Graphs: deduce turning points by completing the square (Higher tier only)

The idea

Completing the square rewrites a quadratic from expanded form into vertex form, which exposes the turning point directly. For y = x² + bx + c, write y = (x + b/2)² + c − (b/2)². The squared bracket is never negative, so its least value is zero when x = −b/2; the y-value there is the constant term c − (b/2)². Hence the turning point is (−b/2, c − (b/2)²). For example, y = x² − 6x + 5 becomes y = (x − 3)² − 4, so the minimum is (3, −4). If the coefficient of x² is not 1, first factor it out of the x-terms, complete the square inside, then multiply back. A negative leading coefficient makes the bracket's contribution greatest at zero, giving a maximum. Deduction means reading the coordinates from the completed-square form, not plotting points.

What earns the marks

• Rewrite the quadratic in the form a(x + p)² + q by completing the square, halving the coefficient of x inside the bracket and subtracting the square of that half from the constant.
• Identify the turning point as (−p, q) when the completed-square form is a(x + p)² + q, since the squared term is zero at x = −p.
• State whether the turning point is a minimum or a maximum: minimum when a > 0, maximum when a < 0.
• Handle a leading coefficient other than 1 by factoring it from the x-terms before completing the square, then restoring it correctly.
• Interpret the completed-square constant as the y-coordinate of the turning point, not the y-intercept.

Common mistakes and how to put them right

• Reading the turning point as (p, q) from (x + p)² + q.

  Put it right: The correct x-coordinate is −p because the bracket equals zero when x = −p.

• Forgetting to subtract (b/2)² when completing the square, which changes the constant and gives the wrong y-coordinate.

• Assuming every turning point is a minimum.

  Put it right: A negative leading coefficient produces a maximum, and the sign of a must be checked.

• Treating the constant term q as the y-intercept.

  Put it right: The y-intercept is found by substituting x = 0 into the original expression.

Exam tips

• Show the completed-square form explicitly before stating the coordinates, so the deduction is visible to the examiner.
• Check the sign of the x-coordinate by substituting it back into the original quadratic and confirming the y-value.
• If the coefficient of x² is not 1, factor it out first and keep track of it through every line to avoid arithmetic slips.
• State 'minimum' or 'maximum' with the coordinates, because the question asks for the turning point and its nature may be required.