Quadratics — roots, intercepts, turning points

A quadratic y = ax² + bx + c has three things you can read off algebraically: the roots (where y = 0), the y-intercept (where x = 0), and the turning point (vertex). OCR Higher demands all three from any given form.

y-intercept

Set x = 0: y = c. So the curve always crosses the y-axis at (0, c).

Roots (x-intercepts)

Set y = 0 and solve ax² + bx + c = 0. Three methods:

1. Factorise: e.g. x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3.
2. Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a).
3. Complete the square (see below).

The discriminant b² − 4ac tells you how many real roots:

• Positive → 2 distinct roots.
• Zero → 1 repeated root (curve touches x-axis).
• Negative → 0 real roots (curve never crosses x-axis).

Turning point via completed square

y = ax² + bx + c → y = a(x − h)² + k. The vertex is at (h, k).

For y = a(x − h)² + k:

• If a > 0, the parabola opens upward → vertex is a minimum.
• If a < 0, opens downward → vertex is a maximum.

Completing the square (a = 1)

x² + 6x + 1 = (x + 3)² − 9 + 1 = (x + 3)² − 8. Half the coefficient of x (= 3), square it (= 9), subtract that and add the constant.

So x² + 6x + 1 has vertex at (−3, −8) and minimum value −8.

Completing the square (a ≠ 1)

2x² + 8x − 5: factor 2 from the x terms first. = 2(x² + 4x) − 5 = 2((x + 2)² − 4) − 5 = 2(x + 2)² − 8 − 5 = 2(x + 2)² − 13.

Vertex at (−2, −13).

Symmetry

The line of symmetry of y = a(x − h)² + k is x = h. The midpoint of the two roots also gives x = h (when there are two real roots).

OCR mark scheme conventions

• M1 for halving the coefficient of x.
• M1 for the −(half)² adjustment.
• A1 for the correctly completed square.
• B1 (independent) for stating the vertex coordinates.
• The minimum/maximum value of y is k; the point is (h, k) — examiners often want both.