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Cambridge IGCSE Maths (0580) · Topic 2: Algebra and graphs · 2.11

Cambridge IGCSE Maths 0580 Sketching Curves Questions

Sketching Curves questions with worked answers for Cambridge IGCSE Mathematics (0580). Each question says whether it is Core or Extended and whether it is for a calculator or non-calculator paper. Try each one before you open the answer.

Practise sketching curves →Sketching Curves notes

What the syllabus asks

Summarised in our own words: check the exact wording in the official 0580 syllabus.

Sketching Curves questions: Core

For everyone: Core students (Papers 1 and 3) and Extended students (Papers 2 and 4).

Question 1 Core non-calculator · easy

For the graph of \(y = (x - 2)(x + 5)\), write down
(a) the \(x\)-coordinates where it crosses the \(x\)-axis
(b) the coordinates of the point where it crosses the \(y\)-axis.

Show the worked answer
  1. (a) \(y = 0\) when \(x - 2 = 0\) or \(x + 5 = 0\).
  2. (b) \(x = 0\): \(y = (-2)(5) = -10\).

Answer: (a) \(2\) and \(-5\) (b) \((0, -10)\)

Question 2 Core non-calculator · easy · 3 marks

Sketch the graph of \(y = (x - 1)(x - 5)\). On your sketch, show the coordinates of the points where the graph meets the axes. Write down the equation of the line of symmetry.

Show the answer and mark scheme
  • B1 U-shaped curve through (1, 0) and (5, 0)
  • B1 (0, 5) marked
  • B1 x = 3

Answer: U-shape through \((1, 0), (5, 0), (0, 5)\); line of symmetry \(x = 3\)

Question 3 Core non-calculator · medium

The graph of \(y = x^2 - 6x + 5\) crosses the \(x\)-axis at \(x = 1\) and \(x = 5\).
Write down the equation of its line of symmetry.

Show the worked answer
  1. The line of symmetry is halfway between the roots: \(\frac{1 + 5}{2} = 3\).

Answer: \(x = 3\)

Question 4 Core non-calculator · medium

For \(y = 4 - x^2\), find
(a) the roots
(b) the \(y\)-intercept.

Show the worked answer
  1. (a) \(4 - x^2 = 0\), so \(x^2 = 4\) and \(x = \pm 2\).
  2. (b) \(x = 0\): \(y = 4\).

Answer: (a) \(x = -2\) and \(x = 2\) (b) \(4\)

Question 5 Core non-calculator · medium · 3 marks

Sketch the graph of \(y = (x - 4)(x + 2)\). On your sketch, show the coordinates of the points where the graph meets the axes, and write down the equation of its line of symmetry.

Show the answer and mark scheme
  • B1 U-shape through (−2, 0) and (4, 0)
  • B1 (0, −8) marked
  • B1 x = 1

Answer: U-shape through \((-2, 0), (4, 0), (0, -8)\); \(x = 1\)

Sketching Curves questions: Extended

Extended content, for Papers 2 and 4. Core students can skip these.

Question 6 Extended non-calculator · medium

By completing the square, find the coordinates of the turning point of \(y = x^2 + 8x + 11\). Say whether it is a maximum or a minimum.

Show the worked answer
  1. \(x^2 + 8x + 11 = (x + 4)^2 - 16 + 11 = (x + 4)^2 - 5\).
  2. The smallest value of \((x + 4)^2\) is \(0\), when \(x = -4\).

Answer: \((-4, -5)\), a minimum

Question 7 Extended non-calculator · medium

Write down the equations of the asymptotes of \(y = \frac{3}{x} + 2\).

Show the worked answer
  1. \(\frac{3}{x}\) is undefined at \(x = 0\): a vertical asymptote.
  2. As \(x\) gets large, \(\frac{3}{x} \to 0\) and \(y \to 2\): a horizontal asymptote.

Answer: \(x = 0\) and \(y = 2\)

Question 8 Extended non-calculator · hard

For the curve \(y = 2^x - 4\), find
(a) the \(y\)-intercept
(b) the \(x\)-intercept
(c) the equation of the asymptote.

Show the worked answer
  1. (a) \(x = 0\): \(1 - 4 = -3\).
  2. (b) \(2^x = 4\), so \(x = 2\).
  3. (c) \(2^x \to 0\) as \(x\) becomes very negative, so \(y \to -4\).

Answer: (a) \(-3\) (b) \(2\) (c) \(y = -4\)

Question 9 Extended non-calculator · hard

Find the points where the curve \(y = x^3 - 9x\) crosses the axes.

Show the worked answer
  1. \(x^3 - 9x = x(x - 3)(x + 3)\), so \(y = 0\) at \(x = -3, 0, 3\).
  2. At \(x = 0\), \(y = 0\).

Answer: \((-3, 0)\), \((0, 0)\) and \((3, 0)\)

Question 10 Extended non-calculator · hard · 3 marks

Sketch the graph of \(y = x^3 - 4x^2 + 3x\). Show the coordinates of the points where it meets the axes.

Show the answer and mark scheme
  • M1 x(x² − 4x + 3) or x(x − 1)(x − 3)
  • A1 (0, 0), (1, 0) and (3, 0)
  • B1 correct cubic shape: rising from bottom left to top right through the three roots

Answer: cubic through \((0, 0), (1, 0), (3, 0)\), rising from bottom left to top right

Examiner Insights: common mistakes in IGCSE Maths 0580, with fixes (our analysis of public sources) →

Every question on this page is original, written for IGCSE Math Revision rather than taken from Cambridge papers, and each answer was re-solved independently before publishing. Questions with a mark scheme come from our 0580 practice bank (M method, A accuracy, B independent marks).

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Questions students ask

Which Cambridge 0580 papers can test sketching curves?

Sketching Curves is on both tiers (syllabus 2.11), so it can come up on Core Papers 1 (non-calculator) and 3 (calculator), and Extended Papers 2 (non-calculator) and 4 (calculator). Extended candidates also need the Extended parts listed on this page. Half of each tier's marks are on a non-calculator paper, so practise the non-calculator questions without one.

Are these Cambridge past-paper questions?

No. Every question here was written for IGCSE Math Revision in the style of the 0580 papers, and each answer was checked independently before publishing. For real exam practice, use the Cambridge past papers linked from our 0580 past papers page.

Where can I practise more sketching curves questions?

In the 0580 practice for topic 2 (Algebra and graphs), where every question is marked and tagged Core or Extended. Practice is part of the IGCSE plan, with a free preview; the questions on this page are free.

More algebra and graphs questions