The formula
vertex at x = −b ÷ 2a · vertex form: y = a(x − h)² + k
Why it works
A parabola is symmetric about a vertical line through its vertex, and that line sits at x = −b/2a. The sign of a decides everything else: positive opens upwards and the vertex is a minimum, negative opens downwards and it is a maximum.
How to solve it by hand
- Find the axis of symmetry: x = −b ÷ 2a
- Substitute that x back into the function for the vertex's y
- The y-intercept is simply c, since x = 0 gives y = c
- The x-intercepts are the roots, from the quadratic formula
What is worth knowing
The vertex form y = a(x − h)² + k is the same function written so you can read the vertex straight off, and it is what makes graphing quick: start at (h, k), then use a to stretch or flip. The minimum or maximum is also why quadratics turn up everywhere in optimisation — the vertex is the answer to any question of the form 'what value gives the largest area, the least cost, the highest point'. Projectile motion under gravity is a parabola for exactly this reason, and the vertex is the top of the arc.