The model is fine. The week you asked about is not

the four weeks measured

3week

14 cmmodel says

believableverdict

The same model throughout. Only the question changes.

The figure needs JavaScript. It plots the four measurements (0, 2), (1, 5), (2, 9) and (3, 14) and the quadratic h = 0.5w² + 2.5w + 2 through all four exactly, then widens the view: at week 13 the model passes 120 cm, which is as tall as this plant grows, and at week 52 it predicts 1484 cm. Everything below stands on its own.

The three stages, and which one failed

Stage	What you do	Did it fail here?
Develop and fit	Choose a family, decide a reasonable domain, find the parameters.	No. The fit is exact.
Test and reflect	Say whether the model is appropriate, and why you chose it.	Yes. Nobody asked how long it stays appropriate.
Use	Read, interpret, predict.	Yes, as a consequence. The prediction is outside the domain.

Almost every lost mark in a modelling question is in the middle stage, and almost every one of them is a sentence rather than a calculation. "This model is reasonable for about the first three months" is worth as much as the algebra that produced it.

Fitting: three equations, three unknowns

A quadratic has three parameters, so three data points pin it down. Use (0, 2), (1, 5) and (2, 9):

At w = 0: c = 2
At w = 1: a + b + c = 5
At w = 2: 4a + 2b + c = 9

Solve the three together, with technology, and you get a = 0.5, b = 2.5, c = 2, so

h = 0.5w² + 2.5w + 2

Then check it against the point you did not use: at w = 3 it gives 14, and the measurement was 14. That check is free and it is the only evidence you have that the family was a reasonable choice.

You are not expected to do non-linear regression at Standard Level. What you are expected to do is set up and solve up to three linear equations in three variables, which is what the above is, and to get parameters from initial conditions or by substituting points. If a question seems to need a curved regression, re-read it: it almost certainly wants equations.

Reflecting: where does it stop?

This kind of plant grows to about 120 cm. So ask when the model reaches that:

0.5w² + 2.5w + 2 = 120, so w² + 5w − 236 = 0
w = 13.06 weeks

So a reasonable domain is roughly 0 ≤ w ≤ 13. Inside it the model is good. Outside it the model is about a parabola rather than about a plant.

And check backwards as well. At w = −1 the model gives a height of 0, and at w = −2 it gives −1 cm. A negative height is the clearest possible signal that the domain has a left-hand end too, and almost nobody checks that side.

Using it: the dangers in order

Reading the model out to where nobody measured:

Week 10, still inside the domain: 77 cm. Believable.
Week 26, half a year: 405 cm, so over four metres. Not a seedling.
Week 52, a year: 1484 cm, nearly fifteen metres.

The arithmetic is right at every step. Extrapolation does not make a model wrong; it asks it a question it was never fitted to answer. Say that in the exam, with the number: "the model predicts 14.8 m after a year, which is not plausible for this plant, so it should not be used beyond about week 13."

On the GDC: three equations, and a prediction

The whole fitting stage is one solver command, and the whole using stage is one substitution. The marks are in the sentences either side of them.

When you may use it. Applications. A calculator is allowed in every paper, and this sub-topic assumes one throughout.

TI-Nspire CX II

menu → Algebra → Solve System of Equations, then Solve System of Linear Equations, with 3 equations and 3 unknowns
Enter c=2, a+b+c=5, 4a+2b+c=9
enter gives a=0.5, b=2.5, c=2
To predict, define it: h(w):=0.5w x² +2.5w+2, then h(52) gives 1484

Casio fx-CG50

MENU → Equation → F1 Simultaneous, unknowns set to 3
Coefficients row by row: 0 0 1 | 2, then 1 1 1 | 5, then 4 2 1 | 9
F1 SOLVE: 0.5, 2.5, 2
To predict, MENU → Table, enter Y1=0.5X²+2.5X+2, set the range and read the values straight off

The mark people lose. Giving a prediction with no comment. "After a year the height is 1484 cm" is arithmetic and earns the arithmetic mark only; the reflection mark needs the next sentence. The other one is entering the equations in the wrong order on the Casio, where c = 2 has to go in as 0 0 1 | 2 and students put the 2 in the first column, which solves a different system without complaining.

Your turn

1. Using h = 0.5w² + 2.5w + 2, what is the predicted height at week 10, in cm?

2. And at week 52, in cm?

3. The plant grows to 120 cm. At what week does the model reach that? Give it to 2 decimal places.

4. What does the model give at week −2, in cm?

5. A question asks you to comment on the model's prediction for week 52. Which answer earns the mark?

Question 5. A question asks you to comment on the model's prediction for week 52. Which answer earns the mark? The height is 1484 cm. The model is wrong, because the answer is too big. The model predicts 14.8 m, which is not plausible for this plant, so it should not be used beyond about week 13. A quadratic is the wrong family and an exponential should have been used.

Where the marks go

1 markThe equations, set up from the data points.

1 markSolving them for the parameters.

1 markA reasonable domain, with a reason.

1 markThe prediction, and a comment on whether it can be trusted.

Two of those four are sentences. Write them even when they feel obvious, because an examiner can only credit what is on the page.