Quadratic Functions and Equations in One Variable
Turning point
The maximum or minimum point of a parabola, found by completing the square.
| English | Turning point |
|---|---|
| Bahasa Melayu | Titik pusingan |
| 中文 | 顶点 |
How it is used
Completing the square on y = x² − 6x + 5 gives y = (x − 3)² − 4, so the turning point is (3, −4), the minimum point of the parabola.
Where it shows up in SPM
A key result in the Quadratic Functions chapter (Form 4). Paper 2 asks you to express y = ax² + bx + c in the form a(x − h)² + k by completing the square, then state the turning point (h, k) and whether it is a maximum or minimum.
Don't confuse it with
Open the chapter: Quadratic Functions and Equations in One Variable →
Frequently asked questions
How do I find the turning point by completing the square?
Write y = a(x − h)² + k. Then the turning point is (h, k).
For y = x² − 6x + 5, take half of −6 to get −3, so y = (x − 3)² − 9 + 5 = (x − 3)² − 4, giving the turning point (3, −4).
How do I tell whether it is a maximum or a minimum?
Check the sign of a. If a is positive the parabola opens upward, so the turning point is a minimum; if a is negative it opens downward, so the turning point is a maximum.
The value k is then the smallest or largest value of y.