2.4 Key features of graphs questions for IB Maths AA

68 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.

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	20	1	21
Paper 2	39	8	47
Every paper	59	9	68

Counted on 10 November 2026

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 · AA-00336Paper 1 · 6 marks

Let $f (x) = 5 \sin (2 x) + 3$ , for $0 \leq x \leq π$ .

(a)
Find the range of $f$ .
2 marks
(b)
Find the set of values of $k$ for which the equation $f (x) = k$ has exactly two solutions.
4 marks

Question 2 · AA-00148Paper 2 · 15 marks

In this question, all angles are in radians.

The point P moves on the unit circle with centre O, so that P has coordinates $(\cos θ, \sin θ)$ , where $0 \leq θ < 2 π$ .

A rod [PQ] of length 2 joins P to a point Q $(q, 0)$ , which slides along the $x$ -axis. The point Q is always to the right of P, so $q > \cos θ$ .

(a)
Show that $q = \cos θ + \sqrt{4 - \sin^{2} θ}$ .
3 marks
(b)
Find the exact value of $q$ when $θ = \frac{5 π}{6}$ .
2 marks
(c)
Write down the maximum value of $q$ and the minimum value of $q$ .
2 marks
(d)
Find the values of $θ$ for which $q = 2.5$ .
3 marks
(e)
For $0 < θ < π$ , show that $\sin (O \hat{Q} P) = \frac{1}{2} \sin θ$ .
3 marks
(f)
Hence, write down the maximum size of $O \hat{Q} P$ and the value of $θ$ at which it occurs.
2 marks

Question 3 · AA-00358Paper 1 · 6 marks

Consider the equation $\sin^{2} x + 2 \sin x = k$ , for $0 \leq x < 2 π$ , where $k \in R$ .

(a)
Show that the equation has solutions only if $- 1 \leq k \leq 3$ .
3 marks
(b)
Find the set of values of $k$ for which the equation has exactly two solutions.
3 marks

Question 4 · AA-00663Paper 2 · HL only · 15 marks

Let $f (x) = \arctan (2 x) - \arctan x$ , for $x \geq 0$ .

(a)
Show that $f^{'} (x) = \frac{1 - 2 x^{2}}{(1 + x^{2}) (1 + 4 x^{2})}$ .
3 marks
(b)
Find the exact $x$ -coordinate of the maximum point on the graph of $y = f (x)$ .
2 marks
(c)
Show that the maximum value of $f (x)$ is $\arctan (\frac{\sqrt{2}}{4})$ .
4 marks
(d)(i)
Find $\lim_{x \to \infty} f (x)$ .
1 mark
(d)(ii)
Hence write down the equation of the horizontal asymptote of the graph of $y = f (x)$ .
1 mark
(e)
Find the set of values of $k$ for which the equation $f (x) = k$ has exactly two solutions.
2 marks
(f)
Find the area enclosed between the curve $y = f (x)$ and the $x$ -axis for $0 \leq x \leq 3$ .
2 marks

Question 5 · AA-00360Paper 1 · 17 marks

Let $f (x) = \cos (2 x) - 3 \cos x + 2$ , for $0 \leq x \leq 2 π$ .

(a)
Show that $f (x) = 2 \cos^{2} x - 3 \cos x + 1$ .
2 marks
(b)
Solve the equation $f (x) = 0$ .
4 marks
(c)(i)
Find $f^{'} (x)$ .
2 marks
(c)(ii)
Hence, show that $f^{'} (x) = \sin x (3 - 4 \cos x)$ .
2 marks
(d)
Hence, find the coordinates of the maximum point on the graph of $f$ , and show that the minimum value of $f$ is $- \frac{1}{8}$ .
4 marks
(e)
Find the set of values of $k$ for which the equation $f (x) = k$ has exactly two solutions.
3 marks

Question 6 · AA-00875Paper 2 · 6 marks

A bakery opens at 06:00. The number of loaves, $n$ , baked $t$ hours after opening is modelled by $n (t) = 48 t - 3 t^{2}$ , for $0 \leq t \leq 8$ .

The profit, in dollars, made from selling $n$ loaves is modelled by $P (n) = 1.8 n - 95$ .

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