IB Maths AA HL Topic 2 — Functions Paper 2 ~9 min read

Modelling with Functions

Modelling questions take a real-world situation — coffee cooling, populations growing, a ball flying through the air — and translate it into a function. Once you have the function, you can predict outputs, find when something equals a given value, or determine limiting behaviour. The trick isn’t usually the maths — it’s reading the question carefully, picking the right model, and translating “after 10 hours” or “limiting value” into the correct calculation.

📘 What you need to know

A model is a function describing how one quantity depends on another (e.g. height vs time, profit vs items sold).
Given the model, you can: substitute to find an output, solve to find an input, or analyse limiting behaviour.
“Initially” means t = 0. Watch out for this phrase — it tells you to substitute zero, not “the smallest value”.
“Limiting value” means t → ∞ (or whatever the variable trends toward). Look for the asymptote.
Find unknown parameters by substituting given data points into the model and solving the resulting equations (usually with a GDC).
Pick the right model by matching the situation: linear → constant rate; quadratic → projectile/trajectory; exponential → growth/decay/compound interest; logarithmic → slow growth (Richter, decibel); rational → cooling/approach to equilibrium; trig → periodic phenomena (tides, daylight).
Domain matters: real-life context restricts the input (time can’t be negative; lengths must be positive). Always check.
Units: read carefully whether quantities are in dollars or thousands of dollars, kg or tonnes, hours or days.

Common models — match the situation to the function

Linear

y = mx + c

constant rate of change — taxi fare, simple wages

Quadratic

y = ax² + bx + c

projectile motion, profit/cost, bridge cables

Exponential

y = Ae^kt

k > 0: growth (compound interest, populations)
k < 0: decay (radioactivity, drug clearance)

Logarithmic

y = a ln x + b

Richter scale, decibels — slow growth that compresses huge ranges

Rational

y = Ae^kt + c

cooling/heating to equilibrium — coffee, room temperature

Trigonometric

y = a sin(bt) + c

tides, daylight hours, periodic ferris-wheel height

If the question hints at “decay”, “decreasing rate”, “approaches a limit”, or shows data falling toward a non-zero value — think exponential decay with an offset: y = Ae^kt + c. The c is the limit; the A sets the starting gap from it.

Three things you do with a model

Predict the output

f(a) = ?

substitute x = a into the model — straightforward calculation

Find the input

f(x) = b

solve the equation (analytically or with the GDC) for x

Limiting behaviour

as x → ∞ or x → 0

find the asymptote — the value the model approaches but never reaches

Translate the wording

Common phrases → maths “Initially” → substitute t = 0
“After k hours/years” → substitute t = k
“Limiting value” / “long-term” → t → ∞ (find the asymptote)
“How long until …” → solve f(t) = (target) for t
“Initial value/amount” → the output when t = 0

Finding unknown parameters

Many questions give you a model with one or two unknowns (often A, k, or both) and provide enough data points for you to solve for them.

🧭 Recipe — finding parameters from data

Substitute each data point (input, output) into the model. This gives you one equation per data point.
Solve simultaneously: simple cases by hand; harder ones using your GDC’s “Solve” function.
For a single unknown, one data point is enough.
For two unknowns, you need two distinct data points.
Check by substituting back: the model should match all the given values.

Special trick: if “initially” gives a value, that’s the easiest equation — just substitute t = 0. For exponentials Ae^kt, this gives A directly. Find k using a second data point.

Worked examples

WE 1

Quadratic model — projectile motion

A ball is kicked upward. Its height (in metres) after t seconds is given by h(t) = −5t² + 20t + 1.5.
(a) Find the height initially. (b) Find when the ball hits the ground. (c) Find the maximum height.

(a) Initially: t = 0 h(0) = 1.5 m (a) 1.5 m (b) Hits ground: h(t) = 0 −5t² + 20t + 1.5 = 0 use GDC or formula: t ≈ −0.0739 or t ≈ 4.07 reject negative time (b) t ≈ 4.07 s (3 sf) (c) Max height at vertex: t = −b/(2a) = −20/(2 × −5) = 2 h(2) = −20 + 40 + 1.5 = 21.5 m (c) max height = 21.5 m at t = 2 s in projectile problems, the constant term is the launch height — here 1.5 m means the ball was kicked from above ground

WE 2

Find unknown parameters — cooling tea

The temperature T °C of a cup of tea is modelled by T(t) = Ae^kt + 22, where t is in minutes. Initially the tea is 90 °C, and after 10 minutes it is 50 °C.
(a) Find A. (b) Find the exact value of k. (c) Find the temperature after 25 minutes.

(a) Initially T(0) = 90 A · e⁰ + 22 = 90 A + 22 = 90 → A = 68 (a) A = 68 (b) After 10 min, T(10) = 50 68 e^(10k) + 22 = 50 68 e^(10k) = 28 e^(10k) = 28/68 = 7/17 10k = ln(7/17) (b) k = (1/10) ln(7/17) (c) Use full model — keep k in symbolic form to avoid rounding T(25) = 68 · e^(25k) + 22 ≈ 68 · 0.247 + 22 ≈ 16.8 + 22 (c) T(25) ≈ 38.8 °C (3 sf) limiting value is 22 °C — that’s the room temperature the tea cools toward but never reaches

WE 3

Exponential growth — population

A bacterial colony grows according to P(t) = 250 e^0.18t, where t is hours and P is population.
(a) Initial population? (b) Population after 12 hours? (c) When does the population reach 5000?

(a) Initial: t = 0 P(0) = 250 · e⁰ = 250 (a) 250 (b) Substitute t = 12 P(12) = 250 · e^(0.18 × 12) = 250 · e^2.16 ≈ 250 × 8.671 ≈ 2168 (b) ≈ 2170 (3 sf) (c) Solve P(t) = 5000 250 e^(0.18t) = 5000 e^(0.18t) = 20 0.18t = ln 20 t = ln 20 / 0.18 ≈ 16.6 (c) ≈ 16.6 hours (3 sf) round populations to a whole number; round time to 3 sf — units depend on what’s being measured

WE 4

Trigonometric model — tide depth

The depth (in metres) of water at a harbour is modelled by D(t) = 4 sin(0.5 t) + 7, where t is hours after midnight.
(a) State the maximum and minimum depths. (b) Find the depth at t = 3. (c) State the period.

(a) sin oscillates between −1 and +1 max: 4(1) + 7 = 11 m min: 4(−1) + 7 = 3 m (a) max 11 m, min 3 m (b) Substitute t = 3 (radians) D(3) = 4 sin(1.5) + 7 ≈ 4(0.9975) + 7 ≈ 11.0 (b) D(3) ≈ 11.0 m (3 sf) (c) Period of sin(bt) = 2π / |b| period = 2π / 0.5 = 4π ≈ 12.57 hours (c) period ≈ 12.6 hours (3 sf) about a 12-hour cycle — matches real tides which complete two cycles per day

WE 5

Choose an appropriate model

The value V (in dollars) of an investment increases by 6% per year. Initially, V = 4000. Suggest a model for V(t) and find V after 8 years.

Step 1: Identify the type — fixed % increase per year → exponential growth V(t) = V₀ × (1.06)^t Step 2: Use initial value V₀ = 4000 V(t) = 4000 × 1.06^t Step 3: Substitute t = 8 V(8) = 4000 × 1.06^8 ≈ 4000 × 1.5938 ≈ 6375 V(8) ≈ $6380 (3 sf) “increases by r% per year” → multiplier (1 + r/100) per year — classic compound interest set-up

WE 6

Limiting value and inverse use

The number of fish in a lake is modelled by N(t) = 800 − 600 e^−0.05t, where t is years from now.
(a) Initially, how many fish? (b) Long-term, how many fish? (c) When are there 700 fish?

(a) t = 0 N(0) = 800 − 600 · 1 = 200 (a) 200 fish (b) Long-term: t → ∞, e^(−0.05t) → 0 N → 800 − 0 = 800 (b) limiting value: 800 fish (c) Solve N(t) = 700 800 − 600 e^(−0.05t) = 700 600 e^(−0.05t) = 100 e^(−0.05t) = 1/6 −0.05t = ln(1/6) = −ln 6 t = ln 6 / 0.05 ≈ 35.8 (c) ≈ 35.8 years (3 sf) the population grows toward a “carrying capacity” of 800 — a typical biological model where resources limit growth

💡 Top tips

“Initially” always means t = 0. Memorise this — it’s the most common opening question.
“Limiting value” / “long-term” → look at what happens as t → ∞. For decay-with-offset (Ae^kt + c), the limit is c.
Match the model to the situation: constant rate → linear; trajectory → quadratic; growth/decay → exponential; cooling/equilibrium → exponential with offset; periodic → trig.
Use the GDC for messy parameter problems. Plug everything into Solve and let it crunch the numbers.
Keep symbolic constants until the end to avoid round-off errors building up.
State units in your answer — degrees, years, dollars, fish, whatever. Examiners deduct marks for forgetting.
Sketch the model on your GDC early — it tells you what’s plausible and catches dumb mistakes.

⚠ Common mistakes

Forgetting “initially” means t = 0. Some students substitute t = 1 instead, getting nonsense.
Choosing the wrong model: e.g. a linear model for compound interest (it’s exponential).
Mixing units: 5 minutes vs 5 hours, kg vs g, dollars vs thousands of dollars.
Round-off mistakes from decimalising parameters too early. Keep them symbolic.
Not checking the domain: time can’t be negative; counts of objects must be non-negative integers.
Skipping the “limiting value” when asked. The question is asking for an asymptote — find what the model trends toward.
Forgetting GDC mode (radians vs degrees) when working with trig models.

And that closes Section 2.7 — Other Functions & Graphs. You’ve now got the complete toolkit for exponential, logarithmic, and modelling problems. Topic 2 next moves into Reciprocal & Rational Functions — graphs of fractions, asymptotes, and rational equations — followed by transformations of graphs, polynomials, and the HL-only modulus and equations sections.

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