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 = Aekt
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 = Aekt + 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 = Aekt + 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 Aekt, this gives A directly. Find k using a second data point.
Worked examples
WE 1Quadratic 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 2Find unknown parameters — cooling tea
The temperature T °C of a cup of tea is modelled by T(t) = Aekt + 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 3Exponential growth — population
A bacterial colony grows according to P(t) = 250 e0.18t, 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 4Trigonometric 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 5Choose 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 6Limiting 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 (Aekt + 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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