In simple terms
A friendly intro before the formal notes — no formulas yet.
From Dots to Curves: Real-World Modelling
This topic is about finding the best mathematical curve — a parabola, an S-shape or a power curve — to describe a set of real-world data points, then reading useful facts off that curve: where it peaks, where it crosses the axes, and what it predicts. The GDC does the fitting; your job is to choose the right shape and interpret the result.
Imagine you are a tailor and the data points are a customer's measurements. You choose the right pattern (a quadratic, cubic or power function) and adjust it to make a garment (the model) that fits. A good fit lets you predict a size for another item — but predicting far beyond the measurements you took risks an outfit that does not fit at all. That is the difference between interpolation and extrapolation.
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Plot and choose. Enter the data into the GDC and make a scatter plot. A single hump or valley (a 'U') suggests a quadratic; an 'S' shape suggests a cubic; a steady curve rising from near the origin suggests a power function.
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Fit the model. Run the matching regression (QuadReg, CubicReg or PwrReg). Write the full equation, keeping the parameters to at least 3 significant figures and storing the exact equation in the GDC.
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Read the features. Use the GDC's graph tools to find the vertex (maximum or minimum), the x-intercepts (roots) and the y-intercept, and to evaluate predictions.
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Interpret in context. Say what each parameter and each feature means for the real situation, attach units, and flag whether a prediction is interpolation (reliable) or extrapolation (question it).
Explore the concept
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Key formulas
Tap any symbol to reveal exactly what it means and its units.
Full topic notes
Formal explanation with the rigour you need for the exam.
The shapes and their key features
Before fitting anything, you need to recognise each shape and name its features, because exam questions ask for those features by name — the vertex, the axis of symmetry, the roots, the intercepts. Start a scatter plot on the GDC and read the shape off the screen.
For the quadratic :\n Axis of symmetry: .\n Vertex: the point on that line, x-coordinate , y-coordinate found by substituting it back in.\n y-intercept: .\n Roots: the solutions of , read from the GDC's 'zero' feature.
Quadratic (): a parabola with exactly one turning point, the vertex. It opens upward when (vertex is a minimum) and downward when (vertex is a maximum). It is symmetric about the axis of symmetry , meets the y-axis at , and has up to two roots where it crosses the x-axis.
Cubic (): typically an 'S' shape with up to two turning points (a local maximum and a local minimum) and up to three roots. It meets the y-axis at . Good for growth that speeds up, levels off, then changes again.
Power (): a single-term curve whose shape is set by . For it rises from the origin with no turning point; for it decays like a reciprocal curve and is undefined at . It is monotonic — it cannot rise then fall — which is the key clue for when NOT to use it.
The sign of the leading coefficient tells the story before you calculate anything. If a quadratic model has a maximum (a thrown ball, peak revenue); if it has a minimum (least cost). Stating this early stops you hunting for a maximum on an upward parabola that does not have one.
Fitting a model with your GDC
Once you have chosen a shape, the GDC finds the specific parameters by regression. The process is the same on every calculator: enter, choose, record, store.
Step 1 — Enter. In the statistics menu, put the independent variable (e.g. time) in one list and the dependent variable (e.g. height) in a second list.
Step 2 — Choose. In the regression sub-menu pick the model: QuadReg for quadratic, CubicReg for cubic, PwrReg for power.
Step 3 — Record. Run it, checking the correct lists are used. Write the full equation, quoting each parameter to at least 3 significant figures. If shown, note the value as a measure of fit.
Step 4 — Store. Save the equation into a function variable (e.g. Y1). Now you can graph it and use max/min, zero and value tools without retyping long decimals — and without introducing rounding error.
Interpretation and prediction
A model earns its keep only when you interpret it and predict sensibly. Two ideas govern how far you can trust a prediction.
Interpolation: predicting a value INSIDE the range of the original data (e.g. the rocket's height at s, since lies between and ). Generally reliable.\n\nExtrapolation: predicting a value OUTSIDE that range (e.g. the height at s). Less reliable — the model may not hold beyond the data, so its validity should be questioned.
Common mistakes examiners penalise
Confusing the vertex with a coefficient — the maximum/minimum of is at (or read from the GDC), NOT at or at the largest coefficient. The vertex is a point ; quote both coordinates with units.
Keeping a non-physical root — a quadratic or cubic model of a real quantity often has two or three roots, but negative times, negative lengths and values outside the modelled window must be rejected with a brief reason ('time cannot be negative'). Take the meaningful root only.
Forcing a power model onto humped data — is monotonic and passes through the origin (for ); it cannot rise then fall. Data with a peak or trough needs a quadratic or cubic, not a power model.
Reading a feature off the wrong place — get the vertex from the GDC's 'maximum'/'minimum' feature and the roots from 'zero'; do not read the largest data value as the maximum or the value of as the vertex.
Extrapolating without comment — predicting outside the data range is allowed, but you must flag it as extrapolation and question its reliability. Treating a far extrapolation as a firm prediction loses the reasoning mark.
Over-rounding the parameters — quoting , , to 3 s.f. and then computing with the rounded values corrupts later answers. Store the exact regression equation and evaluate from it; round only the final answer.
Writing the parameters but not the model — 'find the model' means write the full equation , not a bare list .
Dropping units and context — a height is ' m', a value is '£110 000'. A bare number left unlabelled, or a parameter never interpreted in context, throws away the interpretation marks these questions are designed to test.
Model answer — marked the way our engine marks it
On the calculator paper the marks are analytic: each is tied to a specific line of working — a method mark (M) or an accuracy mark (A) — and an accuracy mark depends on the method mark it follows. Follow-through (FT) means an earlier slip need not cost the marks that depend on it, provided the later step is done correctly on your own figures. The engine also accepts any correctly-rounded value and any equivalent form. But that protection only exists if the method is on the page — a GDC screen described in words earns the method mark, a bare number does not. Study how each mark below is earned.
Where this leads
Modelling with curves is the backbone of the applications course. The regression skill here reappears for exponential and trigonometric models, and the vertex, root and intercept features you read off a parabola are the same features you will locate on any function graph. The discipline of choosing a shape from the data, driving the GDC, rejecting non-physical solutions and interpreting parameters in context is exactly what every modelling question on the calculator paper rewards. Master 'choose the shape, fit it, read the feature, say what it means' and the rest of the modelling course becomes variations on a routine you already own.
Worked examples
See the formulas applied — reveal one step at a time, like the exam.
A small rocket is launched from a platform. Its height, metres, above the ground is recorded at times seconds after launch.\n\n| Time, (s) | 1 | 2 | 3 | 4 | 5 | \n |---|---|---|---|---|---| \n | Height, (m) | 35 | 50 | 55 | 50 | 35 | \n\n(a) A quadratic model is used. Find , and . \n(b) Use the model to find the height after 2.5 seconds. \n(c) Determine the maximum height reached according to the model. [6]
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This is a calculator question. Enter the times in one list and the heights in a second, then run quadratic regression.\n\n**(a) Fit the model.** QuadReg on the two lists gives , , , so\n [M1: regression on the GDC] [A2: all three parameters]\n\n**(b) Prediction at .** Because lies between and this is interpolation. Evaluate the stored equation:\n [M1: substitute into the model]\nThe height is m (3 s.f.). [A1]\n\n**(c) Maximum height.** Since the parabola opens downward, so the vertex is a maximum. Graph the stored equation and use the GDC's 'maximum' feature (or note the vertex is at ):\n [M1: locate the vertex]\nThe maximum height is m, reached at s. [A1]
The concentration of a drug in a patient's bloodstream, (mg/L), is measured hours after a dose.\n\n| Time, (h) | 0 | 1 | 2 | 3 | 4 | 5 | \n |---|---|---|---|---|---|---| \n | Concentration, | 0 | 4.8 | 6.4 | 5.4 | 2.8 | 0.5 | \n\n(a) A cubic model is proposed. Use your GDC to find the model. \n(b) Use the model to estimate the maximum concentration and when it occurs. \n(c) According to the model, at what time does the concentration first return to mg/L after the dose? [6]
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Enter the times in one list and the concentrations in a second, then run cubic regression.\n\n**(a) Fit the model.** CubicReg gives approximately\n (parameters to 3 s.f.). [M1: cubic regression] [A1]\n\n**(b) Maximum concentration.** Graph the stored equation and use the 'maximum' feature over the relevant interval. The local maximum is at about h with mg/L. [M1: use the GDC max feature] [A1]\nSo the concentration peaks at roughly mg/L about hours after the dose.\n\n**(c) Return to zero.** Use the GDC's 'zero' feature on the graph after the peak. The relevant root is at about h. [M1: solve ] [A1]\nThe model predicts the concentration returns to mg/L about hours after the dose. (Reject any negative or pre-peak root — only the first zero after the peak is physically meaningful here.)
The value of a vintage car, in thousands of pounds, is recorded over years since 2000.\n\n| Years, | 2 | 5 | 10 | 15 | 20 | \n |---|---|---|---|---|---| \n | Value, (£1000s) | 12.5 | 14.2 | 20.1 | 35.8 | 68.3 | \n\n(a) A power model is proposed. Find and . \n(b) Estimate the car's value in 2018 (). \n(c) Comment on using this model to predict the value in 2050 (). [6]
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The data rises ever more steeply from near the start, with no turning point — a power model is a sensible choice. Enter in one list and in a second, then run power regression.\n\n**(a) Fit the model.** PwrReg gives and (3 s.f.), so\n [M1: power regression] [A2: both parameters]\n\n**(b) Prediction at .** This is interpolation ( lies between and ). Using the stored equation,\n [M1: substitute into the model]\nThe estimated value is about £110 000 (3 s.f.). [A1]\n\n**(c) Prediction at .** This is extrapolation, far beyond the data (which stops at ). [R1]\nThe power model predicts unbounded growth, but a real car's value may plateau or fall, so a prediction at is unreliable and should be treated with caution. [R1]
The height of a ball is modelled by (height in metres, in seconds). Use your GDC to find the maximum height and the time when the ball hits the ground. [5]
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Model answer — full working.\n\nStore as a function on the GDC and graph it. Since the parabola opens downward, so its vertex is a maximum.\n\nMaximum height. Use the graph's 'maximum' feature:\nthe vertex is at s, giving m.\nSo the maximum height is m (3 s.f.), reached about s after launch.\n\nTime it hits the ground. The ground is . Use the graph's 'zero' feature (or the equation solver):\n\nTime cannot be negative, so reject and take the positive root.\nThe ball hits the ground at s (3 s.f.).\n\n---\nHow our marking engine awards the 5 marks:\n\n- M1 — graph on the GDC. Awarded for setting up the model on the calculator: storing and graphing it (or otherwise driving the GDC toward the maximum). This is the method that everything else stands on.\n- M1 — use the maximum feature. A method mark for using the GDC's 'maximum' (vertex) tool, or equivalently finding and substituting. The engine rewards the approach, so it survives a small slip in the read-off.\n- A1 — 21.9 m. The accuracy mark for the maximum height m (accepting the vertex time s). It depends on the M-mark above and is protected by FT: a candidate whose method is right but who reads a slightly different value still scores on their own figure.\n- M1 — solve . A method mark for setting the height to zero and solving on the GDC (the 'zero' feature or solver). The engine checks that was solved, not some other equation.\n- A1 — 4.15 s. The accuracy mark for s, with the negative root correctly rejected because time cannot be negative. FT applies — a candidate whose regression or figures differed but who correctly took the positive root of THEIR equation keeps this mark.\n\n**'Accept equivalent forms and correct rounding.'** The engine accepts m written as or rounded, and accepts s however the positive root is justified in words. Once a correct final value appears, ISW (ignore subsequent working) means later restatements do not lose marks.\n\nBottom line: of the 5 marks, three are method marks that survive a read-off slip, and the accuracy marks are shielded by follow-through. A student who writes only '21.9 m and 4.15 s' with no method risks losing the method marks if a number is off; a student who shows the graph, the max feature, the solve and the rejection of the negative root keeps the method regardless of a slip in the final digit.
How it all connects
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Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.
Glossary
Try to recall each definition before you reveal it.
Quick check
Answer in your head first — then tap to check. No pressure.
Revision flashcards
Flip the card. Test yourself before the exam.
Quadratic model
Its graph is a parabola with exactly one turning point, the vertex. If the parabola opens upward (vertex is a minimum); if it opens downward (vertex is a maximum). The constant is the y-intercept.
Key takeaways
Review these before you close the topic — retrieval beats re-reading.
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Quadratic (): a parabola with exactly one turning point, the vertex. It opens upward when (vertex is a minimum) and downward when (vertex is a maximum). It is symmetric about the axis of symmetry , meets the y-axis at , and has up to two roots where it crosses the x-axis.
- ✓
Cubic (): typically an 'S' shape with up to two turning points (a local maximum and a local minimum) and up to three roots. It meets the y-axis at . Good for growth that speeds up, levels off, then changes again.
- ✓
Power (): a single-term curve whose shape is set by . For it rises from the origin with no turning point; for it decays like a reciprocal curve and is undefined at . It is monotonic — it cannot rise then fall — which is the key clue for when NOT to use it.
Practice — then mark it
The whole point: a real Cambridge question, marked mark-by-mark.
Get a Paper 2 modelling question marked: fit the model, find the vertex and roots, and predict with full working
Get a Paper 2 modelling question marked: fit the model, find the vertex and roots, and predict with full working
Extra simulations & links
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Frequently asked
Checkpoint
One marked question is worth ten re-reads — close the loop before you move on.
Reading it isn’t knowing it — prove it.
Before you move on: do Get a Paper 2 modelling question marked: fit the model, find the vertex and roots, and predict with full working on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.