Cambridge IGCSE Mathematics 0580: Graphs of functions and sketching curves – Study Guide

This guide teaches 2.10 Graphs of functions and 2.11 Sketching curves from Topic 2 (Algebra and graphs) of the Cambridge IGCSE Mathematics 0580 syllabus for 2025, 2026 and 2027, for exams in the June and November series (and the March series in India). Both subtopics appear in the Core content (C2.10, C2.11) and the Extended content (E2.10, E2.11). Extended adds more function types, exponential growth and decay, asymptotes and turning points; that material is marked Extended only. Core candidates sit Paper 1 (non-calculator) and Paper 3 (calculator); Extended candidates sit Paper 2 (non-calculator) and Paper 4 (calculator).

For the whole topic, see the Algebra and graphs study guide and revision notes. When you have worked through this page, try the graphs of functions and sketching curves practice questions. Course hub: Cambridge IGCSE Mathematics. Printable checklist: 0580 checklist.

What this unit covers

Syllabus ref	What you must be able to do	Tier
C2.10.1	Construct tables of values; draw, recognise and interpret graphs of ax + b, ±x² + ax + b and a/x (x ≠ 0), with a and b integers	Core and Extended
C2.10.2 / E2.10.2	Solve associated equations graphically, including finding and interpreting roots (for example, where a line meets a curve)	Core and Extended
E2.10.1	Tables, graphs and interpretation for axⁿ (and sums of up to three such terms) with n = −2, −1, −½, 0, ½, 1, 2, 3, and for abˣ + c (a, c rational, b a positive integer)	Extended only
E2.10.3	Draw and interpret graphs of exponential growth and decay problems	Extended only
C2.11	Recognise, sketch and interpret linear and quadratic graphs; symmetry and roots required, turning points not	Core and Extended
E2.11	Recognise, sketch and interpret linear, quadratic, cubic, reciprocal and exponential graphs; turning points, roots, symmetry and vertical and horizontal asymptotes required; turning points of quadratics by completing the square	Extended only (beyond Core)

Keep two skills apart. Drawing a graph (2.10) means an accurate plot from a table on the grid you are given. Sketching a curve (2.11) means the right shape with the key features labelled, on axes you draw yourself, with no table.

2.10 Graphs of functions

Tables of values and plotting

1. Work out y for every x in the table. Put negative x-values in brackets: for x = −2, x² = (−2)² = 4, not −4.
2. Plot each point with a small, neat cross.
3. A straight line (ax + b) is drawn with a ruler. Every other graph is a smooth curve through all the points. Never join curve points with straight segments.
4. Do not draw the curve past the x-range of the table.

Check any point that breaks the pattern. A quadratic table is symmetrical about the line of symmetry, so one odd value is usually an arithmetic slip.

Shapes you must recognise (Core)

• y = ax + b: a straight line, gradient a, crossing the y-axis at (0, b).
• y = x² + ax + b: a U-shaped curve (a parabola), symmetrical about a vertical line.
• y = −x² + ax + b: an n-shaped curve, also symmetrical.
• y = a/x: two separate branches. With a > 0 they lie in the first and third quadrants; with a < 0, in the second and fourth. There is no point at x = 0, so the branches never touch the y-axis.

Worked example 1 (Core): drawing a quadratic and solving equations

Draw y = x² − 2x − 3 for −2 ≤ x ≤ 5. Use it to solve (a) x² − 2x − 3 = 0 and (b) x² − 4x − 1 = 0.

x    -2   -1    0    1    2    3    4    5
y     5    0   -3   -4   -3    0    5   12

Plot the points and draw a smooth U-shaped curve. The values are symmetrical about x = 1.

(a) The roots of x² − 2x − 3 = 0 are where the curve meets the x-axis (y = 0): x = −1 and x = 3.

(b) Rearrange so that the left-hand side matches the curve. Add 2x − 2 to both sides:

x^2 - 4x - 1 = 0
x^2 - 2x - 3 = 2x - 2

So draw the line y = 2x − 2 on the same grid. It crosses the curve at about x = −0.2 and x = 4.2. (The exact values are 2 ± √5, roughly −0.24 and 4.24; a reading from your graph is all that is expected.)

The method always works the same way: write the new equation as “curve = something”, and that something is the line to draw.

Worked example 2 (Core): a reciprocal graph meeting a line

Complete a table for y = 12/x and use it, with the line y = x + 1, to solve 12/x = x + 1.

x   -6  -4  -3  -2  -1    1   2   3   4   6
y   -2  -3  -4  -6  -12   12  6   4   3   2

Draw one smooth branch through the positive points and another through the negative points. Draw y = x + 1 with a ruler. The line crosses the curve twice, at x = −4 and x = 3. Check: 12/(−4) = −3 = −4 + 1, and 12/3 = 4 = 3 + 1.

Interpreting. A root is an x-value that makes the equation true. If a line misses a curve, that equation has no real solutions. If it touches the curve at one point, there is one solution.

Extended only: powers and sums of powers

E2.10 widens the list to axⁿ for n = −2, −1, −½, 0, ½, 1, 2, 3, and sums of up to three of these, such as y = x³ − 3x + 1 or y = x² + 4/x. Plot them exactly as before. Three facts help:

• n = 0 gives a constant, so y = 5x⁰ is the horizontal line y = 5.
• n = ½ means a square root, so x ≥ 0 only; n = −½ means a/√x, so x > 0 only.
• Any term with a negative power has no value at x = 0, so the graph has a gap there.

Worked example 3 (Extended): a cubic

The table for y = x³ − 3x + 1, −2 ≤ x ≤ 2:

x    -2   -1.5   -1   -0.5    0    0.5     1    1.5    2
y    -1   2.125   3   2.375   1  -0.375   -1  -0.125   3

The curve rises, turns at (−1, 3), falls, turns at (1, −1), then rises again. It meets the x-axis three times, so x³ − 3x + 1 = 0 has three roots, near x = −1.9, x = 0.35 and x = 1.5. To solve x³ − 3x = 0, rewrite it as x³ − 3x + 1 = 1 and draw the line y = 1. It meets the curve at x = 0 and near x = ±1.7.

Extended only: exponential graphs and growth or decay

A graph of y = abˣ + c, with b a positive integer greater than 1, is a smooth curve that gets steeper and steeper on one side and flattens towards the line y = c on the other. Growth and decay problems use the same idea with a multiplier: a multiplier greater than 1 means growth, between 0 and 1 means decay.

Worked example 4 (Extended, calculator)

A machine is bought for $8000. Its value after t years is V = 8000 × 0.75ᵗ.

t   0      1      2      3      4      5      6
V   8000   6000   4500   3375   2531   1898   1424

(Values to the nearest dollar.) Plot the points and draw a smooth decreasing curve.

• The value halves ($4000) after about 2.4 years; read across from V = 4000 to the curve, then down.
• It halves again ($2000) after about 4.8 years. The halving time is constant, which is a feature of exponential decay.
• The curve never reaches 0. The machine keeps some value, however small.

2.11 Sketching curves

A sketch needs the correct shape, the points where it crosses the axes labelled with coordinates, and any line of symmetry, turning point or asymptote the question requires. It does not need a scale or a table.

Core: linear and quadratic

• Line. Find the two intercepts. For 2x + 5y = 10: x = 0 gives y = 2, and y = 0 gives x = 5. Join (0, 2) and (5, 0) and extend.
• Quadratic. Decide U or n from the sign of x². Factorise to find the roots. The y-intercept is the constant term. The line of symmetry is halfway between the roots. Core does not need the turning point.

Worked example 5: sketching a quadratic

Sketch y = x² + 2x − 15.

x^2 coefficient positive  ->  U-shape
Roots: (x + 5)(x - 3) = 0  ->  x = -5, x = 3
y-intercept: x = 0  ->  y = -15
Symmetry: x = (-5 + 3) / 2 = -1

Draw a U through (−5, 0), (0, −15) and (3, 0), symmetrical about x = −1.

Extended only: the turning point by completing the square.

x^2 + 2x - 15 = (x + 1)^2 - 1 - 15 = (x + 1)^2 - 16

(x + 1)² is never negative, so the least value of y is −16, when x = −1. The minimum point is (−1, −16). Generally, y = (x + p)² + q has turning point (−p, q). For an n-shape, the turning point is a maximum.

Extended only: cubic, reciprocal and exponential sketches

The syllabus lists the forms you will meet.

Form	Key features to show
y = ax³ + b	Same shape as y = x³, moved up by b. Crosses the y-axis at (0, b). One root. If a < 0 the curve falls from left to right.
y = ax³ + bx² + cx	Factorise as x(ax² + bx + c): one root is always x = 0. Up to three roots and two turning points.
y = a/x + b	Vertical asymptote x = 0. Horizontal asymptote y = b. Two branches.
y = arˣ + b	Horizontal asymptote y = b. Crosses the y-axis at (0, a + b). Growth if r > 1, decay if 0 < r < 1.

Worked example 6 (Extended): three quick sketches

(a) y = x³ − 4x² + 3x. Factorise: x(x − 1)(x − 3). Roots x = 0, 1, 3. The x³ coefficient is positive, so the curve comes up from the bottom left, crosses at 0, turns, crosses at 1, turns, crosses at 3 and rises.

(b) y = 2/x + 1. Asymptotes x = 0 and y = 1. Root: 2/x + 1 = 0 gives x = −2, so it crosses at (−2, 0). The branches sit in the top-right and bottom-left of the point where the asymptotes cross.

(c) y = 3 × 2ˣ − 6. Asymptote y = −6, drawn as a dashed line. y-intercept 3 × 1 − 6 = −3. Root: 3 × 2ˣ = 6, so 2ˣ = 2, x = 1. The curve flattens towards y = −6 on the left and climbs steeply on the right.

Also note: y = −x³ + 8 has a single root at x = 2 (because 2³ = 8) and y-intercept 8, and falls from left to right.

Common errors

• Joining plotted points of a curve with a ruler, or drawing a flat bottom on a parabola.
• Working out (−3)² as −9 in a table; a slip like this shows up as a value that breaks the symmetry.
• Joining the two branches of a reciprocal graph across x = 0.
• Drawing the wrong line for a graphical solution because the equation was not rearranged to “curve = line” first. Check by subtracting: curve minus line must give the equation you were asked to solve.
• Reading the y-value instead of the x-value at an intersection when the question asks you to solve for x.
• Letting a sketch of an exponential or reciprocal curve cross or touch its asymptote (Extended).
• Giving the turning point of (x + 1)² − 16 as (1, −16): the sign of p changes (Extended).
• Leaving a root or intercept unlabelled on a sketch. The labels carry the marks.

Next steps

• Graphs of functions and sketching curves practice questions
• Algebra and graphs practice questions and Extended algebra practice
• Free 10-minute diagnostics: Core and Extended

Official syllabus

Cambridge IGCSE Mathematics 0580 syllabus for 2025, 2026 and 2027 (Version 3), Cambridge University Press & Assessment: subject content C2.10, C2.11, E2.10 and E2.11.