2.4 Key features of graphs questions for IB Maths AI

53 questions on Key features of graphs, 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	HL only	All
Paper 1	22	4	26
Paper 2	25	2	27
Every paper	47	6	53

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-00084Paper 1 · HL only · 5 marks

Consider the function $f (x) = x^{2} + 6 x + 34$ , $x \in R$ .

(a)
Calculate the discriminant of $f (x)$ .
1 mark
(b)
Hence explain why the graph of $y = f (x)$ does not intersect the $x$ -axis.
1 mark
(c)
Solve the equation $f (x) = 0$ , giving your answers in the form $a + b i$ .
2 marks
(d)
Write down the product of the two solutions.
1 mark

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Question 2 · AI-00183Paper 2 · 15 marks

Nadia serves a volleyball. The path of the ball is modelled by

y = a x^{2} + b x + c, x \geq 0,

where $y$ is the height of the ball above the floor, in metres, and $x$ is the horizontal distance of the ball from Nadia, in metres.

The ball passes through the points $(2, 3.9)$ , $(5, 4.5)$ and $(8, 4.02)$ .

(a)
Use the point $(2, 3.9)$ to write down an equation in $a$ , $b$ and $c$ .
1 mark
(b)
Write down two more equations in $a$ , $b$ and $c$ .
2 marks
(c)
Find the equation of the path of the ball.
2 marks
(d)
Write down the height of the ball when Nadia hits it.
1 mark
(e)
Find the maximum height reached by the ball.
2 marks
(f)
The net is 9 m horizontally from Nadia and its top is 2.43 m above the floor. Determine whether the ball passes over the net.
3 marks
(g)
The far end of the court is 18 m horizontally from Nadia. Determine whether the ball lands inside the court.
3 marks
(h)
State one limitation of this model.
1 mark

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Question 3 · AI-00085Paper 1 · HL only · 7 marks

Ingrid knows that $4 - 3 i$ is one solution of the equation $x^{2} + b x + c = 0$ , where $b, c \in R$ .

(a)
Write down the other solution of the equation.
1 mark
(b)
Find the value of $b$ and the value of $c$ .
4 marks
(c)
Find the coordinates of the vertex of the graph of $y = x^{2} + b x + c$ .
2 marks

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Question 4 · AI-00184Paper 2 · 14 marks

The water level in a reservoir is measured against a marker on the dam wall. The level, $L$ metres above the marker, $t$ months after the start of a year is modelled by

L (t) = a t^{3} + b t^{2} + c t + d, 0 \leq t \leq 12.

A negative value of $L$ means that the water is below the marker. The table shows four measurements.

$t$ (months)	0	2	5	9
$L$ (m)	−3.2	1.8	1.05	−2.75

(a)
Write down the value of $d$ .
1 mark
(b)
Write down a system of three equations in $a$ , $b$ and $c$ .
2 marks
(c)
Find the values of $a$ , $b$ and $c$ .
2 marks
(d)
Find the maximum water level between $t = 0$ and $t = 5$ , and the value of $t$ at which it occurs.
2 marks
(e)
Find the values of $t$ at which the water level is at the marker.
2 marks
(f)
Hence find the total length of time during the year for which the water is below the marker.
3 marks
(g)
The engineers consider using the model to predict the water level 18 months after the start of the year. Find the level predicted by the model and comment on the reliability of this prediction.
2 marks

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Question 5 · AI-00258Paper 1 · 7 marks

Kiri makes and sells hoodies. When she sells $x$ hoodies in a month, her revenue is $R (x) = 45 x - 0.3 x^{2}$ New Zealand dollars (NZD) and her costs are $C (x) = 600 + 12 x$ NZD, for $0 \leq x \leq 150$ .

Her monthly profit is $P (x) = R (x) - C (x)$ .

(a)
Show that $P (x) = - 0.3 x^{2} + 33 x - 600$ .
1 mark
(b)
Sketch the graph of $y = P (x)$ for $0 \leq x \leq 150$ . Label the maximum point and the intercepts with the axes with their coordinates.
4 marks
(c)
Find the numbers of hoodies Kiri could sell in a month to make a profit.
2 marks

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Question 6 · AI-00275Paper 2 · HL only · 15 marks

A fountain in a hotel lobby shoots a jet of water from a nozzle. The height of the jet, $h$ metres, above the floor is modelled by

h (x) = - 0.35 x^{2} + 2.1 x + 0.8, x \geq 0,

where $x$ is the horizontal distance, in metres, from the nozzle.

(a)
Write down the height of the nozzle above the floor.
1 mark
(b)
Find the maximum height of the jet and the horizontal distance from the nozzle at which it occurs.
3 marks
(c)
A decorative beam crosses the lobby at a height of $4.5$ metres. To check whether the water reaches the beam, the hotel engineer, Rosa, writes the equation $- 0.35 x^{2} + 2.1 x - 3.7 = 0$ .
Show that this equation has no real solutions.
2 marks
(d)
Solve the equation $- 0.35 x^{2} + 2.1 x - 3.7 = 0$ , giving your answers in the form $a + b i$ .
3 marks
(e)
State, with reference to your answer to part (b), how the real part of the solutions in part (d) relates to the shape of the jet.
1 mark
(f)
Rosa adjusts the pump so that the jet is modelled by $h (x) = - 0.35 x^{2} + b x + 0.8$ , where $b > 0$ .
Find the least value of $b$ for which the jet reaches the beam.
3 marks
(g)
Using the original model, find the horizontal distance from the nozzle at which the jet hits the floor.
2 marks

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