drink cools toward its surroundings, never below them.
Setting T = 50 gives e−0.05t = ½, so t = ln 2/0.05 = 13.9 minutes.
A bare decay model T = 80e−kt would predict the drink approaching 0°, which no room allows. The added constant is a refinement doing exactly what refinements do, encoding a physical fact the simple model missed.
TRY IT UNSEEN
Criticise and refine
Hours of daylight in a town are modelled by D = 12 + 4.2 sin (30m)°, with m in months after the spring equinox. State what the model predicts for the longest day, and give one limitation and one refinement.
Show the working
The longest day is 12 + 4.2 = 16.2 hours, three months in, at midsummer.
One limitation: real daylight is not perfectly sinusoidal, and month lengths are unequal, so the model drifts against the calendar over the year.
One refinement: fit the period in days rather than months, or adjust the amplitude to the town's latitude. Naming a specific, checkable improvement is what the mark scheme wants; “collect more data” on its own is not one.
THE EXAM BIT
- Diagnose the family from behaviour before fitting constants; a sentence naming the signature earns the first mark.
- Interpret every constant with units and a physical meaning; unexplained numbers are unfinished answers.
- “Initial” means t = 0; long-term behaviour means the asymptote; both are one-line evaluations.
- Limitation answers name the assumption that fails and where; refinement answers change the model, not the data collection.
- Combined models, a constant plus a trig or exponential term, shift the centre line; read the constant first and the oscillation or decay about it second.
CHECK YOURSELF
The value of a machine is modelled by V = 500 + 7500e−0.25t pounds after t years. State the initial value, the long-term value, and what the 500 represents.
Show a hint
Evaluate at t = 0, then let the exponential die.
Show the answer
At t = 0, V = 500 + 7500 = £8000.
As t grows, the exponential vanishes and V flattens onto £500.
The 500 is the machine's scrap or residual value, the floor the model refuses to fall through, and it is the refinement that separates this from bare exponential decay.
Behaviour picks the family: repetition is trig, proportional rate is exponential, constant product is reciprocal.
Fit the constants, interpret them with units, and say where the model stops being true.
CHECK YOUR PROGRESS
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- Match a situation's behaviour to the right function family.
- Fit and interpret a model's constants in context, with units.
- State limitations honestly and suggest refinements.
No animated video for this topic yet; these notes stand alone.