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More about Graphs of Functions: Paper 1 Long Questions

Ten Paper 1 questions on what happens to the graph of y = f(x) under a translation, a reflection or an enlargement, for cubic, exponential, logarithmic and sine graphs. The set opens with single points and function values, moves on to intercepts and turning points, and ends with solving f(x) = k from a graph. Detailed solutions can be requested by email.

10 questions · 48 marks · about 55 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 — where one point goes under a translation and a reflection
Question 2 — transformations read from function values, not from a graph
Question 3 — a horizontal translation read off a sketch
Question 4 — an upward translation of an exponential graph
Question 5 — recognising f(x + 2) and f(−x) in factorised form
Question 6 — enlargement along the x-axis and along the y-axis, read from intercepts
Question 7 — solving f(x) = k and f(x) > k from a graph on a restricted domain
Question 8 — reflect then translate a logarithmic graph; where the two graphs meet
Question 9 — enlarging and translating y = sin x°; greatest value, least value and roots
Question 10 — a cubic translated leftwards; counting roots from turning points

Question 1 · Modelled on 2021 P1 Q10(b) — the vertical translation read through a single point, then extended to a horizontal shift and a reflection

The point $P (3, - 2)$ lies on the graph of $y = f (x)$ . Write down the coordinates of the point corresponding to $P$ on each of the following graphs.

(a) $y = f (x) + 5$

(b) $y = f (x - 4)$

(c) $y = - f (x)$

(3 marks)

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(a) $(3, 3)$ (b) $(7, - 2)$ (c) $(3, 2)$

Adding 5 outside f moves every point up. Replacing x by x − 4 moves every point 4 units to the right, not the left.

Question 2 · New sub-type, no direct past-paper model — the tabular view of a transformation, which the syllabus lists beside the graphical one

It is given that $f (- 1) = 2$ , $f (0) = 0$ , $f (1) = - 6$ and $f (2) = - 10$ . Let $g (x) = f (x + 1) - 2$ and $h (x) = - f (- x)$ .

(a) Find $g (0)$ and $g (1)$ .

(b) Find $h (1)$ and $h (- 2)$ .

(4 marks)

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(a) $g (0) = - 8$ , $g (1) = - 12$ (b) $h (1) = - 2$ , $h (- 2) = 10$

Work inside the bracket first: g(0) needs f(1), and h(−2) needs f(2).

Question 3 · Modelled on 2025 P1 Q18(b) — the same rightward translation, applied to a sketched graph instead of a quadratic formula

The figure shows the graph of $y = f (x)$ . The graph cuts the $x$ -axis at $x = - 2$ , touches the $x$ -axis at $x = 3$ and cuts the $y$ -axis at $y = 4$ . The graph of $y = g (x)$ is obtained by translating the graph of $y = f (x)$ rightwards by 2 units.

(a) Express $g (x)$ in terms of $f$ .

(b) Write down the $x$ -intercepts and the $y$ -intercept of the graph of $y = g (x)$ .

Graph of y = f(x) cutting the x-axis at −2, touching it at 3 and cutting the y-axis at 4

(4 marks)

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(a) $g (x) = f (x - 2)$ (b) $x$ -intercepts $0$ and $5$ ; $y$ -intercept $0$

The point (−2, 0) moves to (0, 0), so the new graph passes through the origin. Its y-intercept is 0, not 4.

Question 4 · Modelled on 2021 P1 Q10(b) — the graph of y = f(x) + 3, with an exponential function in place of a variation

Let $f (x) = 2^{x} - 8$ .

(a) Find the $x$ -intercept of the graph of $y = f (x)$ .

(b) The graph of $y = g (x)$ is obtained by translating the graph of $y = f (x)$ upwards by 6 units. Find the $x$ -intercept and the $y$ -intercept of the graph of $y = g (x)$ .

(4 marks)

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(a) $3$ (b) $x$ -intercept $1$ ; $y$ -intercept $- 1$

An upward translation changes every y-value, so the x-intercept has to be found again from g(x) = 0.

Question 5 · Modelled on 2016 P1 Q18(c) — the same ‘describe the geometric meaning’ task, with a cubic, and the second change disguised by a minus sign

Let $f (x) = (x - 1) (x - 4) (x - 6)$ .

(a) Under a transformation, $f (x)$ is changed to $(x + 1) (x - 2) (x - 4)$ . Describe the geometric meaning of the transformation.

(b) Under another transformation, $f (x)$ is changed to $- (x + 1) (x + 4) (x + 6)$ . Describe the geometric meaning of the transformation.

(4 marks)

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(a) the leftward translation of 2 units (b) the reflection with respect to the $y$ -axis

In (b), f(−x) = (−x − 1)(−x − 4)(−x − 6), and the three minus signs combine into one.

Question 6 · Modelled on 2017 P2 Q31 and 2024 P1 Q19(b) — the stretch read from intercepts, written out with both directions of enlargement

The figure shows the graph of $y = f (x)$ . The graph cuts the $x$ -axis at $x = - 4$ , $x = 6$ and $x = 8$ , and cuts the $y$ -axis at $y = 3$ . Let $g (x) = f (\frac{x}{2})$ and $h (x) = 3 f (x)$ .

(a) Describe the geometric meaning of transforming $f (x)$ to $g (x)$ .

(b) Write down the $x$ -intercepts and the $y$ -intercept of the graph of $y = g (x)$ .

(c) Write down the $x$ -intercepts and the $y$ -intercept of the graph of $y = h (x)$ .

Graph of y = f(x) cutting the x-axis at −4, 6 and 8 and the y-axis at 3

(5 marks)

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(a) the enlargement to 2 times of the original along the $x$ -axis (b) $x$ -intercepts $- 8, 12, 16$ ; $y$ -intercept $3$ (c) $x$ -intercepts $- 4, 6, 8$ ; $y$ -intercept $9$

An enlargement along the x-axis moves the x-intercepts and leaves the y-intercept alone. One along the y-axis does the opposite.

Question 7 · New sub-type, no direct past-paper model — solving an equation and an inequality from a given graph, as the syllabus asks

The figure shows the graph of $y = f (x)$ for $- 3 \leq x \leq 5$ . The graph starts at $(- 3, - 2)$ , rises to $(- 1, 4)$ , falls to $(3, - 2)$ and rises to $(5, 4)$ , turning only at $(- 1, 4)$ and $(3, - 2)$ .

(a) How many roots does the equation $f (x) = 0$ have for $- 3 \leq x \leq 5$ ?

(b) Find the range of values of $k$ such that the equation $f (x) = k$ has exactly three distinct roots for $- 3 \leq x \leq 5$ .

(c) Solve the inequality $f (x) > - 2$ for $- 3 \leq x \leq 5$ .

Graph of y = f(x) from (−3, −2) up to (−1, 4), down to (3, −2) and up to (5, 4)

(5 marks)

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(a) 3 (b) $- 2 < k < 4$ (c) $- 3 < x < 3$ or $3 < x \leq 5$

At k = 4 and k = −2 the line y = k passes through a turning point and an end point together, giving only two roots.

Question 8 · Modelled on 2019 P1 Q19(b) — the same reflect-then-translate instruction, applied to a logarithmic graph instead of a parabola

Let $f (x) = lo g_{2} (x + 4)$ . The graph of $y = g (x)$ is obtained by reflecting the graph of $y = f (x)$ with respect to the $x$ -axis and then translating the resulting graph upwards by 3 units.

(a) Express $g (x)$ in terms of $x$ .

(b) Find the $x$ -intercept of the graph of $y = g (x)$ .

(c) The graphs of $y = f (x)$ and $y = g (x)$ intersect at the point $P$ . Find the coordinates of $P$ .

(6 marks)

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(a) $g (x) = 3 - lo g_{2} (x + 4)$ (b) $4$ (c) $(22 - 4, \frac{3}{2})$

The two graphs are reflections of each other in the line y = 3/2, so they can only meet on that line.

Question 9 · Modelled on 2024 P1 Q19(b) — an enlargement along the y-axis and a vertical translation, applied to the sine graph

Let $f (x) = sin x^{\circ}$ . The graph of $y = g (x)$ is obtained by enlarging the graph of $y = f (x)$ to 3 times of the original along the $y$ -axis and then translating the resulting graph downwards by 1 unit.

(a) Express $g (x)$ in terms of $x$ . Hence write down the greatest value and the least value of $g (x)$ .

(b) How many roots does the equation $g (x) = 0$ have for $0 \leq x < 360$ ?

(c) Someone claims that the graph of $y = g (x)$ and the straight line $y = 2$ have exactly one common point for $0 \leq x < 360$ . Is the claim correct? Explain your answer.

(6 marks)

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(a) $g (x) = 3 sin x^{\circ} - 1$ ; greatest value $2$ , least value $- 4$ (b) 2 (c) Correct: $g (x) = 2$ only when $sin x^{\circ} = 1$ , that is $x = 90$ .

The line y = 2 is at the greatest value of g(x), so it only touches the graph at the top of the wave.

Question 10 · Modelled on 2016 P1 Q18(b) — a translated graph that touches the x-axis, with a cubic, and a claim about roots added

Let $f (x) = x^{3} - 6 x^{2} + 9 x$ . It is given that the graph of $y = f (x)$ has turning points at $(1, 4)$ and $(3, 0)$ .

(a) Factorize $f (x)$ . Hence explain why the graph of $y = f (x)$ touches the $x$ -axis at $(3, 0)$ .

(b) The graph of $y = g (x)$ is obtained by translating the graph of $y = f (x)$ leftwards by 3 units. Express $g (x)$ in the form $x^{3} + p x^{2} + q x + r$ , and write down the coordinates of the turning points of the graph of $y = g (x)$ .

(c) Someone claims that the equation $x^{3} + 3 x^{2} = 5$ has three distinct real roots. Do you agree? Explain your answer.

(7 marks)

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(a) $x (x - 3)^{2}$ ; the repeated factor $(x - 3)^{2}$ means $f (x)$ does not change sign at $x = 3$ , and $(3, 0)$ is a turning point on the $x$ -axis (b) $g (x) = x^{3} + 3 x^{2}$ ; $(- 2, 4)$ and $(0, 0)$ (c) Disagree: $5 > 4$ , so $y = 5$ meets the graph of $y = g (x)$ only once.

A translation moves the turning points but does not change their heights.

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

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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