2.3 Graphing functions with technology questions for IB Maths AI

33 questions on Graphing functions with technology, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Applications and Interpretation
Part of 2 Functions
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	All
Paper 1	17	17
Paper 2	16	16
Every paper	33	33

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AI-00254Paper 1 · 6 marks

A museum opens a new dinosaur exhibition. The number of visitors to the exhibition, $N$ thousand per week, $t$ weeks after it opens, can be modelled by

N (t) = 12 t e^{- 0.4 t}, 0 \leq t \leq 12.

(a)
Sketch the graph of $N$ for $0 \leq t \leq 12$ , labelling the maximum point with its coordinates.
3 marks
(b)
Write down the greatest number of visitors in one week, according to the model.
1 mark
(c)
Find the values of $t$ for which there are $10 000$ visitors per week.
2 marks

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Question 2 · AI-00354Paper 2 · 19 marks

After Lukas takes one dose of a painkiller, the concentration of the drug in his blood, $C_{1}$ milligrams per litre ( ${mg L}^{- 1}$ ), $t$ hours after the dose, is modelled by

C_{1} (t) = 15 t e^{- 0.5 t}, 0 \leq t \leq 12.

(a)
Find the concentration one hour after the dose.
1 mark
(b)
Find the maximum concentration and the time at which it occurs.
2 marks
(c)
Sketch the graph of $C_{1}$ for $0 \leq t \leq 12$ , labelling the maximum point and the endpoints with their coordinates.
3 marks
(d)
The painkiller is effective when its concentration is at least $6 {mg L}^{- 1}$ .
Find the length of time for which one dose is effective.
3 marks
(e)
Lukas takes a second, identical dose 4 hours after the first. For $t \geq 4$ the total concentration is
$C (t) = 15 t e^{- 0.5 t} + 15 (t - 4) e^{- 0.5 (t - 4)} .$
Show that $C (6) = 15.5$ , correct to three significant figures.
2 marks
(f)
Find the maximum total concentration for $4 \leq t \leq 12$ and the time at which it occurs.
3 marks
(g)
The concentration should not be above $15 {mg L}^{- 1}$ . Find the length of time for which the total concentration is above this level.
3 marks
(h)
Find the total length of time, from the first dose, for which the concentration is at least $6 {mg L}^{- 1}$ .
2 marks

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Question 3 · AI-00255Paper 1 · 6 marks

Femi is investigating how bright a lamp appears at different distances. The intensity of the light, $I$ lux, at a distance of $d$ metres from the lamp is given by $I = \frac{36}{d^{2}}$ , for $d > 0$ .

The following table shows some values of $I$ .

$d$ (m)	1	1.5	2	3	4	5	6
$I$ (lux)	36	$p$	9	$q$	2.25	1.44	1

(a)
Find the value of $p$ and the value of $q$ .
2 marks
(b)
On the following grid, draw the graph of $I = \frac{36}{d^{2}}$ for $1 \leq d \leq 6$ .
3 marks
(c)
Use your graph to estimate the distance from the lamp at which the intensity is 6 lux.
1 mark

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Question 4 · AI-00355Paper 2 · 15 marks

A swimming pool has a horizontal floor and is in the shape of a cuboid, $25 m$ long and $12 m$ wide. The pool is empty.

At 07:00, pipe A starts to fill the pool and the depth of the water increases at a constant rate of $0.3 m$ per hour. At 10:00, pipe B is also turned on, and from then the depth increases at a constant rate of $0.5 m$ per hour. When the depth reaches $1.9 m$ , both pipes are turned off.

Let $d$ be the depth of the water, in metres, $t$ hours after 07:00.

(a)
Write down the depth of the water at 10:00.
1 mark
(b)
Find the time of day at which the pipes are turned off.
2 marks
(c)
Sketch the graph of $d$ against $t$ for $0 \leq t \leq 7$ , labelling the coordinates of the points where the rate of change of the depth changes.
3 marks
(d)
Find an equation for $d$ in terms of $t$ for $3 \leq t \leq 5$ .
2 marks
(e)
A swimming lesson can begin once the depth is at least $1.4 m$ . Find the earliest time of day at which the lesson can begin.
2 marks
(f)
Find the volume of water in the pool when the pipes are turned off.
2 marks
(g)
Find the rate, in $m^{3}$ per hour, at which pipe B delivers water.
3 marks

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Question 5 · AI-00256Paper 1 · 6 marks

Hannah pours a cup of tea at $85 ° C$ and leaves it on her desk in a room kept at a constant $21 ° C$ . The tea cools quickly at first and then more and more slowly, getting closer and closer to the temperature of the room.

Let $T$ be the temperature of the tea, in $° C$ , $t$ minutes after it was poured.

(a)
Sketch a graph of $T$ against $t$ for $0 \leq t \leq 60$ . Label clearly the value where the graph meets the $T$ -axis and the equation of any asymptote.
3 marks
(b)
The temperature of the tea can be modelled by $T (t) = 21 + 64 e^{- 0.08 t}$ .
Find the time it takes for the tea to cool to $40 ° C$ .
2 marks
(c)
Hannah claims that, according to the model, the tea will eventually cool to $20 ° C$ . Explain why she is wrong.
1 mark

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Question 6 · AI-00356Paper 2 · 16 marks

The temperature inside a greenhouse in Almería, $T ° C$ , $t$ hours after 06:00, is modelled by

T (t) = 0.04 t^{3} - 0.9 t^{2} + 5.4 t + 14, 0 \leq t \leq 12.

(a)
Write down the temperature at 06:00.
1 mark
(b)
Find the coordinates of
(i) the local maximum point of the graph of $T$ ;
(ii) the local minimum point of the graph of $T$ .
4 marks
(c)
Sketch the graph of $T$ for $0 \leq t \leq 12$ , labelling the endpoints and the points found in part (b).
3 marks
(d)
Write down the range of temperatures in the greenhouse during this period.
2 marks
(e)
A fan switches on whenever the temperature is above $21 ° C$ . Find the times of day at which the fan switches on and off. Give your answers to the nearest minute.
4 marks
(f)
Ines wants to use the model to predict the temperature at 20:00. Find $T (14)$ and comment on her plan.
2 marks

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