The vertex form
Any quadratic y = ax² + bx + c can be rewritten as y = a(x − h)² + k. This is called completing the square. The point (h, k) is the vertex, and x = h is the line of symmetry.
Key idea
For x² + bx, add and subtract (b⁄2)²: x² + bx = (x + b⁄2)² − (b⁄2)². If a > 0 the vertex is a minimum; if a < 0 it is a maximum.
Reading off the turning point
Once in the form a(x − h)² + k, the least (or greatest) value of y is k, reached when x = h. This is the quickest way to find the minimum or maximum without calculus.
Worked example
Complete the square for y = x² + 6x + 5. Half of 6 is 3, so x² + 6x = (x + 3)² − 9. Then y = (x + 3)² − 9 + 5 = (x + 3)² − 4. The vertex is (−3, −4), a minimum value of −4.
Remember
- Take out the coefficient of x² before completing the square if a ≠ 1.
- The constant to add inside is (half the x-coefficient)².
- Vertex (h, k): x = h is the axis of symmetry, k is the min/max value.