Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who need to sketch common function graphs, read values from curves and handle simple transformations without guessing.
What query it owns: how to sketch and interpret graphs of functions in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the revision-guide angle, while Tutopiya’s Graphs of Functions subtopic page owns the learning resource and the free Graphs of Functions quiz owns the practice.

Graphs of functions turn algebra into pictures — and Cambridge IGCSE Mathematics (0580/0607) tests both directions. You must sketch curves from equations and read solutions from graphs. This guide covers the standard graphs you need, how to apply transformations, and the exact command words that appear on past papers.

Key takeaways

• Every point on a function graph satisfies y = f(x) — the x-value is the input, the y-value is the output.
• Know the shapes of linear, quadratic, cubic, reciprocal and exponential graphs cold.
• f(x) + k shifts the graph up by k; f(x + k) shifts left by k.
• To solve f(x) = g(x) graphically, find where the two curves intersect.

What are graphs of functions in Cambridge IGCSE Maths?

The graph of a function y = f(x) is the set of all points (x, f(x)) plotted on coordinate axes. Sketching graphs is not art — examiners want correct shape, labelled intercepts and key points. Reading graphs means using a drawn curve to estimate roots, gradients or intersection points. Cambridge IGCSE Extended also tests reflections and translations of known graphs.

Read the full notes on Tutopiya’s Graphs of Functions subtopic page before you attempt questions.

The standard graphs you must recognise

Function type	Typical equation	Key shape features
Linear	y = mx + c	Straight line; gradient m; y-intercept c
Quadratic	y = ax² + bx + c	Parabola; U or ∩ depending on sign of a
Cubic	y = x³ or y = ax³ + …	S-shape through the origin (basic cubic)
Reciprocal	y = k/x	Two branches; asymptotes on both axes
Exponential	y = aˣ	Rapid growth or decay; passes through (0, 1) if aˣ

How to sketch a function graph — step by step

1. Identify the function type (linear, quadratic, cubic, etc.).
2. Find intercepts — set x = 0 for the y-intercept; set y = 0 for x-intercepts if they exist.
3. Mark the vertex or turning points for quadratics and cubics.
4. Note asymptotes for reciprocal graphs (where the denominator would be zero).
5. Draw a smooth curve through the points; label axes and key coordinates.

Test your sketching with the free Graphs of Functions quiz.

Graph transformations: what moves where

Transformation	Effect on the graph
y = f(x) + k	Shift up k units
y = f(x) − k	Shift down k units
y = f(x + k)	Shift left k units
y = f(x − k)	Shift right k units
y = −f(x)	Reflect in the x-axis
y = f(−x)	Reflect in the y-axis

Graphs of functions in past-paper wording: command words that matter

Command word / phrase	What the question wants	Typical stem
Sketch the graph	Correct shape with labelled features	”Sketch the graph of y = x² − 4.”
Draw	Accurate plot on given axes	”Draw the graph of y = 2ˣ for 0 ≤ x ≤ 3.”
Write down	State a value read from the graph	”Write down the coordinates of the point where the graph crosses the y-axis.”
Use your graph to solve	Read roots or intersections	”Use your graph to solve x² − 4 = 0.”
Describe the transformation	Explain how one graph maps to another	”Describe fully the transformation that maps y = x² onto y = x² + 3.”
Estimate	Read an approximate value	”Estimate the gradient of the curve at x = 2.”

Worked exam-style stems (how to answer the wording)

1. “Sketch the graph of y = 1/x for x ≠ 0.” Two branches in quadrants 1 and 3 (for positive k); asymptotes at x = 0 and y = 0. Reward: both branches, asymptotes indicated, no line drawn on the axes.

2. “The graph of y = f(x) passes through (2, 5). Write down the coordinates of the point on the graph of y = f(x) + 3.” Adding 3 shifts up: (2, 8). Reward: correct x unchanged, y increased by 3.

3. “Use your graph to solve x² − 2x − 3 = 0.” Roots are x-intercepts of y = x² − 2x − 3. Read from sketch: x = −1 or x = 3. Reward: both values from the graph.

When you can recognise the wording instantly, work the full set on the Functions topical past papers and the Graphs of Functions quiz.

How graphs connect to the rest of Functions

Function graphs build on Quadratic Functions and Logarithms, where exponential and log curves appear as inverse pairs. Use the Cambridge IGCSE Maths resource hub to revise weak areas.

Common mistakes students make

• Drawing straight lines instead of curves for quadratics and cubics.
• Getting transformation direction wrong (f(x + 2) shifts left, not right).
• Forgetting asymptotes on reciprocal graphs.
• Labelling intercepts with the wrong coordinate (mixing up x and y).

When you need more support

If graph sketching or transformations keep costing marks, work through the Quadratic Functions quiz and the Functions topical past papers, then get focused help from a Cambridge IGCSE Maths tutor.

Frequently asked questions

Which graphs do I need to know for IGCSE Maths? Linear, quadratic, cubic, reciprocal (y = k/x), exponential (y = aˣ) and simple transformations of these. Know intercepts, shape and asymptotes where relevant.

How do I solve an equation using a graph? Sketch or use the given graph of y = f(x). The solutions to f(x) = 0 are the x-intercepts. For f(x) = g(x), find intersection points.

What does f(x) + 2 do to a graph? It shifts the entire graph up by 2 units. Every y-coordinate increases by 2; x-coordinates stay the same.

How should I revise graphs of functions? Learn the standard shapes, practise three sketches by hand, then take the Graphs of Functions quiz. Revisit transformations you confuse before moving on.

Ready to master Cambridge IGCSE Maths graphs of functions?

Start with the Graphs of Functions subtopic page, then book a free trial with a Cambridge IGCSE Maths specialist to turn graph questions into guaranteed marks.

Ready to Excel in Your Studies?

Get personalised help from Tutopiya's expert tutors. Whether it's IGCSE, IB, A-Levels, or any other curriculum — we match you with the perfect tutor and your first session is free.

Book Your Free Trial