The amount of a particular medicine in a patient's bloodstream, $M$ in milligrams, $t$ hours after administration is modelled by $M(t) = 250(0.85)^t$. Write down the initial amount of medicine administered. Calculate the exact half-life of the medicine in the bloodstream.

$M(0) = 250 \text{ mg}$ A1 Set the amount equal to half the initial amount: $125 = 250(0.85)^t$ M1 Divide by 250: $0.85^t = 0.5$ M1 Take logs: $t = \frac{\ln(0.5)}{\ln(0.85)} \approx 4.27 \text{ hours}$ A1

IB Maths AI SL · Unit 2: Functions

Exponential Models (IB Maths AI SL): notes, exam questions and worked examples

Updated October 2026 · By Pete Bromfield

An exponential model $f (x) = k a^{x} + c$ (or $k e^{r x} + c$ ) grows or decays by the same factor each step and levels off at the horizontal asymptote $y = c$ . In AI SL you evaluate a model, find its parameters from given information, solve for the time with your GDC, and say what each parameter means in context.

16 questions
Paper 1: 1
Paper 2: 15
3 worked examples

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Notes
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Worked examples
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FAQ

Exponential Models notes

Growth and decay

In $f (x) = k a^{x} + c$ , the graph grows when $a > 1$ and decays when $0 < a < 1$ ; $f (x) = k a^{- x} + c$ with $a > 1$ also decays. A population of 150 growing by 18% a month is $P (t) = 150 (1.18)^{t}$ .

Initial value and asymptote

At $x = 0$ , $a^{0} = 1$ , so the initial value is $k + c$ . As $x$ grows in a decay model the $k a^{x}$ term shrinks towards 0, so $y = c$ is the horizontal asymptote: the value the quantity settles at. For $T (t) = 25 + 850 e^{- 0.15 t}$ the initial temperature is $875$ and the asymptote $T = 25$ is room temperature.

Finding the parameters

Substitute the information you are given. If $V (0) = 5000$ and $V (1) = 4000$ in $V (t) = k a^{t}$ , then $k = 5000$ and $a = 0.8$ . With an asymptote, use the starting value and the long-run value to find $k$ and $c$ first.

Solving for the time

To find when a model reaches a value, graph both sides on your GDC and find the intersection (or use the solver). For $18 000 (0.92)^{t} = 9000$ the value halves after $t \approx 8.31$ years. Read the question for the form of the answer: a time, or the first whole year.

Interpreting the model

Say what each number means with units: in $P (t) = 150 (1.18)^{t}$ , 150 is the starting population and 1.18 means it grows by 18% each month. A model is only reliable over the range of data it was built from.

Key formulas

Exponential models Learn it: $f (x) = k a^{x} + c, f (x) = k a^{- x} + c, f (x) = k e^{r x} + c$
Horizontal asymptote of $k a^{x} + c$ Learn it: $y = c$
Exponents & logs In the booklet: $a^{x} = b ⟺ x = \log_{a} b$

The IB formula booklet is provided in every exam paper. In the booklet = printed there, so know where to find it; Learn it = not in the booklet, so learn it by heart. All AI SL formulas →

Exponential Models worked examples

Worked example 1: Evaluating an exponential growth model · easy

The rabbit population is modelled by $P (t) = 150 (1.18)^{t}$ , $t$ in months. Calculate the expected population after $8$ months.

Solution

5 steps

Examiner tip: Populations of living things (animals, bacteria, people) must be rounded to the nearest integer. Fractional organisms don't exist.

Worked example 2: Half-life of a depreciating asset · medium

A stamp depreciates: $V (t) = 18 000 (0.92)^{t}$ , $t$ in years. Using your GDC, find how many years for the stamp's value to halve.

Solution

5 steps

Examiner tip: Half-life of

y = a b^{t}

is INDEPENDENT of the starting amount

a

. Just solve

0.5 = b^{t}

each time.

Worked example 3: Cooling model and horizontal asymptote · hard

An ingot cools according to $T (t) = 25 + 850 e^{- 0.15 t}$ ( $t$ in minutes). Find the initial temperature and state the equation of the horizontal asymptote, explaining its meaning.

Solution

5 steps

Examiner tip: In cooling models

y = c + a e^{- k t}

, the constant

c

is the ambient temperature and mathematically acts as the horizontal asymptote.

Play with it Exponential growth model: drag the sliders

The table shows the number of bacteria (in thousands) after x hours. Change a and b so that y = a·bˣ fits the data (the check wants R² of at least 0.99). What do a and b mean in this context?

Want more? Open the full Grapher.

Exponential Models exam-style questions

3 questions from our AI SL bank, written in the style of IB papers, easiest first. One has its full worked solution open here. Open the worked answer, or check your answer in the practice area.

Question 1 · medium · 5 marks · Paper 1

A heated liquid is placed in a room. Its temperature, $T$ in $^{\circ} C$ , after $t$ minutes is modelled by the function $T (t) = 20 + 75 e^{- 0.08 t}$ for $t \geq 0$ .

Write down the initial temperature of the liquid.
Calculate the temperature of the liquid after exactly 10 minutes.
Write down the equation of the horizontal asymptote of the graph of $T (t)$ and explain what it represents in the context of the problem.

Attempt it and see the mark scheme →

Question 2 · hard · 4 marks · Paper 2

The amount of a particular medicine in a patient's bloodstream, $M$ in milligrams, $t$ hours after administration is modelled by $M (t) = 250 (0.85)^{t}$ .

Write down the initial amount of medicine administered.
Calculate the exact half-life of the medicine in the bloodstream.

Full worked solution Free, no sign-in

$M (0) = 250 mg$ A1
Set the amount equal to half the initial amount: $125 = 250 (0.85)^{t}$ M1
Divide by 250: ${0.85}^{t} = 0.5$ M1
Take logs: $t = \frac{\ln (0.5)}{\ln (0.85)} \approx 4.27 hours$ A1

Every question here has a full worked solution. See them all →

Question 3 · very hard · 8 marks · Paper 2

A heated metal object is placed in a cooling bath. Its temperature, $T$ in $^{\circ} C$ , after $t$ minutes is modelled by $T (t) = 18 + 105 e^{- 0.12 t}$ .

Write down the ambient temperature of the cooling bath.
Find the exact time it takes for the object to reach $40^{\circ} C$ .
A second object's temperature is modelled by $S (t) = 15 + 120 e^{- 0.15 t}$ . Using your GDC, find the exact time when both objects are at the identical temperature.

Check your answer in practice →

Free account — sign in to download Topic PDF: 7 questions with worked answers · 4 pages · 263 KB

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Where students lose marks on exponential models

Examiner's noteExponential and log models: e⁰ = 1, the equation vs the inequality, reading the model

Where marks are lost: Candidates treated a term with exponent 0 as zero, rounded b to 1 in f(t) = abᵗ, misread log₁₀x in a model, set an equation where the context needed an inequality (and then rounded the wrong way), gave calculator mantissas as answers, and commented on a model without using its equation.

Examiners flagged this in 3 exam series (May 2024, November 2024, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Exponential Models on your calculator

A GDC is allowed on every AI paper. Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire, with a checked worked example:

Graphs: intersections, zeros, max and min on your calculator

Exponential Models FAQ

What is the horizontal asymptote of an exponential model?

For f(x) = k a^x + c the asymptote is y = c: the ka^x part tends to 0 as x grows (for decay), so the quantity levels off at c. In context it is the long-run value, such as room temperature for a cooling drink.

Do I need logarithms to solve exponential equations in AI SL?

You can, but the GDC is quicker and enough: graph both sides and find the intersection, or use the equation solver. Write down the equation you solved.

How many IB Maths AI SL exponential modelling questions are there?

There are 16 exam-style exponential modelling questions in the AI SL question bank (Paper 1: 1 · Paper 2: 15), graded 3 very hard, 8 hard, 5 medium. Every question has a full IB-style mark scheme (M, A and R marks).

Is exponential modelling on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 1 · Paper 2: 15. Practise with your GDC — AI papers expect calculator methods throughout.

Next steps

More AI SL Unit 2: Applications of Functions · Direct/Inverse Variation and Cubic Models · Functions: Domain, Range & Inverses · Graphing Linear, Quadratic, Exponential and Logarithmic Functions · Linear Modelling · Quadratic Modelling · Sinusoidal Modelling · Solving Equations

Teaching this topic? Free ready-to-teach lessons (slides and teacher notes): Exponential growth models · Exponential decay models.

The ideas behind it, in the notes: Linear functions and modelling

Keep revising AI SL: AI SL revision notes · IB Maths AI SL past papers · AI SL common exam mistakes · AI SL syllabus checklist · AI SL functions knowledge organiser

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Exponential Models (IB Maths AI SL): notes, exam questions and worked examples

Try it first

Exponential Models notes

Growth and decay

Initial value and asymptote

Finding the parameters

Solving for the time

Interpreting the model

Key formulas

Exponential Models worked examples

Worked example 1: Evaluating an exponential growth model · easy

Solution

Worked example 2: Half-life of a depreciating asset · medium

Solution

Worked example 3: Cooling model and horizontal asymptote · hard

Solution

Exponential Models exam-style questions

Question 1 · medium · 5 marks · Paper 1

Question 2 · hard · 4 marks · Paper 2

Full worked solution Free, no sign-in

Question 3 · very hard · 8 marks · Paper 2

Where students lose marks on exponential models

Exponential Models on your calculator

Exponential Models FAQ

Next steps

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