A-level · Beta

Modelling with functions

Compare linear, exponential and cooling models against synthetic sample values.

y=20+(2)ty=20+(2)t
00821642463284010WeeksSavings balance (£)

Scroll the diagram sideways to see every label.

Synthetic teaching data — these are constructed values, not real measurements.

Root mean square error on t = 0 to 5: 5.7009. This is the square root of the mean squared residual.

Prediction at t = 5: 30.

Residual = sample value − model prediction
TimeSamplePredictionResidual
01020-10
11522-7
22024-4
32526-1
430282
535305
Assumptions and limitations

Assumes a £10 starting balance, £5 added each week, no withdrawals and no interest.

A smaller sample error does not establish causation or guarantee future accuracy. The reveal button shows the rule used to construct this dataset; it does not run a fitting algorithm.

This prediction is inside the sample time range.

Red points: samples; red segments: residuals; blue: selected model; green: prediction. The dashed boundary marks the last sample time. Values are rounded to 4 decimal places. Scale changes with model parameters.

Predict, test and explain

Step 1 of 3

Predict

Does a perfect fit to these sample values prove a model will predict forever?

InvestigateDoes this context suggest a constant difference or a constant ratio?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Does a perfect fit to these sample values prove a model will predict forever?

N(t)=20(1.5)tN(t)=20(1.5)^t

Hint

Look for constant differences or ratios, then check context and limitations.

Reveal answer

A fitted model describes the sample under assumptions. Extrapolation needs justification beyond a low sample error.

N(2)=45N(2)=45

Build an explanation

Hint 1

Look for constant differences or ratios, then check context and limitations.

Hint 2

A fitted model describes the sample under assumptions. Extrapolation needs justification beyond a low sample error.

Hint 3

Explain why the rule is valid, including its assumptions.

Worked example

This example uses fixed values, separate from the diagram controls.

A synthetic culture starts at 20 units and grows by 50% each hour. Model the first few hours.

  1. A 50% increase means multiplying by 1.5 each hour.
  2. The initial value is 20.
  3. Use N(t) = 20 × 1.5^t, assuming the multiplier stays constant.

Answer: N(t) = 20 × 1.5^t; limited by resources in a real culture.

Connect this idea

Watch out: A fitted model describes the sample under assumptions. Extrapolation needs justification beyond a low sample error.

Specification and learning route

AQA B11 · Edexcel Pure 2.11

AS and A-level. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

Useful starting knowledge: Linear functions; Exponential functions; Interpreting graphs.

AQA specification · Edexcel specification