The vertex sits at x = −b/(2a). For y = x² − 4x + 3 that is x = 2, and substituting back gives y = −1. The same x is the axis of symmetry, because a parabola is symmetric about the vertical line through its turning point.
Finding the vertex
x = −b/(2a) is worth understanding rather than memorising, because two derivations make it obvious.
From the roots. The quadratic formula gives roots at (−b ± √disc)/(2a). They sit symmetrically either side of −b/(2a), so their midpoint — which by symmetry is the vertex — is exactly that.
From the slope. The derivative of ax² + bx + c is 2ax + b, and the turning point is where that is zero: x = −b/(2a). The same answer, arrived at without needing the roots to be real.
The second derivation is the more useful one, because it works even when the curve never crosses the x-axis and there are no roots to take a midpoint of.
The focus and directrix
This is what a parabola actually is, and it is usually taught as an afterthought.
Fix a point — the focus — and a line — the directrix. The set of points equally far from both is a parabola. Every other property follows from that one definition.
The focal length is 1/(4a). The focus sits that far above the vertex, and the directrix that far below. A large a means a small focal length, which is the same statement as “a large a makes a narrow curve”.
The latus rectum is the width of the curve at the focus — the chord through the focus parallel to the directrix — and it is |4p|, or |1/a|. It is the single most convenient number for sketching one by hand: mark the vertex, go up one focal length, and the curve is that wide there.
The reflective property comes straight from the definition: every ray arriving parallel to the axis reflects through the focus. That is why a satellite dish is a parabola, and why the point of a car headlight sits at one.
Two forms, one curve
Standard form, y = ax² + bx + c, is easy to differentiate and easy to feed into the quadratic formula. It hides the vertex.
Vertex form, y = a(x − h)² + k, shows the vertex at (h, k) directly. It hides the y-intercept.
Converting from standard to vertex form is completing the square, and it is the same operation that derives the quadratic formula — which is why the −b/(2a) turns up in both.
Vertex form also makes the transformations obvious: h shifts the curve sideways, k shifts it up and down, and a stretches it. Reading a graph off vertex form takes no arithmetic at all.
What the discriminant decides
b² − 4ac decides how the curve meets the x-axis, and nothing else about its shape.
Positive: two real roots, and the curve crosses twice.
Zero: one repeated root, and the vertex sits exactly on the axis.
Negative: no real roots. The curve stays entirely above the axis if a is positive, or entirely below it if a is negative.
Worth being clear about: a negative discriminant is a statement about where the parabola sits, not about what shape it is. The curve is exactly as much a parabola as any other; it simply does not happen to reach zero.
Where it gets used
Projectiles. Ignoring air resistance, a thrown object follows a parabola. The vertex is the highest point and the roots are launch and landing.
Optimisation. Any quadratic cost or revenue model has its optimum at the vertex, which is why −b/(2a) turns up so often in business problems.
Reflectors. Dishes, headlights, telescope mirrors and solar concentrators are all parabolic, and all for the same reflective property.
Suspension bridges. A cable carrying a uniform load along its span hangs in a parabola. A cable carrying only its own weight hangs in a catenary, which looks similar and is not the same curve — a distinction that took mathematicians a long time to settle.
Sources and methodology
The conic definitions are standard; these are the references.
Method. The vertex is computed from −b/(2a) rather than by completing the square symbolically, and the suite checks it a second way — by sampling the curve on both sides and confirming the reported vertex really is the extreme value, which a sign error in the formula would fail. Every root the page reports is substituted back into the original quadratic and required to evaluate to zero. That engine is verified on every change against 119 hand-written assertions, including that every reported root evaluates the parabola to zero, and that each vertex is genuinely the extreme value of its own curve rather than merely a point on it. The count and the per-case breakdown are published on the formula verification page.
Read the guide
For the roots alone with the quadratic formula worked through step by step, the Quadratic Formula Calculator is the focused version.