IB Maths AI SL Topic 3 — Modelling with Functions Paper 1 & 2 Chapter synthesis ~8 min read

Strategy for Modelling Functions

This is the synthesis note for the whole chapter. Given any real-world scenario — a falling stone, a savings account, a tide, a vibrating string — you choose the right model type, find its parameters from the given information, and use it to make predictions. Three skills: recognise the model from the shape or description; find parameters by substituting data points; and state a meaningful domain that keeps the model physically sensible.

📘 What you need to know

Seven model types: linear, piecewise linear, quadratic, cubic, exponential, direct/inverse variation, sinusoidal. Each has a unique shape and signature behaviour.
Initial value f(0): read directly from the constant term for linear/quadratic/cubic; from k + c for exponentials; from d for sinusoidals (when sin is used).
Parameters from data points: substitute (x, y) pairs into the chosen model form. Each data point gives one equation; n unknowns need n equations. AI SL covers systems up to 3×3.
Choose the model from the data shape: constant rate ⇒ linear; one turning point ⇒ quadratic; two turning points ⇒ cubic; constant percentage rate (or asymptote) ⇒ exponential; repeating pattern ⇒ sinusoidal.
Domain must match the real-world scenario: time ≥ 0; physical quantities can’t go negative; the model only applies until the scenario ends (object lands, tank empties, etc.).
Sketching on the GDC is essential: it confirms the shape, locates turning points, finds zeros, and lets you verify your answers against the visual.

The model gallery — which to pick

The first decision is always: which type of model? Either the question tells you (“model as f(x) = asin(bx) + d“) or you have to choose based on the data. The visual below summarises the six distinctive shapes.

Six common model shapes. The first step in any modelling problem is to look at the data (or read the description) and decide which signature matches: constant rate, single turning point, two turning points, asymptotic curve, hyperbolic, or wave.

Finding parameters from data

Once you’ve chosen the model, the unknowns (m, c; a, b, c; k, a, c; etc.) are found by substituting known data points. Each data point gives one equation; the number of unknowns tells you how many you need.

For each model, the constant term has a special meaning that often gives one parameter for free:

Initial value — one parameter for free linear, quadratic, cubic: f(0) = constant term
exponential: f(0) = k + c
sinusoidal: f(0) = d (for sin) or d + a (for cos)

Then use one or two more data points for any remaining unknowns. For a quadratic with three unknowns (a, b, c), you typically know c (initial) plus two more points — that’s a 2×2 system after substitution. AI SL’s GDC has a system-of-equations solver for 2×2 and 3×3 systems — use it.

Domain in context

Mathematical models extend in both directions of x to ±∞, but real-world scenarios don’t. State the meaningful domain as part of your answer:

Time-based models: usually 0 ≤ t ≤ t_end. The end-time is whenever the scenario stops being valid — the object lands, the tank empties, the heater is switched off, the day ends.

Physical quantities: distance > 0, mass > 0, count > 0. Restrict the domain to keep outputs sensible.

Periodic models: usually one full period (or whatever the question covers). E.g. 0 ≤ t < 24 hours for daily models.

The model is a simplification: even when fitted perfectly to data, the model is an approximation. Outside the range of the original data, predictions become less reliable. Always note that the model “assumes” certain conditions are maintained.

🧭 Recipe — any modelling problem

Identify the model type: from the description (constant rate, percentage change, periodic, etc.) or from the data shape (one turning point, asymptote, etc.).
Write the general form with unknown parameters left as letters (m, c, a, etc.).
Substitute the initial value if given: f(0) usually pins one parameter immediately.
Substitute remaining data points to form equations. Solve simultaneously (use GDC for 2×2 or 3×3 systems).
State the model with parameters plugged in, the domain in context, and use it for any required predictions. Sketch on the GDC to verify the shape matches expectations.

Worked examples

WE 1

Pick the model and find it — constant rate

A water tank is being filled at a constant rate. After 2 seconds it contains 15 mL, and after 5 seconds it contains 30 mL. (a) Explain why a linear model is appropriate. (b) Find the model V(t). (c) State the rate at which the tank is being filled and the initial volume.

(a) “constant rate” ⇒ linear slope (rate of change) is constant, so V(t) = mt + c (b) find m and c from the two data points m = (30 − 15)/(5 − 2) = 15/3 = 5 use V(2) = 15: 15 = 5(2) + c ⇒ c = 5 V(t) = 5t + 5 (c) interpret rate = m = 5 mL per second initial volume = c = 5 mL (already in tank at t = 0) (a) linear · (b) V(t) = 5t + 5 · (c) 5 mL/s, initial 5 mL always state what m and c MEAN in the context, not just the numbers. The exam often awards a separate mark for interpretation.

WE 2

Find a parameter then the domain — quadratic

A stone is dropped from rest from a tall platform. Its height (m) above the ground at time t (s) is given by h(t) = c − 5t². After 2 seconds the stone is 25 m above the ground. (a) Find c, the initial height. (b) Find the time when the stone hits the ground. (c) State the domain of the model.

(a) substitute h(2) = 25 25 = c − 5(4) 25 = c − 20 ⇒ c = 45 initial height: 45 m (b) hits ground when h(t) = 0 45 − 5t² = 0 ⇒ t² = 9 t = 3 or t = −3 (reject negative) stone lands at t = 3 s (c) domain in context 0 ≤ t ≤ 3 (from drop to landing) (a) c = 45 m · (b) t = 3 s · (c) 0 ≤ t ≤ 3 notice c IS the initial height — for any quadratic h(t) = at² + bt + c, the constant term is f(0). One data point gives c immediately.

WE 3

Find one unknown in a cubic from one data point

The volume V (m³) of water in a reservoir at time t months is modelled by V(t) = at³ − 6t² + 20t + 5. After 4 months the volume is 21 m³. (a) Find a. (b) Use the model to predict the volume after 8 months.

(a) substitute V(4) = 21 a(64) − 6(16) + 20(4) + 5 = 21 64a − 96 + 80 + 5 = 21 64a − 11 = 21 64a = 32 ⇒ a = 0.5 (b) V(t) = 0.5t³ − 6t² + 20t + 5 V(8) = 0.5(512) − 6(64) + 20(8) + 5 = 256 − 384 + 160 + 5 = 37 (a) a = 0.5 · (b) V(8) = 37 m³ when only ONE parameter is unknown, you only need ONE data point. The general rule: number of equations = number of unknowns.

WE 4

Find two unknowns from two data points — exponential

A cup of tea cools according to the model T(t) = ka^t + 20, where T is the temperature in °C and t is the time in hours. The initial temperature is 100°C, and after 1 hour the temperature is 60°C. (a) Find k and a. (b) Predict the temperature after 2 hours.

(a) Step 1 — use T(0) = 100 to find k k(1) + 20 = 100 k = 80 Step 2 — use T(1) = 60 to find a 80a + 20 = 60 80a = 40 ⇒ a = 0.5 T(t) = 80(0.5)ᵗ + 20 (b) predict T(2) T(2) = 80(0.5)² + 20 = 80(0.25) + 20 = 20 + 20 = 40 °C (a) k = 80, a = 0.5 · (b) T(2) = 40 °C the boundary c = 20 was GIVEN here (room temperature). If the question said “the long-term temperature is 20°C”, you’d extract c = 20 yourself. Then k + c = initial, giving k.

WE 5

Identify the model from a data table

The following data is collected for a quantity f(t):

t = 0: f = 100 · t = 1: f = 50 · t = 2: f = 25 · t = 3: f = 12.5

(a) Suggest a suitable model. Justify. (b) Find an equation. (c) Predict f(5).

(a) check successive ratios 50/100 = 0.5 25/50 = 0.5 12.5/25 = 0.5 constant ratio ⇒ EXPONENTIAL decay (b) f(t) = kaᵗ (no boundary needed: data tends to 0) f(0) = k = 100 a = ratio = 0.5 f(t) = 100(0.5)ᵗ (c) predict f(5) f(5) = 100(0.5)⁵ = 100/32 = 3.125 (a) exponential (constant ratio 0.5) · (b) f(t) = 100(0.5)ᵗ · (c) f(5) = 3.125 “constant ratio” is the giveaway for exponential. “Constant difference” between successive values would mean LINEAR instead. Always check ratios first when the data looks like it’s decreasing or increasing fast.

WE 6

Build a sinusoidal model from observed features

A Ferris wheel has a maximum passenger height of 32 m above the ground, a minimum of 2 m, and one revolution takes 60 seconds. A passenger boards at the height of the principal axis (the midline) as the carriage moves upward. The height h (m) at time t (s) is modelled by h(t) = a sin(bt°) + d. (a) Find a, b, d. (b) Find the first time the passenger reaches the maximum height.

(a) Step 1 — amplitude and principal axis a = (max − min)/2 = (32 − 2)/2 = 15 d = (max + min)/2 = (32 + 2)/2 = 17 Step 2 — b from the period period = 60 s ⇒ b = 360/60 = 6 h(t) = 15 sin(6t°) + 17 (b) maximum when sin(6t) = 1 6t = 90° ⇒ t = 15 s (check: h(15) = 15(1) + 17 = 32 ✓) (a) a = 15, b = 6, d = 17 · (b) max at t = 15 s sinusoidal modelling always boils down to three numbers: amplitude (half the range), principal axis (mean of max + min), and period (used to find b = 360°/period). Choosing sin vs cos depends on where the cycle “starts” in the data.

💡 Top tips

Look at the data SHAPE first: a quick GDC sketch shows the model type immediately. Don’t guess from words alone.
Use the initial value for one free parameter: f(0) usually gives c (or d) in one step.
Count unknowns and data points: each unknown needs one equation. Three unknowns need three data points.
Use the GDC system solver for 2×2 and 3×3: don’t do the algebra by hand for cubic with three unknowns — type the equations into the GDC.
Always state the domain in real-world units: 0 ≤ t ≤ 3 seconds; 0 < x < 15 cm. Maths without context loses marks.
Verify with a sketch on the GDC: plot the final model and check it passes through the given data points before declaring victory.

⚠ Common mistakes

Choosing the wrong model: data that looks “curved upward” could be quadratic OR exponential. Check constant differences (linear), constant second differences (quadratic), or constant ratios (exponential) before deciding.
Treating c as the initial value in an exponential: for f(x) = ka^x + c, the initial value is k + c, NOT just c.
Not stating the domain: the question often gives marks just for “t ≥ 0″ or “0 < x < 15″. Easy marks if you remember.
Predicting too far outside the data: models are only validated within the range of the original data. Predicting 50 years ahead from 2 data points is mathematical wishful thinking.
Forgetting to use the GDC: AI SL is a GDC-allowed course. Doing cubic algebra by hand wastes time and risks errors. Use the system solver, max/min tools, and intersect feature liberally.

This was the final note of the Modelling with Functions chapter. You now have a complete toolbox: linear, piecewise, quadratic, cubic, exponential, direct/inverse variation, and sinusoidal. Every IB AI SL exam question on modelling is some combination of “choose the model + find parameters + state domain + predict”. With practice, you’ll spot the model from a couple of data points within seconds.

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