Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who need to sketch parabolas, find vertices and solve problems involving y = ax² + bx + c with confidence.
What query it owns: how to understand and revise quadratic functions in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the revision-guide angle, while Tutopiya’s Quadratic Functions subtopic page owns the learning resource and the free Quadratic Functions quiz owns the practice.

Quadratic functions appear throughout Cambridge IGCSE Mathematics (0580/0607) — in algebra, graphs and optimisation-style word problems. A quadratic has the form y = ax² + bx + c and graphs as a parabola. This guide explains how to read the shape, locate the vertex and roots, and answer the command words that actually appear on papers.

Key takeaways

• A quadratic function has the form y = ax² + bx + c; its graph is a parabola.
• If a > 0 the parabola is U-shaped (minimum); if a < 0 it is ∩-shaped (maximum).
• The vertex is at x = −b/(2a); substitute back to find the y-coordinate.
• Roots are where y = 0 — solve by factorising, the formula or completing the square.

What is a quadratic function in Cambridge IGCSE Maths?

A quadratic function is any function that can be written as y = ax² + bx + c where a ≠ 0. Its graph is a smooth curve called a parabola. Cambridge IGCSE Extended papers ask you to sketch the graph, find the vertex, solve ax² + bx + c = 0, and interpret features such as the line of symmetry. The sign of a tells you whether the vertex is a minimum or a maximum.

You can read the full explanation and worked examples on Tutopiya’s Quadratic Functions subtopic page before you attempt questions.

The core features you must master

Feature	How to find it	What it tells you
y-intercept	Set x = 0 → c	Where the graph crosses the y-axis
x-intercepts (roots)	Solve ax² + bx + c = 0	Where the graph crosses the x-axis
Vertex	x = −b/(2a), then find y	Turning point (min or max)
Line of symmetry	x = −b/(2a)	Vertical line through the vertex
Shape	Sign of a	U-shape if a > 0; ∩-shape if a < 0

How to sketch a quadratic — step by step

1. Identify a, b, c from the equation y = ax² + bx + c.
2. Plot the y-intercept at (0, c).
3. Find the vertex using x = −b/(2a) and substitute for y.
4. Find the roots by factorising or using the quadratic formula.
5. Draw a smooth curve through the points; add the line of symmetry if asked.

Once you have sketched a few, test yourself with the free Quadratic Functions quiz.

Completing the square: when the exam asks for vertex form

Completing the square rewrites y = ax² + bx + c as y = a(x − h)² + k, which reveals the vertex (h, k) directly.

Example: y = x² − 6x + 5 → y = (x − 3)² − 4. Vertex at (3, −4), minimum because a = 1 > 0.

Form	Best for
y = ax² + bx + c	Roots via factorising or formula
y = a(x − h)² + k	Vertex and sketching quickly
Factorised form a(x − p)(x − q)	Roots at x = p and x = q

Quadratic functions in past-paper wording: command words that matter

Command word / phrase	What the question wants	Typical stem
Sketch the graph	Shape, intercepts, vertex labelled	”Sketch the graph of y = x² − 4x + 3.”
Write down the coordinates of the vertex	Turning point	”Write down the coordinates of the minimum point.”
Solve	Find roots (x-values where y = 0)	“Solve x² − 5x + 6 = 0.”
Complete the square	Rewrite in vertex form	”Express x² + 6x + 2 in the form (x + a)² + b.”
Find the maximum value	Vertex y-coordinate when a < 0	”Find the maximum value of 12x − x².”
Show that	Prove a given result with working	”Show that the line of symmetry is x = 2.”

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

1. “y = x² − 4x + 3. Write down the coordinates of the vertex.” x = −(−4)/(2×1) = 2; y = 4 − 8 + 3 = −1. Vertex: (2, −1). Reward: method for x, then correct y.

2. “Solve x² − 5x + 6 = 0.” Factorise: (x − 2)(x − 3) = 0 → x = 2 or x = 3. Reward: factorisation or correct use of the formula.

3. “Sketch the graph of y = 2 − x².” a = −1 so ∩-shape; y-intercept (0, 2); vertex (0, 2) is the maximum; roots when 2 − x² = 0 → x = ±√2. Label key points.

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

How quadratic functions connect to the rest of Functions

Quadratic graphs are a major part of Graphs of Functions, and completing the square links to Composite and Inverse of Functions when quadratics appear inside function notation. Use the Cambridge IGCSE Maths resource hub to move between subtopics.

Common mistakes students make

• Confusing the vertex x-coordinate (−b/2a) with a root.
• Sketching a U-shape when a is negative (or vice versa).
• Forgetting the y-intercept at (0, c) on sketches.
• Stopping after factorising when the question asks to complete the square.

When you need more support

If quadratic sketches or completing the square keep tripping you up, work through the Graphs of Functions quiz and the Functions topical past papers, then get focused help from a Cambridge IGCSE Maths tutor.

Frequently asked questions

What is the vertex of a quadratic? The vertex is the turning point of the parabola — the minimum if a > 0, the maximum if a < 0. Its x-coordinate is −b/(2a).

How do I know if a parabola opens up or down? Look at the sign of a in y = ax² + bx + c. Positive a gives a U-shape (minimum); negative a gives a ∩-shape (maximum).

When should I complete the square? When the question asks for vertex form, the minimum/maximum value, or the coordinates of the turning point. It is also useful for solving when the coefficient of x² is 1.

How should I revise quadratic functions? Read the subtopic notes, sketch three different quadratics by hand, then take the Quadratic Functions quiz. Revisit any completing-the-square steps you got wrong.

Ready to master Cambridge IGCSE Maths quadratic functions?

Start with the Quadratic Functions subtopic page, then book a free trial with a Cambridge IGCSE Maths specialist to turn quadratic functions 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