Mathematics 0580 · syllabus C2.10, C2.11, E2.10, E2.11
Cambridge IGCSE Mathematics 0580 — Graphs of functions topical past papers
Completing tables of values, plotting quadratic, cubic, reciprocal and exponential curves, sketching their shapes, and solving equations from the graph.
- Questions
- 212
- Years
- 2009–2026
- With mark scheme
- 93%
- Questions 2022–2026
- 65
Anyone can browse these pages. Opening the questions and their mark schemes takes a free Momentum account.
What Graphs of functions covers
This topic is about drawing a graph accurately from its equation and then using it. You complete a table of values, plot the points, and join them with a smooth curve for a non-linear function or a ruled line for a linear one. Core candidates draw linear graphs, quadratics of the form ±x² + ax + b and reciprocal graphs a/x; Extended candidates draw sums of up to three terms of the form axⁿ, with n = −2, −1, −½, 0, ½, 1, 2 or 3, and exponential graphs such as y = 2ˣ + 1.
Solving equations graphically is the second half. The x-coordinates where a curve crosses the x-axis solve f(x) = 0, and where it meets a horizontal line y = k they solve f(x) = k. Where it meets another line, they solve the equation formed by setting the two expressions equal, so the line to draw is found by rearranging the given equation into 'curve = line'. Sketching is a separate skill: recognising the shape of each type of graph and marking its key features without plotting. Core sketches linear and quadratic graphs with their roots and line of symmetry; Extended adds cubic, reciprocal and exponential graphs, with turning points and asymptotes.
Plotting and graphical solutions are Core content, so they can be set on all four papers, with a wider range of curves on Papers 2 and 4. Extended candidates also draw tangents to estimate gradients, which leads into differentiation, and draw graphs of exponential growth and decay. The topic connects to quadratic and simultaneous equations, whose solutions are the crossing points read from the graph.
In the syllabus
The headings this topic has in the Cambridge International 2025–2027 syllabus, as the board prints them.
- C2.10 Graphs of functions
- C2.11 Sketching curves
- E2.10 Graphs of functions
- E2.11 Sketching curves
What you should be able to do
- Complete a table of values, including negative and fractional x-values
- Plot points accurately and draw a smooth curve through them
- Solve f(x) = 0 or f(x) = k by reading where the curve meets the x-axis or a horizontal line
- Find the line to draw to solve a rearranged equation, and read off the solutions
- Recognise and sketch linear and quadratic graphs, and on the Extended route cubic, reciprocal and exponential graphs
- Draw a tangent to estimate the gradient of a curve at a point (Extended)
Key ideas
- The solutions of f(x) = 0 are the x-coordinates where y = f(x) crosses the x-axis
- To solve f(x) = g(x), draw y = g(x) on the graph of y = f(x) and read the x-coordinates of the crossing points
- y = ax² + bx + c is a ∪ shape when a > 0 and a ∩ shape when a < 0, symmetrical about the vertical line through its turning point
- y = k/x has the axes as its asymptotes; y = abˣ + c has the horizontal asymptote y = c
- Each point is plotted, and each value read, to an accuracy of half a small grid square
How Graphs of functions has been examined
From 2022 to 2026, Graphs of functions was on Paper 3 · Calculator (Core) in 12 of 14 exam series, worth about 9% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 1 · Non-calculator (Core) | 7 | 3% | 5 of 14 |
| Paper 2 · Non-calculator (Extended) | 14 | 3% | 8 of 14 |
| Paper 3 · Calculator (Core) | 23 | 9% | 12 of 14 |
| Paper 4 · Calculator (Extended) | 21 | 6% | 11 of 14 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 10.
Filed in Momentum under: Plotting Curves, Graphical Solution of Equations.
A worked example
The graph of y = x² − 2x − 3 has been drawn for −2 ≤ x ≤ 4. Find the equation of the straight line to draw on it to solve x² − 4x + 1 = 0, and estimate the solutions.
- The graph shows y = x² − 2x − 3, so rewrite the new equation with that expression on the left.
- Adding 2x − 4 to both sides of x² − 4x + 1 = 0 gives x² − 2x − 3 = 2x − 4.
- So draw the line y = 2x − 4: where it crosses the curve, x² − 2x − 3 = 2x − 4, which is the equation to solve.
- The line passes through (0, −4) and (3, 2), and it crosses the curve at about x = 0.3 and x = 3.7.
- Check with the quadratic formula: x = (4 ± √12)/2 = 2 ± √3, which is 0.27 or 3.73.
Answer: Draw y = 2x − 4; the solutions are about x = 0.3 and x = 3.7
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Squaring a negative x wrongly in a table of values: for x = −3, x² is 9 and −x² is −9, so check the signs before plotting.
- Joining the points of a curve with ruled straight segments: a non-linear graph needs a smooth curve through every point.
- Drawing the wrong line for a graphical solution: rearrange the given equation so that one side is exactly the plotted function, and the other side is the line to draw.
- Giving y-values when the solutions of an equation are the x-coordinates of the crossing points.
- Joining the two branches of a reciprocal graph across x = 0: the graph is not defined there, so the branches stay separate.
Exam technique
- Recalculate any point that looks out of line with the rest of the curve; one wrong value spoils the whole graph.
- To find the line to draw, write the plotted equation above the given equation and subtract to see what is left.
- Read crossing points to within half a small square, and give every solution in the range drawn.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.