Back to Mathematical Modelling

Higher Applications of Mathematics

Quadratic models

Modelling curved relationships and turning points.

Modelling referenceModelling mixed quiz

Modelling reference

Check variables, formulae, model types, units, errors and assumptions.

Mixed revision quiz

Review Mathematical Modelling with adaptive feedback.

Before you start

  • Be confident substituting numbers into simple formulae.
  • Check the units in the question before calculating.
  • Be ready to explain what an answer means in the real situation.

Method helper

Which model or method do I use?

Does the output change by a fixed amount each step?

Use a linear model.

Is there a starting fee plus a rate per unit?

Use y = mx + c and interpret the intercept.

Does the situation involve area, a turning point or a squared variable?

Use a quadratic model.

Is the amount increasing or decreasing by a percentage repeatedly?

Use an exponential model.

Have you been given a formula and values?

Substitute into the formula carefully.

Do you need to make one variable the subject?

Rearrange the formula one step at a time.

Are measurements rounded or allowed within a range?

Calculate tolerance or error.

Do the units look inconsistent?

Convert units and check dimensional sense.

Are you testing several assumptions or scenarios?

Use technology to compare model outputs.

Does the prediction seem unrealistic?

Comment on model limitations.

Modelling lesson

Key idea

  • This topic focuses on using curved models with squared terms, turning points and changing rates. Mathematical modelling is about using maths to represent a real situation well enough to support a decision.
  • A good model identifies the variables, uses a sensible formula or graph, and states the assumptions being made.
  • For Higher Applications, the interpretation matters. You should explain whether the output is realistic, what it means in context, and what might make the model less reliable.

Key formulae, definitions and methods

  • Quadratic models include a squared term, such as y = ax² + bx + c
  • A turning point can represent a maximum or minimum in context.
  • Use a quadratic model when the change is not constant.

Worked examples

Modelling walkthrough 1

Build the model

A rectangular garden design changes width and length while keeping a path around the edge.

  1. Identify the variable that changes, such as width.
  2. Write each dimension in terms of that variable.
  3. Multiply the dimensions to form an area model.

Quadratic models are useful when area or a changing rate is involved.

Modelling walkthrough 2

Use the model

A rectangular garden design changes width and length while keeping a path around the edge.

  1. Substitute a practical value into the quadratic formula.
  2. Calculate the predicted area.
  3. Check that the answer has square units.

Units are important because area should be in square units.

Modelling walkthrough 3

Evaluate the model

A rectangular garden design changes width and length while keeping a path around the edge.

  1. Use a graph or table to find a maximum or minimum if needed.
  2. Interpret the turning point in the real situation.
  3. Check whether negative or unrealistic inputs should be excluded.

The turning point only matters if it makes sense in the context.

Watch out

  • Using a model without defining the variables and units.
  • Choosing a linear model when the rate of change is not constant.
  • Treating a model prediction as an exact fact rather than an estimate.
  • Forgetting to convert units before substituting values.
  • Giving a calculation without commenting on assumptions or limitations.

Technology and data connection

Related Higher Applications topics

Spreadsheet: Using spreadsheets for modelling

Next step

Move into practice

Use the learning notes to identify variables, assumptions and units, then try varied formulas, model types and reasonableness checks.

Modelling mixed quiz