Roots, intercepts and turning points of quadratics

This is exclusively a Higher topic on Edexcel 1MA1 — Paper 1H or 2H. It marries algebra (completing the square, factorising) with graph reading.

Key features of y = ax² + bx + c

• y-intercept: substitute x = 0 → y = c.
• Roots / x-intercepts: solve ax² + bx + c = 0 (factorise, formula, or completing the square).
• Turning point: the minimum if a > 0, maximum if a < 0. The x-coordinate is x = −b/(2a).

Completing the square

Rewrite x² + bx + c as (x + b/2)² − (b/2)² + c.

Example: x² − 6x + 4 = (x − 3)² − 9 + 4 = (x − 3)² − 5.

So the turning point is (3, −5).

For ax² + bx + c with a ≠ 1, factor out a first:

2x² + 8x + 5 = 2(x² + 4x) + 5 = 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 8 + 5 = 2(x + 2)² − 3.

Turning point at (−2, −3).

Reading turning points from completed-square form

If y = a(x − p)² + q, the turning point is (p, q). Note the sign flip on p (because the bracket has x − p).

Roots from completed-square form

(x − 3)² − 5 = 0 ⇒ (x − 3)² = 5 ⇒ x − 3 = ±√5 ⇒ x = 3 ± √5.

This is often the cleanest path to surd-form roots without using the quadratic formula.

Common Edexcel mark-scheme phrasing

• M1 for halving b correctly.
• M1 for adjusting the constant.
• A1 for the correct (x ± p)² + q form.
• B1 for the turning point coordinates.