Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Quadratic & Other Curves

In one sentence. Parabolas, cubics and reciprocal curves — reading intercepts, turning points and symmetry, and solving equations where curves meet lines.

Graphs — one of 2 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with Galileo Galilei. Learn Quadratic & Other Curves, teach it, then defend it against Galileo Galilei.

Where this sits in the specification

Specification 4MA1, 3.3 Graphs — quadratic, cubic, reciprocal; graphical solution of equations. Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 3.3 — Graphs — quadratic, cubic, reciprocal; graphical solution of equations

1 · Understand it

What it is

y = x² − 4x + 3 draws a parabola: a symmetric valley (or hill, if the x² is negative). It crosses the y-axis where x = 0, crosses the x-axis at the roots, and turns exactly halfway between them. Cubics make an S-shape; y = 1/x makes two branches that never touch the axes.

Why it works

The y-intercept is just x = 0 substituted (leaving the constant term). The x-intercepts are where y = 0 — the same roots factorising finds. Symmetry happens because (x − a)(x − b) treats its two roots identically, so the turning point must sit at their average. And 1/x can never be zero because no number of the form 1-over-something is nothing.

When a question is asking for it

Sketching questions ('mark where the curve crosses the axes'), turning-point questions, and 'use your graph to solve…' — where the solutions of x² = 2x + 8 are literally the x-values where the parabola meets the line.

How to hold it in your head

A parabola is a thrown ball's path. Where it lands (roots), how high it starts (y-intercept), the top of its flight (turning point, midway between launch and landing) — every feature of the graph answers a physical question.

2 · The mistakes that cost marks

  • Reading the y-intercept as a root. The curve crosses the y-axis at x = 0, the x-axis at y = 0 — different events.
  • Placing the turning point at a root. It sits at the MIDPOINT of the roots, by symmetry.
  • Thinking every curve crosses every axis. y = 1/x approaches the axes forever without touching: y = 0 is impossible.
  • Solving x² = 2x + 8 by reading only one intersection. A line usually cuts a parabola TWICE.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

For y = x² − 4x + 3: find where it crosses the y-axis, where it crosses the x-axis, and its line of symmetry.

  1. y-axis (x = 0): y = 3 → crosses at (0, 3).
  2. x-axis (y = 0): x² − 4x + 3 = 0 → (x − 1)(x − 3) = 0 → x = 1 and x = 3.
  3. Symmetry: midway between the roots, x = 2. (Turning point: (2, −1).)

Answer: (0, 3); x = 1 and 3; symmetry at x = 2

Solve x² = 2x + 8 (the x-values where the parabola y = x² meets the line y = 2x + 8).

  1. Rearrange to zero: x² − 2x − 8 = 0.
  2. Factorise: (x − 4)(x + 2) = 0.
  3. x = 4 or x = −2 — the two crossing points.

Answer: x = 4 and x = −2

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — The curve y = x² − 4x + 3 crosses the y-axis at the point (0, k). Write down the value of k.
  2. Grade 7 — The curve y = x² − 4x + 3 crosses the x-axis at two points. Work out the smaller x-value.
  3. Grade 8 — Write down the x-coordinate of the turning point of y = x² − 4x + 3.
  4. Grade 9 — The graphs of y = x² and y = 2x + 8 intersect at two points. Work out the positive x-value where they meet.
  5. 9+ — Consider the curve y = 1/x. There is exactly one value that y can never take, no matter what x is. Write down that value.

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Quadratic & Other Curves, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Quadratic & Other Curves works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Quadratic & Other Curves against Galileo Galilei — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 10 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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