How to Find the Vertex of a Parabola

The formula for the x-coordinate

x = −b ÷ 2a

That is the same −b and 2a that appear in the quadratic formula, without the square root part. Which is not a coincidence — it is the middle of the two roots.

For x² − 2x − 15: x = −(−2) ÷ 2(1) = 2 ÷ 2 = 1.

Getting the y-coordinate

Substitute your x back into the original equation.

y = 1² − 2(1) − 15 = 1 − 2 − 15 = −16.

So the vertex is at (1, −16).

This is the step people forget. Finding x = 1 is only half an answer — a vertex is a point, and it needs both coordinates.

The axis of symmetry

The axis of symmetry is the vertical line through the vertex, written as x = 1 for our example.

A parabola is a mirror image of itself across this line. Anything on one side has a matching point on the other at the same height.

It is the same value as the vertex's x-coordinate, which means answering "find the axis of symmetry" needs only the first half of the work above.

Why it sits between the roots

Because the parabola is symmetrical, the vertex must sit exactly halfway between the two roots.

For x² − 2x − 15 the roots are 5 and −3. Their midpoint is (5 + −3) ÷ 2 = 1. Which matches.

That gives you a second route: if you already know the roots, average them. It is quicker than the formula and it is a useful cross-check when you have both.

Maximum or minimum?

Whether the vertex is the highest or lowest point depends entirely on the sign of a.

If a is positive, the parabola opens upward and the vertex is a minimum.

If a is negative, it opens downward and the vertex is a maximum.

For our example a = 1, so (1, −16) is the lowest point on the curve — the value of the expression never drops below −16.

This is what makes vertices useful outside algebra class. Questions about maximum profit, maximum height of a thrown object, or minimum cost are all vertex questions.

Vertex form

A quadratic can also be written as y = a(x − h)² + k, where (h, k) is the vertex. This is called vertex form.

Our example becomes y = (x − 1)² − 16. You can read the vertex straight off it — though note the sign flip: (x − 1) means h = +1.

Converting to vertex form is what completing the square does. If a question gives you vertex form, you already have the vertex with no work at all.