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More about Graphs of Functions: Paper 2 Multiple-Choice

Ten multiple-choice questions at Paper 2 pace. Each one turns on reading a transformation correctly: which way a graph moves, which coordinate changes, and what stays fixed. Detailed solutions can be requested by email.

10 questions · about 15 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 — a point under a horizontal translation
Question 2 — the x-intercept created by a downward translation
Question 3 — reflect in the y-axis, then translate: the order matters
Question 4 — identifying a translation from a pair of graphs
Question 5 — working back from the roots to the translation
Question 6 — features of y = 1 − 3ˣ
Question 7 — solving f(x) ≥ 2 from a graph
Question 8 — a horizontal translation does not change the number of roots
Question 9 — two ways to get y = 8x³ from y = x³
Question 10 — intercepts after an enlargement along the x-axis

Question 1 · Modelled on 2012 P2 Q38 — the f(x − 2) + 1 idea reduced to one point and one shift, in the direction students most often reverse

The point $(2, 5)$ lies on the graph of $y = f (x)$ . Which of the following points must lie on the graph of $y = f (x + 3)$ ?

$(5, 5)$
$(- 1, 5)$
$(2, 8)$
$(2, 2)$

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B. f(x + 3) shifts the graph 3 units to the left. Check: at x = −1, f(−1 + 3) = f(2) = 5.

Question 2 · Modelled on 2021 P1 Q10(b), as an MC item — a vertical translation that creates a new x-intercept

The graph of $y = f (x)$ passes through the points $(1, 6)$ and $(5, 4)$ . Let $g (x) = f (x) - 4$ . Which of the following must be an $x$ -intercept of the graph of $y = g (x)$ ?

$5$
$1$
$9$
$- 4$

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A. g(5) = f(5) − 4 = 0, so the graph of y = g(x) cuts the x-axis at x = 5.

Question 3 · New sub-type — a reflection in the y-axis combined with a horizontal translation, where the order of the two steps changes the answer

The graph of $y = f (x)$ is reflected with respect to the $y$ -axis, and the resulting graph is then translated rightwards by 2 units. The equation of the new graph is

$y = f (- x - 2)$ .
$y = f (2 - x)$ .
$y = - f (x - 2)$ .
$y = f (x + 2)$ .

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B. Reflecting gives y = f(−x). Translating rightwards replaces x by x − 2, giving f(−(x − 2)) = f(2 − x).

Question 4 · Modelled on 2018 P2 Q31 — matching a transformed graph to its equation, with a cubic and a translation in two directions

The figure shows the graph of $y = f (x)$ and the graph of $y = g (x)$ . The graph of $y = g (x)$ is obtained by translating the graph of $y = f (x)$ , and the point $(- 1, 2)$ on the graph of $y = f (x)$ corresponds to the point $(2, 3)$ on the graph of $y = g (x)$ . Then $g (x) =$

Graphs of y = f(x) (solid) and y = g(x) (dashed); f turns at (−1, 2), g turns at (2, 3)

$f (x - 3) + 1$ .
$f (x + 3) + 1$ .
$f (x - 3) - 1$ .
$f (x + 3) - 1$ .

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A. The graph moves 3 units right and 1 unit up. Moving right means replacing x by x − 3.

Question 5 · New sub-type, no direct past-paper model — the roots of f(x) = 0 given, the translation to be found

It is given that the roots of the equation $f (x) = 0$ are $- 6$ and $4$ only. Which of the following equations has roots $- 3$ and $7$ only?

$f (x + 3) = 0$
$f (x - 3) = 0$
$f (x) - 3 = 0$
$f (x) + 3 = 0$

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B. The roots move 3 units to the right, which is the graph of y = f(x − 3). Check: f(−3 − 3) = f(−6) = 0.

Question 6 · Modelled on 2014 P2 Q38 and 2026 P2 Q35 — the y = 1 − f(x) transformation, asked as statements about an exponential graph

Which of the following statements about the graph of $y = 1 - 3^{x}$ are true?

I. The graph passes through the origin.
II. $y$ decreases when $x$ increases.
III. The $x$ -intercept of the graph is $1$ .

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

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A. y = 1 − 3ˣ is y = 3ˣ reflected in the x-axis and moved up 1 unit. It meets the x-axis where 3ˣ = 1, that is at x = 0, not x = 1.

Question 7 · New sub-type, no direct past-paper model — reading the solution of an inequality from where a graph lies above a line

The figure shows the graph of $y = f (x)$ and the straight line $y = 2$ . They intersect at the points where $x = - 2$ , $x = 1$ and $x = 4$ . The solution of $f (x) \geq 2$ is

Graph of y = f(x) and the line y = 2 meeting at x = −2, 1 and 4

$x \geq 4$ .
$x \leq - 2$ or $1 \leq x \leq 4$ .
$- 2 \leq x \leq 4$ .
$- 2 \leq x \leq 1$ or $x \geq 4$ .

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D. Read off the x-values where the curve is on or above the line y = 2: from −2 to 1, and from 4 onwards.

Question 8 · New sub-type, no direct past-paper model — counting roots from turning points, with a translation that changes nothing

The figure shows the graph of $y = f (x)$ , which has turning points at $(- 1, 3)$ and $(2, - 1)$ . How many distinct real roots does the equation $f (x - 5) = 1$ have?

Graph of y = f(x) with turning points (−1, 3) and (2, −1)

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D. y = f(x − 5) is the same curve moved 5 units right. Its turning points stay at heights 3 and −1, and 1 lies between them.

Question 9 · Modelled on 2024 P1 Q19(b), as an MC item — an enlargement along the y-axis and a reduction along the x-axis giving the same graph

Which of the following transformations change the graph of $y = x^{3}$ to the graph of $y = 8 x^{3}$ ?

I. Enlarging the graph to 8 times of the original along the $y$ -axis
II. Reducing the graph to $\frac{1}{2}$ of the original along the $x$ -axis
III. Reducing the graph to $\frac{1}{8}$ of the original along the $x$ -axis

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

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C. Reducing to 1/2 along the x-axis replaces x by 2x, and (2x)³ = 8x³. Reducing to 1/8 gives (8x)³ = 512x³.

Question 10 · Modelled on 2017 P2 Q31 — the same stretch, with the options written as intercepts instead of sketches

The figure shows the graph of $y = f (x)$ . The graph cuts the $x$ -axis at $x = - 2$ , $x = 6$ and $x = 9$ , and cuts the $y$ -axis at $y = 4$ . Let $g (x) = f (\frac{x}{2})$ . Which of the following is true about the graph of $y = g (x)$ ?

Graph of y = f(x) cutting the x-axis at −2, 6 and 9 and the y-axis at 4

The $x$ -intercepts are $- 1, 3, 4.5$ and the $y$ -intercept is $4$ .
The $x$ -intercepts are $- 4, 12, 18$ and the $y$ -intercept is $8$ .
The $x$ -intercepts are $- 4, 12, 18$ and the $y$ -intercept is $4$ .
The $x$ -intercepts are $- 1, 3, 4.5$ and the $y$ -intercept is $2$ .

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C. Replacing x by x/2 doubles every x-coordinate. The y-intercept is g(0) = f(0) = 4, unchanged.

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

More about Graphs of Functions is senior Number and Algebra content. From 2012 to 2026, Paper 2 set a graph-transformation question in at least six papers, and Paper 1 has several times placed a translation, reflection or enlargement inside a longer question on a quadratic function, most recently in 2024 and 2025. The marks are usually lost on direction: f(x − 4) moves a graph right, not left, and in a two-step transformation each step acts on the equation you already have.

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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