Practice › Number and Algebra › Functions and Graphs of Quadratic Functions

Quadratic functions and graphs — Paper 1 long questions

Eighteen long questions on quadratic functions, in the shape Paper 1 actually uses: complete the square, read the vertex, then use it. The set starts with a single completed square and finishes with a Section B question carrying a translation, a circumcentre and an explain-your-answer part. Detailed solutions can be requested by email.

18 questions · about 2 hours · answers below each question

All questions are original, written for Math and AI Academy and modelled on past-paper style. They are not reproductions of HKEAA examination questions.

Jump to

Question 1 — completing the square; reading the vertex off the completed form
Question 2 — completing the square with a leading coefficient; axis of symmetry and y-intercept
Question 3 — maximum value of a downward parabola; x-intercepts
Question 4 — recovering a quadratic from its vertex and one more point
Question 5 — translating a graph horizontally and vertically; effect on the vertex
Question 6 — reflection in the y-axis and in the x-axis; writing the new function
Question 7 — the graph touching the x-axis; discriminant equal to zero
Question 8 — a graph lying entirely above the x-axis; range of a constant
Question 9 — building a quadratic model from a constraint; maximising an area
Question 10 — maximum of a quadratic in a physical context; solving for a given height
Question 11 — vertex in terms of a constant; using a condition on the vertex to find that constant
Question 12 — reading zeros and maximum from a quadratic model
Question 13 — finding a quadratic through three given points
Question 14 — a straight line tangent to a parabola; discriminant of the combined equation
Question 15 — intercepts and vertex as coordinates; area of a triangle; translating to touch the axis
Question 16 — vertex in terms of k for a downward parabola; a condition placing the vertex on a given line
Question 17 — reflection in the y-axis; area of a triangle in terms of k; solving a cubic that factorises
Question 18 — translation as a transformation of the function; circumcentre; testing whether a triangle can be equilateral

Question 1 · 3 marks · Modelled on 2022 P1 Q16

Let

f (x) = x^{2} - 6 x + 11

(a) Using the method of completing the square, express $f (x)$ in the form $(x - h)^{2} + k$ . (2 marks)

(b) Write down the coordinates of the vertex of the graph of $y = f (x)$ and the least value of $f (x)$ . (1 mark)

Show answer

(a) $f (x) = (x - 3)^{2} + 2$ . (b) Vertex $(3, 2)$ ; least value $2$ .
A square is never negative, so the smallest $f (x)$ can be is $2$ , reached when $x = 3$ .

Question 2 · 4 marks · Modelled on 2020 P1 Q17(a)

Let

f (x) = 2 x^{2} + 12 x + 7

(a) Using the method of completing the square, express $f (x)$ in the form $a (x - h)^{2} + k$ . (2 marks)

(b) Write down the equation of the axis of symmetry of the graph of $y = f (x)$ and the $y$ -intercept of the graph. (2 marks)

Show answer

(a) $f (x) = 2 (x + 3)^{2} - 11$ . (b) Axis of symmetry $x = - 3$ ; $y$ -intercept $7$ .
Take the $2$ out of the first two terms before halving the coefficient of $x$ ; halving $12$ instead of $6$ is where most marks go.

Question 3 · 5 marks · Modelled on 2012 P2 Q24, rewritten as a long question

Let

g (x) = - x^{2} + 10 x - 21

(a) Using the method of completing the square, express $g (x)$ in the form $a (x - h)^{2} + k$ . (2 marks)

(b) Write down the greatest value of $g (x)$ . (1 mark)

(c) Find the $x$ -intercepts of the graph of $y = g (x)$ . (2 marks)

Show answer

(a) $g (x) = - (x - 5)^{2} + 4$ . (b) Greatest value $4$ . (c) $x = 3$ and $x = 7$ .
The two intercepts sit $2$ units either side of $x = 5$ — a useful check that the axis of symmetry is right.

Question 4 · 4 marks · Modelled on 2014 P2 Q35, extended

The vertex of the graph of

y = f (x)

(2, - 9)

, where

f

is a quadratic function. The graph passes through the point

(5, 0)

(a) Express $f (x)$ in the form $a (x - h)^{2} + k$ . (2 marks)

(b) Hence express $f (x)$ in the form $a x^{2} + b x + c$ . (2 marks)

Show answer

(a) $f (x) = (x - 2)^{2} - 9$ . (b) $f (x) = x^{2} - 4 x - 5$ .
Substituting $(5, 0)$ gives $9 a - 9 = 0$ , so $a = 1$ .

Question 5 · 5 marks · Modelled on 2025 P1 Q18(b)

Let

f (x) = x^{2} - 8 x + 13

. The graph of

y = g (x)

is obtained by translating the graph of

y = f (x)

leftwards by

3

units and then upwards by

5

units.

(a) Find the coordinates of the vertex of the graph of $y = f (x)$ . (2 marks)

(b) Find the coordinates of the vertex of the graph of $y = g (x)$ . (1 mark)

(c) Express $g (x)$ in the form $a x^{2} + b x + c$ . (2 marks)

Show answer

(a) $(4, - 3)$ . (b) $(1, 2)$ . (c) $g (x) = x^{2} - 2 x + 3$ .
A leftward translation replaces $x$ by $x + 3$ ; writing $x - 3$ sends the vertex the wrong way.

Question 6 · 6 marks · Modelled on 2019 P1 Q19(b)

Let

h (x) = 2 x^{2} - 12 x + 23

(a) Using the method of completing the square, find the coordinates of the vertex of the graph of $y = h (x)$ . (2 marks)

(b) The graph of $y = p (x)$ is obtained by reflecting the graph of $y = h (x)$ with respect to the $y$ -axis. Express $p (x)$ in the form $a x^{2} + b x + c$ and write down the coordinates of its vertex. (2 marks)

(c) The graph of $y = q (x)$ is obtained by reflecting the graph of $y = h (x)$ with respect to the $x$ -axis. Write down the coordinates of the vertex of the graph of $y = q (x)$ . (2 marks)

Show answer

(a) $h (x) = 2 (x - 3)^{2} + 5$ , vertex $(3, 5)$ . (b) $p (x) = 2 x^{2} + 12 x + 23$ , vertex $(- 3, 5)$ . (c) $(3, - 5)$ .
Reflection in the $y$ -axis flips the sign of the $x$ -coordinate only; reflection in the $x$ -axis flips the $y$ -coordinate only.

Question 7 · 4 marks · Modelled on 2017 P1 Q18(a)

It is given that the graph of

y = x^{2} + k x + 9

touches the

x

-axis at exactly one point, where

k

is a constant.

(a) Find the values of $k$ . (2 marks)

(b) For each value of $k$ found in (a), write down the coordinates of the vertex of the graph. (2 marks)

Show answer

(a) $k = 6$ or $k = - 6$ . (b) $(- 3, 0)$ when $k = 6$ ; $(3, 0)$ when $k = - 6$ .
Touching the $x$ -axis means the vertex sits on it, so $Δ = k^{2} - 36 = 0$ .

Question 8 · 5 marks · Modelled on 2017 P1 Q18, reversed

The graph of

y = 2 x^{2} + 4 x + c

lies entirely above the

x

-axis, where

c

is a constant.

(a) Find the range of values of $c$ . (3 marks)

(b) If $c$ is the least integer satisfying (a), find the least value of $2 x^{2} + 4 x + c$ . (2 marks)

Show answer

(a) $c > 2$ . (b) $c = 3$ , least value $1$ .
Entirely above means no real root, so $Δ = 16 - 8 c < 0$ . Note $c > 2$ is strict — at $c = 2$ the graph touches the axis.

Question 9 · 6 marks · Modelled on 2017 P1 Q19, with a different real-world system

A farmer uses

60

m of fencing to enclose a rectangular plot against a long straight wall. The wall forms one side of the plot, so fencing is needed for the other three sides only. The two sides perpendicular to the wall are each

x

m long.

(a) Express the area $A$ m² of the plot in terms of $x$ . (2 marks)

(b) Using the method of completing the square, find the greatest possible area of the plot and the value of $x$ at which it occurs. (4 marks)

Show answer

(a) $A = 60 x - 2 x^{2}$ . (b) $A = - 2 (x - 15)^{2} + 450$ , greatest area $450$ m² when $x = 15$ .
The side along the wall is $60 - 2 x$ , not $60 - x$ — that single slip changes every later answer.

Question 10 · 6 marks · Modelled on 2017 P1 Q19, changed context

A ball is thrown upwards from a platform. Its height above the ground,

h

m, at time

t

seconds after it is thrown is given by

h = - 5 t^{2} + 20 t + 1

(a) Find the height of the platform. (1 mark)

(b) Using the method of completing the square, find the greatest height reached by the ball and the time at which this occurs. (3 marks)

(c) Find the time at which the ball returns to the height of the platform. (2 marks)

Show answer

(a) $1$ m. (b) $h = - 5 (t - 2)^{2} + 21$ , greatest height $21$ m at $t = 2$ . (c) $t = 4$ .
Part (c) needs no new work: the path is symmetric about $t = 2$ , so it returns at $t = 4$ .

Question 11 · 7 marks · Modelled on 2020 P1 Q17

Let

f (x) = x^{2} - 6 k x + 13 k^{2} - 4 k

, where

k

is a positive constant. Denote the vertex of the graph of

y = f (x)

V

(a) Using the method of completing the square, express the coordinates of $V$ in terms of $k$ . (3 marks)

(b) It is given that $V$ lies on the $x$ -axis. Find $k$ . (2 marks)

(c) Hence find the $x$ -intercept of the graph of $y = f (x)$ . (2 marks)

Show answer

(a) $f (x) = (x - 3 k)^{2} + 4 k^{2} - 4 k$ , so $V = (3 k, 4 k^{2} - 4 k)$ . (b) $k = 1$ . (c) $x = 3$ .
$4 k^{2} - 4 k = 0$ gives $k = 0$ or $k = 1$ ; $k = 0$ is rejected because $k$ is positive.

Question 12 · 6 marks · Modelled on 2016 P1 Q17, changed context

A shop finds that its daily profit

P

dollars from selling a drink at

x

dollars per cup is given by

P = - 2 x^{2} + 48 x - 160

(a) Find the two prices at which the shop makes no profit. (3 marks)

(b) Find the price that gives the greatest daily profit, and find that profit. (3 marks)

Show answer

(a) $x = 4$ and $x = 20$ . (b) Price $$12$ , greatest profit $$128$ .
The maximising price is the midpoint of the two break-even prices, which is worth using as a check.

Question 13 · 6 marks · Modelled on 2024 P1 Q19(a), reversed

The graph of

y = a x^{2} + b x + c

passes through the points

(0, 5)

(2, - 3)

and

(- 1, 12)

, where

a

b

and

c

are constants.

(a) Find $a$ , $b$ and $c$ . (4 marks)

(b) Find the coordinates of the vertex of the graph. (2 marks)

Show answer

(a) $a = 1$ , $b = - 6$ , $c = 5$ . (b) $(3, - 4)$ .
Start with $(0, 5)$ : it gives $c = 5$ immediately and leaves only two unknowns.

Question 14 · 6 marks · Modelled on 2021 P1 Q19(c)

The straight line

y = x + 4

is a tangent to the graph of

y = x^{2} + 3 x + k

, where

k

is a constant.

(a) Find $k$ . (3 marks)

(b) Find the coordinates of the point of contact. (3 marks)

Show answer

(a) $k = 5$ . (b) $(- 1, 3)$ .
Tangent means the equation $x^{2} + 2 x + (k - 4) = 0$ has a double root, so $Δ = 0$ and the root is $x = - 1$ .

Question 15 · 8 marks · Modelled on 2015 P1 Q18

Let

f (x) = x^{2} - 4 x - 5

. The graph of

y = f (x)

cuts the

x

-axis at the points

A

and

B

, where

A

is on the left of

B

. Denote the vertex of the graph by

V

(a) Find the coordinates of $A$ , $B$ and $V$ . (4 marks)

(b) Find the area of $△ A B V$ . (2 marks)

(c) The graph of $y = f (x)$ is translated upwards by $m$ units so that the resulting graph touches the $x$ -axis. Find $m$ . (2 marks)

Show answer

(a) $A = (- 1, 0)$ , $B = (5, 0)$ , $V = (2, - 9)$ . (b) $27$ . (c) $m = 9$ .
In (b) take $A B = 6$ as the base; the height is $∣ - 9 ∣ = 9$ , a length, so it is positive.

Question 16 · 8 marks · Modelled on 2026 P1 Q19(a)–(b)

Let

f (x) = - 2 x^{2} + 12 k x - 16 k^{2}

, where

k

is a positive constant. Denote the vertex of the graph of

y = f (x)

P

(a) Using the method of completing the square, express the coordinates of $P$ in terms of $k$ . (3 marks)

(b) It is given that $P$ lies on the straight line $y = 2 x$ . Find $k$ . (3 marks)

(c) Hence write down the greatest value of $f (x)$ . (2 marks)

Show answer

(a) $f (x) = - 2 (x - 3 k)^{2} + 2 k^{2}$ , so $P = (3 k, 2 k^{2})$ . (b) $k = 3$ . (c) $18$ .
$2 k^{2} = 6 k$ gives $k = 0$ or $k = 3$ , and $k = 0$ is rejected.

Question 17 · 10 marks · Modelled on 2019 P1 Q19

Let

f (x) = x^{2} - 8 k x + 20 k^{2} + 6 k

, where

k

is a positive constant. Denote the vertex of the graph of

y = f (x)

P

. The graph of

y = g (x)

is obtained by reflecting the graph of

y = f (x)

with respect to the

y

-axis. Denote the vertex of the graph of

y = g (x)

Q

. Denote the origin by

O

(a) Using the method of completing the square, express the coordinates of $P$ in terms of $k$ . (3 marks)

(b) Write down the coordinates of $Q$ in terms of $k$ . (1 mark)

(c) It is given that the area of $△ O P Q$ is $40$ . Find $k$ , and hence find the coordinates of $P$ and $Q$ . (6 marks)

Show answer

(a) $f (x) = (x - 4 k)^{2} + 4 k^{2} + 6 k$ , so $P = (4 k, 4 k^{2} + 6 k)$ . (b) $Q = (- 4 k, 4 k^{2} + 6 k)$ . (c) $k = 1$ , $P = (4, 10)$ , $Q = (- 4, 10)$ .
$P Q = 8 k$ is horizontal and the height from $O$ is $4 k^{2} + 6 k$ , giving $16 k^{3} + 24 k^{2} = 40$ ; $k = 1$ is the only positive root.

Question 18 · 12 marks · Modelled on 2026 P1 Q19

Let

f (x) = x^{2} + 6 k x + 13 k^{2}

, where

k

is a positive constant. Denote the vertex of the graph of

y = f (x)

A

. Let

g (x) = f (x - 6 k)

and denote the vertex of the graph of

y = g (x)

B

. Denote the origin by

O

(a) Using the method of completing the square, express the coordinates of $A$ in terms of $k$ . (3 marks)

(b) Describe the geometric meaning of transforming the graph of $y = f (x)$ into the graph of $y = g (x)$ , and hence write down the coordinates of $B$ in terms of $k$ . (2 marks)

(c) It is given that $k = 1$ . Find the coordinates of the circumcentre of $△ O A B$ . (4 marks)

(d) Is it possible that $△ O A B$ is an equilateral triangle? Explain your answer. (3 marks)

Show answer

(a) $f (x) = (x + 3 k)^{2} + 4 k^{2}$ , so $A = (- 3 k, 4 k^{2})$ . (b) A translation of $6 k$ units in the positive $x$ -direction; $B = (3 k, 4 k^{2})$ . (c) $(0, \frac{25}{8})$ . (d) Yes, when $k = \frac{3 3}{4}$ .
$A$ and $B$ are symmetric about the $y$ -axis, so the circumcentre lies on it; for (d) set $O A^{2} = A B^{2}$ , which gives $16 k^{4} = 27 k^{2}$ .

Continue practising this topic: Try the Paper 1 Challenge for harder variants, or move to Paper 2 multiple-choice — also available as Paper 2 Challenge MC.

Where this sits in the syllabus

Quadratic functions sit in the Number and Algebra strand and lean on quadratic equations, which is why the discriminant appears here too. Nearly every Paper 1 question on this topic opens with completing the square, so the coordinates of the vertex in terms of k are worth getting exactly right before anything else.

Full coverage of this topic, and the rest of the course, is on the Mathematics page. More sets are listed on the practice index.

Start with a free discovery session

No cost, no obligation. We look at where your child actually is, and show you what a term of real progress would look like.

Book a free discovery session See the programmes