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More about Graphs of Functions: Paper 1 Challenge

This is the Challenge tier: the questions state a condition and leave the method to you, and several ask you to test a claim. Work the standard Paper 1 set first. The ideas here include recovering a translation from one point, reflection in the line y = 3, the order of two steps, and how an enlargement changes an area. Detailed solutions can be requested by email.

10 questions · 56 marks · about 70 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 — points a graph shares with its reflections
Question 2 — recovering a translation from where one point lands
Question 3 — the same exponential graph from a translation and from an enlargement
Question 4 — choosing a horizontal translation so the graph passes through the origin
Question 5 — number of roots after a translation inside f and a sign outside
Question 6 — reflection in the horizontal line y = 3
Question 7 — order of operations: a translation and a reduction along the x-axis
Question 8 — how enlargements along both axes change an area
Question 9 — recovering a and b from the greatest and least values; roots at the ends of the range
Question 10 — every horizontal translation that sends a cubic through the origin

Question 1 · New sub-type, no direct past-paper model — the points a reflection leaves in place, found by solving f(x) = f(−x) and f(x) = −f(x)

Technique not in the standard set: points left fixed by a reflection.

Let $f (x) = x^{3} - 4 x + 3$ .

(a) Find the coordinates of all the points that lie on both the graph of $y = f (x)$ and the graph of $y = f (- x)$ .

(b) How many points lie on both the graph of $y = f (x)$ and the graph of $y = - f (x)$ ? Explain your answer.

(4 marks)

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(a) $(- 2, 3)$ , $(0, 3)$ , $(2, 3)$ (b) 3: a common point needs $f (x) = 0$ , and $x^{3} - 4 x + 3 = (x - 1) (x^{2} + x - 3) = 0$ has three distinct real roots.

A common point of y = f(x) and y = −f(x) needs f(x) = −f(x), so f(x) = 0. Those points are exactly the x-intercepts.

Question 2 · Modelled on 2025 P1 Q18(b), run backwards — the translation is not given; it has to be read from where the maximum point lands

Condition stated, method not named: the student must recover the translation from one image point.

The graph of $y = f (x)$ has a maximum point $A (1, 5)$ and passes through the point $B (4, - 1)$ . The graph of $y = g (x)$ is obtained by translating the graph of $y = f (x)$ , and the maximum point of the graph of $y = g (x)$ corresponding to $A$ is $(- 2, 7)$ .

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

(b) Find the coordinates of the point corresponding to $B$ on the graph of $y = g (x)$ .

(c) Someone claims that the graph of $y = g (x)$ can also be obtained by reflecting the graph of $y = f (x)$ with respect to the $y$ -axis and then translating the resulting graph vertically. Is the claim correct? Explain your answer.

(5 marks)

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(a) $g (x) = f (x + 3) + 2$ (b) $(1, 1)$ (c) Not correct: after the reflection the maximum point is at $x = - 1$ , and a vertical translation keeps it at $x = - 1$ , not $x = - 2$ .

A vertical translation can fix the height of the maximum point, never its x-coordinate.

Question 3 · Modelled on 2024 P1 Q19(b) — describing a transformation, where one exponential change has two correct descriptions and a third that fails

New idea: two different transformations giving the same graph, and a proof that a third cannot.

Let $f (x) = 2^{x}$ and $g (x) = 8 \cdot 2^{x}$ .

(a) The graph of $y = g (x)$ can be obtained by translating the graph of $y = f (x)$ leftwards by $h$ units. Find $h$ .

(b) Describe another single transformation that changes the graph of $y = f (x)$ to the graph of $y = g (x)$ .

(c) Someone claims that the graph of $y = g (x)$ can also be obtained by translating the graph of $y = f (x)$ vertically. Is the claim correct? Explain your answer.

(5 marks)

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(a) $h = 3$ (b) the enlargement to 8 times of the original along the $y$ -axis (c) Not correct: $g (x) - f (x) = 7 \cdot 2^{x}$ is not a constant (7 at $x = 0$ , 14 at $x = 1$ ).

A vertical translation adds the same number to every y-value. Here the gap between the graphs keeps growing.

Question 4 · Modelled on 2016 P1 Q18(b) — a translation fixed by a condition on the graph, made horizontal and given three candidate answers

Condition stated, not the translation: three candidates appear and a second condition picks one.

Let $f (x) = x^{3} - 7 x + 6$ .

(a) Factorize $f (x)$ .

(b) The graph of $y = f (x + k)$ passes through the origin, where $k$ is a constant. Find all possible values of $k$ .

(c) It is further given that the other two $x$ -intercepts of the graph of $y = f (x + k)$ are both negative. Find $k$ .

(5 marks)

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(a) $(x - 1) (x - 2) (x + 3)$ (b) $- 3, 1, 2$ (c) $k = 2$

The x-intercepts of y = f(x + k) are those of y = f(x) moved k units: 1 − k, 2 − k and −3 − k.

Question 5 · New sub-type, no direct past-paper model — the equation f(x) = k read from turning points, then disguised by a translation and a sign

New technique: counting roots of a transformed equation without redrawing the graph.

The figure shows the graph of $y = f (x)$ , which has turning points at $(- 2, 5)$ and $(2, - 3)$ .

(a) How many real roots does the equation $f (x) = 0$ have?

(b) Find the range of values of $k$ such that the equation $f (x) = k$ has exactly one real root.

(c) Find the range of values of $c$ such that the equation $f (x - 4) + c = 0$ has three distinct real roots.

Graph of y = f(x) with turning points (−2, 5) and (2, −3)

(5 marks)

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(a) 3 (b) $k < - 3$ or $k > 5$ (c) $- 5 < c < 3$

f(x − 4) + c = 0 means f(x − 4) = −c. The shift by 4 changes nothing; the sign of c changes everything.

Question 6 · Modelled on 2015 P1 Q18(c) — the y = 2 − f(x) idea, with the mirror line stated and the formula left to be derived

Technique not in the standard set: reflection in a horizontal line other than the x-axis.

Let $f (x) = lo g_{3} x$ . The graph of $y = g (x)$ is the reflection of the graph of $y = f (x)$ in the straight line $y = 3$ .

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

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

(c) Find the coordinates of the point of intersection of the graph of $y = f (x)$ and the graph of $y = g (x)$ .

(6 marks)

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(a) $g (x) = 6 - f (x)$ : the mid-point of $(x, f (x))$ and $(x, g (x))$ lies on $y = 3$ (b) $729$ (c) $(27, 3)$

Reflecting in y = 3 keeps x and sends a height y to 6 − y, because 3 is halfway between y and 6 − y.

Question 7 · Modelled on 2024 P1 Q19(b) — two claims about the same pair of steps in different orders, only one of which survives

Trap: the same two steps in the other order give a clean, plausible and wrong graph.

Let $f (x) = x^{3}$ .

(a) Someone claims that the graph of $y = (2 x + 4)^{3}$ can be obtained by translating the graph of $y = f (x)$ leftwards by 4 units and then reducing the resulting graph to $\frac{1}{2}$ of the original along the $x$ -axis. Is the claim correct? Explain your answer.

(b) Another person claims that the graph of $y = (2 x + 4)^{3}$ can be obtained by reducing the graph of $y = f (x)$ to $\frac{1}{2}$ of the original along the $x$ -axis and then translating the resulting graph leftwards by 4 units. Is the claim correct? Explain your answer.

(c) Find the $x$ -intercept of the graph of $y = (2 x + 4)^{3}$ .

(6 marks)

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(a) Correct: $f (x + 4)$ then $x \to 2 x$ gives $(2 x + 4)^{3}$ (b) Not correct: $f (2 x)$ then $x \to x + 4$ gives $(2 x + 8)^{3}$ (c) $- 2$

Each step replaces x in what you already have. Reduce first, and the later shift gets multiplied by 2.

Question 8 · New sub-type, no direct past-paper model — enlargements along both axes combined, and a claim about area tested

New idea: enlargements along both axes scale an area by the product of the two factors.

The figure shows the graph of $y = f (x)$ for $- 2 \leq x \leq 4$ . It consists of the two line segments $A B$ and $B C$ , where the coordinates of $A$ , $B$ and $C$ are $(- 2, 0)$ , $(1, 3)$ and $(4, 0)$ respectively.

(a) Find the area of the region bounded by the graph of $y = f (x)$ and the $x$ -axis.

(b) Let $g (x) = 2 f (\frac{x}{2})$ . Write down the coordinates of the points corresponding to $A$ , $B$ and $C$ on the graph of $y = g (x)$ . Hence find the area of the region bounded by the graph of $y = g (x)$ and the $x$ -axis.

(c) Someone claims that for any positive constant $m$ , the area of the region bounded by the graph of $y = m f (\frac{x}{m})$ and the $x$ -axis is $9 m$ . Is the claim correct? Explain your answer.

Graph of y = f(x) made of line segments from A(−2, 0) to B(1, 3) to C(4, 0)

(6 marks)

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(a) $9$ (b) $(- 4, 0)$ , $(2, 6)$ , $(8, 0)$ ; area $36$ (c) Not correct: the base becomes $6 m$ and the height $3 m$ , so the area is $9 m^{2}$ .

Both the base and the height are enlarged m times, so the area grows by m², not m.

Question 9 · Modelled on 2024 P1 Q19(b) — the transformation recovered from its effect on the greatest and least values of a sine graph

Condition stated (greatest and least values), transformation to be recovered; trap at the ends of the range.

Let $f (x) = sin x^{\circ}$ . The graph of $y = g (x)$ is obtained by enlarging the graph of $y = f (x)$ along the $y$ -axis and then translating the resulting graph vertically, so that $g (x) = a sin x^{\circ} + b$ , where $a$ and $b$ are constants and $a > 0$ . The greatest value and the least value of $g (x)$ are $7$ and $- 1$ respectively.

(a) Find $a$ and $b$ .

(b) How many roots does the equation $g (x) = 5$ have for $0 \leq x < 360$ ?

(c) Find all values of $k$ such that the equation $g (x) = k$ has exactly one root for $0 \leq x < 360$ .

(7 marks)

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(a) $a = 4, b = 3$ (b) 2 (c) $k = 7$ or $k = - 1$

Every value strictly between −1 and 7 is reached twice in one cycle. Only the greatest and least values are reached once.

Question 10 · Modelled on 2016 P1 Q18(b) — a translation fixed by a condition, made to have two answers, then a claim about roots

Condition stated, not the translation; two candidate graphs, and a claim decided at a turning point.

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

(a) Factorize $p (x)$ .

(b) The graph of $y = q (x)$ is obtained by translating the graph of $y = p (x)$ horizontally so that it passes through the origin. Find all possible expressions for $q (x)$ , giving each in factorized form.

(c) For one of the expressions in (b), the graph of $y = q (x)$ touches the $x$ -axis at the origin. For this $q (x)$ , someone claims that the equation $q (x) = 4$ has exactly two distinct real roots. Is the claim correct? Explain your answer.

(7 marks)

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(a) $(x - 1)^{2} (x + 2)$ (b) $q (x) = x^{2} (x + 3)$ or $q (x) = x (x - 3)^{2}$ (c) Correct: $q (x) = x^{2} (x + 3)$ has turning points $(- 2, 4)$ and $(0, 0)$ , and $x^{3} + 3 x^{2} - 4 = (x - 1) (x + 2)^{2} = 0$ gives only $x = 1$ and $x = - 2$ .

y = 4 is the height of a turning point, so the line touches the graph there and crosses it once more.

Continue practising this topic: Try the Paper 1 long questions 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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