Vertex Form of a Quadratic

Vertex form writes a parabola around its own turning point instead of around the origin: a stretches it and decides which way it opens, h slides it sideways, k slides it up and down, and the vertex sits at (h, k) where you can read it off without touching a pencil. Worked example: y = 2(x − 3)² − 5 at x = 6 gives 2(6 − 3)² − 5 = 2(9) − 5 = 13. Expand the same expression and you get 2x² − 12x + 13 — the standard form, whose vertex formulas h = −b/2a = 12/4 = 3 and k = c − b²/4a = 13 − 18 = −5 hand back exactly the (3, −5) you started with. The leading coefficient a survives the change of form unchanged; only b and c move.

The two forms answer different questions, which is why both survive. Standard form ax² + bx + c makes the y-intercept obvious and feeds the quadratic formula; vertex form makes the maximum or minimum obvious, which is what an optimisation problem actually wants — the peak of a projectile, the least cost, the largest enclosed area. Getting from one to the other is completing the square, the manoeuvre a Babylonian scribe was already performing on clay around 1800 BCE, long before there was any notation to write it in. The trap is the sign of h: the bracket reads (x − h), so y = a(x + 4)² + k has its vertex at h = −4, four units to the left. Written the other way round it catches almost everybody once.