Mathématiques

What Is Vertex Form?

Vertex form puts the vertex in plain sight: y = 2(x − 3)² − 4 has its minimum at (3, −4). The sign trap, the SAT question it exists for, and its British name.

Vertex form

A quadratic written as y = a(x − h)² + k, an arrangement in which (h, k) is the vertex of its parabola.

Aussi appelé: Completed-square form
Où les élèves le rencontrent: Grade 9 or 10 mathematics as completing the square, and on the Digital SAT, where quadratics sit in the Advanced Math content domain.

La réponse en bref
Vertex form writes a quadratic as y = a(x − h)² + k, where (h, k) is the vertex of the parabola. For y = 2(x − 3)² − 4 the vertex is (3, −4), and because a is positive, −4 is the minimum value. British textbooks call the same rewriting completed-square form.

Un exemple

y = 2(x − 3)² − 4 → vertex (3, −4), minimum value −4

The squared bracket is zero when x = 3 and positive everywhere else, and multiplying by 2 keeps it positive, so the smallest the whole expression can be is −4. Read h as the value that makes the bracket zero, not as the number printed inside it: y = 2(x + 3)² − 4 has its vertex at (−3, −4). The a in front does not move the vertex at all — it decides how steeply the curve rises away from it, and its sign decides whether −4 is a minimum or a maximum.

Read the coordinates, and mind one sign

The form is y = a(x − h)² + k and the vertex is (h, k). Because the form is written with a subtraction, the number you can see inside the bracket is h only when the sign in front of it is a minus. (x − 3)² gives h = 3; (x + 3)² is really (x − (−3))², so h = −3.

That single sign accounts for most of the errors on this topic, and there is a way to stop making it. Ask what value of x makes the bracket equal zero, and use that. It works whatever the form looks like, including awkward cases such as (2x − 6)².

The Digital SAT question this exists for

A recognisable item type hands you a quadratic already in vertex form and asks for the minimum value of the function, or for the x value at which the minimum occurs. There is no algebra to do. The whole question is whether you can read the form, and a student who starts differentiating or completing the square has misread what is being tested.

The two versions want different halves of the same pair, which is where marks are lost under time pressure. Minimum value means k, the y coordinate, so −4. The x value at which it occurs means h, so 3. Reading the question twice costs less than answering the other one.

Quadratic and other nonlinear functions belong to the Advanced Math content domain on the digital test, so this is not a rare shape — it is a standard one worth being fluent in rather than merely able to work out.

The same form under two names

British specifications almost never use the phrase vertex form. They ask you to complete the square, and the answer to write x² − 6x + 5 in the form (x + p)² + q is (x − 3)² − 4 — which is vertex form with a = 1 and the letters relabelled.

This matters for anyone preparing for a British qualification and the SAT in the same year, which is a common pattern in our classes. Two different textbooks, two different names, one identical piece of algebra. Recognising them as the same thing saves relearning a method you already have.

Questions fréquentes

How do you convert a quadratic into vertex form?

Complete the square. For x² − 6x + 5, halve the coefficient of x to get −3, write (x − 3)², which expands to x² − 6x + 9, then correct the constant by subtracting 4. The result is (x − 3)² − 4. When there is a coefficient in front of x², factorise it out first.

What does a do in y = a(x − h)² + k?

It stretches the parabola vertically and sets its direction, without moving the vertex. Positive a opens the curve upwards, making k a minimum; negative a opens it downwards, making k a maximum. A larger value of a makes the curve narrower, so a = 2 rises twice as fast as a = 1.

Why is the vertex at (h, k) rather than (−h, k)?

Because the form contains a subtraction. The vertex is at whatever value of x makes the bracket zero, and x − h = 0 gives x = h. If the expression reads (x + 3)², the bracket is zero at x = −3, so the vertex is at −3. Solving the bracket rather than reading it avoids the trap.

Is vertex form the same as completed-square form?

Yes. American and SAT material says vertex form and writes a(x − h)² + k; British papers say complete the square and write (x + p)² + q. The algebra is identical and so is what it tells you — the coordinates of the turning point and the minimum or maximum value of the expression.

Sources

The Math Section — SAT Suite of Assessments — College Board

Dernière mise à jour 21 août 2026