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

Quadratic functions and graphs — Paper 2 multiple-choice

Twenty multiple-choice questions at Paper 2 pace. Most are one line of work once you see which form of the quadratic to use, so the thing being tested is recognition, not algebra. Detailed solutions can be requested by email.

20 questions · about 30 minutes · 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.

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Question 1 — reading the vertex from the completed-square form
Question 2 — axis of symmetry from the standard form
Question 3 — completing the square
Question 4 — least value of an upward parabola
Question 5 — greatest value of a downward parabola
Question 6 — y-intercept of a quadratic graph
Question 7 — vertical translation of a graph
Question 8 — horizontal translation written as a change of variable
Question 9 — reflection with respect to the y-axis
Question 10 — reflection with respect to the x-axis
Question 11 — the graph touching the x-axis
Question 12 — two distinct x-intercepts as a discriminant inequality
Question 13 — reading the signs of a, b and c from a sketch
Question 14 — vertex in terms of a constant; testing several statements
Question 15 — least value from the two x-intercepts
Question 16 — finding a constant from a stated minimum value
Question 17 — recovering the function from the vertex and one point
Question 18 — number of x-intercepts
Question 19 — a vertex on the y-axis fixes the linear coefficient
Question 20 — maximum of a quadratic model

Question 1 · Modelled on 2023 P2 Q35

The coordinates of the vertex of the graph of

y = (x - 3)^{2} + 4

are

$(- 3, 4)$ .
$(3, 4)$ .
$(3, - 4)$ .
$(- 3, - 4)$ .

Show answer

B. The form $(x - h)^{2} + k$ has vertex $(h, k)$ , and $x - 3$ gives $h = 3$ , not $- 3$ .

Question 2 · Modelled on 2015 P2 Q6

The equation of the axis of symmetry of the graph of

y = 2 x^{2} - 8 x + 5

$x = - 4$ .
$x = - 2$ .
$x = 2$ .
$x = 4$ .

Show answer

C. The axis is $x = - \frac{b}{2 a} = - \frac{- 8}{4} = 2$ . Forgetting the $2 a$ gives $4$ .

Question 3 · Modelled on 2022 P1 Q16(a), as an MC item

x^{2} + 6 x + 11 = (x + h)^{2} + k

for all real values of

x

, then

h + k =

$2$ .
$5$ .
$8$ .
$11$ .

Show answer

B. $x^{2} + 6 x + 11 = (x + 3)^{2} + 2$ , so $h = 3$ and $k = 2$ .

Question 4 · Modelled on 2014 P2 Q35

The least value of

3 x^{2} - 12 x + 7

$- 5$ .
$- 2$ .
$2$ .
$7$ .

Show answer

A. $3 x^{2} - 12 x + 7 = 3 (x - 2)^{2} - 5$ . The least value is the constant $- 5$ , not the $x$ -value $2$ at which it occurs.

Question 5 · Modelled on 2012 P2 Q24

The greatest value of

- 2 x^{2} + 8 x - 3

$2$ .
$3$ .
$5$ .
$11$ .

Show answer

C. $- 2 x^{2} + 8 x - 3 = - 2 (x - 2)^{2} + 5$ .

Question 6 · Modelled on 2015 P2 Q6

The

y

-intercept of the graph of

y = 4 x^{2} - 5 x + 9

$- 5$ .
$4$ .
$5$ .
$9$ .

Show answer

D. Put $x = 0$ . The $y$ -intercept is always the constant term.

Question 7 · Modelled on 2025 P1 Q18(b), as an MC item

The graph of

y = x^{2} - 4

is translated upwards by

3

units. The equation of the resulting graph is

$y = x^{2} - 7$ .
$y = (x + 3)^{2} - 4$ .
$y = x^{2} - 1$ .
$y = (x - 3)^{2} - 4$ .

Show answer

C. Upwards by $3$ adds $3$ to $y$ . Options C and D are horizontal translations.

Question 8 · Modelled on 2024 P1 Q19(b)

Let

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

. The graph of

y = f (x)

is translated rightwards by

1

unit. The equation of the resulting graph is

$y = x^{2} - 1$ .
$y = x^{2} + 1$ .
$y = x^{2} + 4 x + 3$ .
$y = x^{2} + 2 x + 1$ .

Show answer

A. A rightward translation gives $y = f (x - 1) = (x - 1)^{2} + 2 (x - 1) = x^{2} - 1$ . Replacing $x$ by $x + 1$ instead produces option C.

Question 9 · Modelled on 2019 P1 Q19(b)

The graph of

y = x^{2} - 6 x + 5

is reflected with respect to the

y

-axis. The equation of the resulting graph is

$y = - x^{2} + 6 x - 5$ .
$y = x^{2} + 6 x + 5$ .
$y = x^{2} + 6 x - 5$ .
$y = - x^{2} - 6 x - 5$ .

Show answer

B. Replace $x$ by $- x$ ; only the odd-power term changes sign. Option A is the reflection in the $x$ -axis.

Question 10 · Modelled on 2019 P1 Q19(b)

The graph of

y = 2 x^{2} + 3 x - 1

is reflected with respect to the

x

-axis. The equation of the resulting graph is

$y = 2 x^{2} - 3 x - 1$ .
$y = 2 x^{2} - 3 x + 1$ .
$y = - 2 x^{2} - 3 x + 1$ .
$y = - 2 x^{2} + 3 x + 1$ .

Show answer

C. Replace $y$ by $- y$ , so every term changes sign.

Question 11 · Modelled on 2017 P1 Q18(a)

If the graph of

y = x^{2} + k x + 16

touches the

x

-axis, then

k =

$4$ only.
$8$ only.
$4$ or $- 4$ .
$8$ or $- 8$ .

Show answer

D. $Δ = k^{2} - 64 = 0$ , which has two solutions. A graph touching the axis on the left and one touching it on the right both qualify.

Question 12 · Modelled on 2017 P1 Q18

The graph of

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

cuts the

x

-axis at two distinct points. The range of values of

c

$c < 1$ .
$c \leq 1$ .
$c > 1$ .
$c \geq 1$ .

Show answer

A. $Δ = 4 - 4 c > 0$ . The inequality is strict — at $c = 1$ the graph only touches the axis.

Question 13 · Modelled on 2012 P2 Q38

The figure shows the graph of

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

. Which of the following must be true?

$a > 0$ and $c > 0$
$a < 0$ and $c > 0$
$a < 0$ and $c < 0$
$a > 0$ and $c < 0$

Show answer

B. The graph opens downwards, so $a < 0$ ; it cuts the $y$ -axis above the origin, so $c > 0$ .

Question 14 · Modelled on 2023 P2 Q35

Let

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

, where

m

is a real constant. Which of the following must be true?
I. The

x

-coordinate of the vertex of the graph of

y = f (x)

m

.
II. The

y

-coordinate of the vertex of the graph of

y = f (x)

3 m^{2}

.
III. The graph of

y = f (x)

has no

x

-intercept when

m \neq = 0

I and II only
I and III only
II and III only
I, II and III

Show answer

D. $f (x) = 2 (x - m)^{2} + 3 m^{2}$ . The least value $3 m^{2}$ is positive whenever $m \neq = 0$ , so the graph stays above the $x$ -axis.

Question 15 · Modelled on 2014 P2 Q35

The graph of

y = f (x)

cuts the

x

-axis at

(2, 0)

and

(6, 0)

, where

f (x) = x^{2} + p x + q

. The least value of

f (x)

$- 8$ .
$- 4$ .
$4$ .
$12$ .

Show answer

B. $f (x) = (x - 2) (x - 6)$ and the vertex sits at $x = 4$ , the midpoint, giving $f (4) = - 4$ .

Question 16 · Modelled on 2014 P2 Q35

If the least value of

x^{2} - 4 x + k

3

, then

k =

$- 1$ .
$3$ .
$5$ .
$7$ .

Show answer

D. $x^{2} - 4 x + k = (x - 2)^{2} + k - 4$ , so $k - 4 = 3$ .

Question 17 · Modelled on 2014 P2 Q35, run backwards

The vertex of the graph of

y = f (x)

(- 1, 4)

and the graph passes through

(0, 2)

, where

f

is a quadratic function. Then

f (x) =

$- 2 x^{2} - 4 x + 2$ .
$2 x^{2} + 4 x + 2$ .
$- 2 x^{2} + 4 x + 2$ .
$- x^{2} - 2 x + 3$ .

Show answer

A. $f (x) = a (x + 1)^{2} + 4$ and $f (0) = 2$ give $a = - 2$ .

Question 18 · Modelled on 2017 P1 Q18(a)

The number of

x

-intercepts of the graph of

y = - 3 x^{2} + 6 x - 3

$0$ .
$1$ .
$2$ .
$3$ .

Show answer

B. $- 3 x^{2} + 6 x - 3 = - 3 (x - 1)^{2}$ , so the graph touches the $x$ -axis at $x = 1$ only.

Question 19 · Modelled on 2023 P2 Q35

The vertex of the graph of

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

lies on the

y

-axis and the graph passes through

(2, 1)

. Then

c =

$- 3$ .
$- 1$ .
$1$ .
$3$ .

Show answer

A. A vertex on the $y$ -axis means the axis of symmetry is $x = 0$ , so $b = 0$ ; then $4 + c = 1$ .

Question 20 · Modelled on 2017 P1 Q19, condensed to an MC item

The height of a stone above the ground,

h

m, at time

t

seconds is given by

h = - 4 t^{2} + 16 t

. The greatest height reached by the stone is

$2$ m.
$4$ m.
$16$ m.
$32$ m.

Show answer

C. $h = - 4 (t - 2)^{2} + 16$ . The greatest height is $16$ m, reached at $t = 2$ seconds — $2$ is the time, not the height.

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

Where this sits in the syllabus

Paper 2 asks about this topic in three recurring shapes: read something off the completed square, move a graph and say where it lands, or turn a statement about intercepts into a condition on the discriminant. Being fluent with all three is worth several marks a year.

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

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