Quadratic features and completing the square

Quadratics y = ax² + bx + c have several features WJEC Higher candidates must extract: roots, y-intercept, turning point, and line of symmetry.

Y-intercept

Set x = 0. The y-intercept is simply c. For y = x² − 4x + 3, the curve cuts the y-axis at (0, 3).

Roots (x-intercepts)

Roots are where y = 0. Find them by:

• factorising (if possible),
• using the quadratic formula, or
• completing the square.

For y = x² − 4x + 3: factor as (x − 1)(x − 3) = 0, giving roots x = 1 and x = 3.

Turning point via completing the square

Completing the square rewrites x² + bx + c as (x + b/2)² − (b/2)² + c.

Example: x² − 4x + 3 = (x − 2)² − 4 + 3 = (x − 2)² − 1.

The minimum value of (x − 2)² is 0 (when x = 2), so the minimum of the whole expression is −1. Turning point: (2, −1).

For y = a(x − p)² + q the turning point is (p, q). Minimum if a > 0; maximum if a < 0.

Line of symmetry

The line of symmetry passes through the turning point: x = p. For our example, x = 2.

Full sketch checklist

1. y-intercept (set x = 0).
2. Roots (factor or formula).
3. Turning point (complete the square).
4. Shape (∪ if a > 0, ∩ if a < 0).

Coefficient a not 1

For 2x² − 8x + 5: factor a out of the x-terms first: 2(x² − 4x) + 5 = 2[(x − 2)² − 4] + 5 = 2(x − 2)² − 3. Turning point: (2, −3).

WJEC exam tip

If a question says "express in the form (x + p)² + q", give the values of p and q exactly — and then USE that form to read off the turning point in the next part. Both marks are gifted to candidates who write the completed-square form correctly.