Three points cannot choose the model

the three points

n/aquadratic

n/aexponential

identical so farverdict

Both curves pass through every data point. That is what makes the choice impossible from the data alone.

The figure needs JavaScript. It plots the points (0, 2), (1, 4) and (2, 8), draws both x² + x + 2 and 2 × 2ₓ through all three exactly, then extends them: at hour 5 they give 32 and 64, at hour 10 they give 112 and 2048. A line at 180 students is crossed by the exponential at hour 7. Everything below stands on its own.

What just happened

Both models were fitted to the same three points and both fit perfectly.

Quadratic. y = ax² + bx + c through the three points gives y = x² + x + 2.
Exponential. y = k × aₓ gives y = 2 × 2ₓ.
Check at hour 2: 4 + 2 + 2 = 8, and 2 × 4 = 8. Both exact.

And then at hour 10: 112 against 2048. A factor of more than eighteen, from models that agreed on every number you had.

A perfect fit is not evidence. Through any three points you can pass a quadratic, and usually an exponential, and always a cubic. The question "which model fits?" has more than one answer, so it is the wrong question.

What does choose it

Three things, in this order, and none of them is the fit.

The context. There are 180 students in the year. The exponential predicts more than that by hour 7, which is impossible. The quadratic does not until hour 13. Neither is right for ever, and one is wrong much sooner.
The mechanism. Ask what is actually happening. If each person who signs up tells two more, growth is proportional to the number already signed up, which is exponential. If a fixed number of posters go up each hour, it is linear. The shape follows from the story.
The shape of more data. Three points cannot tell you. Ten can.

A straight line was ruled out, and that is the one thing the data did do. A line through (0, 2) and (2, 8) passes through (1, 5), and the real value is 4. So the data can eliminate a family even when it cannot choose between the survivors. Say so in a modelling question: ruling something out is a mark.

The families, and what each is for

Family	Form	Reach for it when
Linear	f(x) = mx + c	A fixed amount is added each step. A fixed cost plus a rate.
Piecewise linear	different lines on different intervals	The rule changes at a boundary: tax bands, phone plans, the depth of a pool.
Quadratic	f(x) = ax² + bx + c	There is a single turning point: a thrown ball, a profit that peaks, an area.
Exponential	f(x) = kaₓ + c, or ka⁻ₓ + c, or keⁿₓ + c	Growth or decay by a constant FACTOR, with a floor or a ceiling at y = c.
Direct variation	f(x) = axⁿ, n > 0	One quantity is a fixed power of another, through the origin.
Inverse variation	f(x) = axⁿ, n < 0	One goes up as the other goes down, with the y-axis as an asymptote.
Cubic	f(x) = ax³ + bx² + cx + d	Two turning points, or a volume built from a length.
Sinusoidal	f(x) = a sin(bx) + d	Something repeats: tides, temperature over a year, a wheel.

Three of them, with the numbers

Exponential with a ceiling. A cup of coffee cooling in a room at 28°C: T = 60(0.8)ₜ + 28, with t in minutes.

At t = 0: 60 + 28 = 88°C. The 60 is how far above the room it starts.
At t = 5: 47.66°C. At t = 10: 34.44°C.
The horizontal asymptote is y = 28, the room. The coffee never goes below it, which is why the +28 is in the model and not an afterthought.

The gap above the room halves every 3.11 minutes, whatever the starting temperature. That constant is what "exponential" means.

Sinusoidal. Water depth at a pier: d = 1.2 sin(30t) + 2.5, with t in hours and the angle in degrees.

Amplitude 1.2 m: how far it swings either way.
Period 360/30 = 12 hours: one full cycle.
Principal axis y = 2.5 m: the middle, which is the mean depth.
So the depth runs between 1.3 m and 3.7 m.

Inverse variation. Light from a single lamp: I = 90/d², which is 90d⁻².

At 1 m, 90 units. At 2 m, 22.5. At 3 m, 10.
Doubling the distance divides the intensity by 4, not by 2, because n is −2.
The y-axis is a vertical asymptote: at d = 0 the model has no value, which is also a statement about where the model stops being about lamps.

On the GDC: fitting a model to points

At Standard Level you find parameters by solving equations, not by regression, and the machine will solve three equations in three unknowns in one go.

When you may use it. Applications. A calculator is allowed in every paper, and this sub-topic is built on the assumption that you have one.

TI-Nspire CX II

For the quadratic through three points: menu → Algebra → Solve System of Equations
Three unknowns a, b, c and three equations: c=2, a+b+c=4, 4a+2b+c=8
enter gives a=1, b=1, c=2, so y = x² + x + 2
To check a model against data, graph both: the points in a Lists & Spreadsheet page shown as a scatter, and the function over the top

Casio fx-CG50

MENU → Equation → F1 Simultaneous, and set the unknowns to 3
Enter the coefficients row by row: 0 0 1 | 2, 1 1 1 | 4, 4 2 1 | 8
F1 SOLVE: 1, 1, 2
Or MENU → Statistics, lists in, then F2 CALC → F3 REG → F3 X^2 for a quadratic fit, which is the same answer by a different road

The mark people lose. Reporting r² = 1 as evidence that the model is right. Fit a quadratic to three points and r² is 1 every time, because three points determine a quadratic exactly. It is arithmetic, not agreement. An exponential through the same three points also fits perfectly and predicts eighteen times as much by hour ten. If you quote a fit, quote what else fits as well.

Your turn

1. Using y = x² + x + 2, what is the prediction at hour 10?

2. Using y = 2 × 2ₓ, what is the prediction at hour 10?

3. For T = 60(0.8)ₜ + 28, what is the temperature after 10 minutes, to 2 decimal places?

4. For d = 1.2 sin(30t) + 2.5, what is the period in hours?

5. Both models fit the three data points exactly. What should you write?

Question 5. Both models fit the three data points exactly. What should you write? The exponential, because growth in sign-ups is usually exponential. Either fits the data, so the choice has to come from the context: the year group is 180, which the exponential exceeds by hour 7. The quadratic, because r² = 1 for it. Neither; three points are too few to model at all.

Where the marks go

1 markNaming an appropriate family, with a reason from the context.

1 markSetting up the equations to find the parameters.

1 markSolving them, usually with technology.

1 markUsing the model, and saying where it stops being believable.

The first and last marks are for sentences, not arithmetic, and they are the two most often left blank. A model with no stated reason and no stated limit is half an answer.