What is the vertex of a parabola, and how do you find it?

The graph of every quadratic, y = ax² + bx + c, is a parabola: a smooth, symmetrical curve shaped like a U. The vertex is the point where the curve stops going down and starts going up (or the other way round). In the UK it is often called the turning point.

Which way the parabola opens depends on the sign of a, the number in front of x²:

• a > 0: the parabola opens upwards (∪) and the vertex is its minimum, the lowest point.
• a < 0: the parabola opens downwards (∩) and the vertex is its maximum, the highest point.

The vertical line through the vertex is the axis of symmetry. Fold the graph along it and the two halves match exactly.

The vertex formula. For y = ax² + bx + c, the vertex is at

x = −b ÷ 2a, and the y-coordinate is what you get when you put that x back into the equation.

Where does −b ÷ 2a come from? Look at the points where y = c. Then ax² + bx = 0, so x(ax + b) = 0, which gives x = 0 or x = −b ÷ a. These two points are at the same height, so by symmetry the vertex sits exactly halfway between them: x = −b ÷ 2a. If you prefer a formula for y too, it is y = c − b² ÷ 4a.

You can find the vertex from any of the three usual forms of a quadratic:

• Standard form y = ax² + bx + c: use x = −b ÷ 2a, then work out y.
• Vertex form y = a(x − h)² + k: the vertex is simply (h, k). Watch the sign: y = (x + 4)² − 7 is y = (x − (−4))² − 7, so the vertex is (−4, −7). You turn standard form into vertex form by completing the square.
• Factored (intercept) form y = a(x − p)(x − q): the parabola crosses the x-axis at p and q, so the vertex is halfway between them, at x = (p + q) ÷ 2.

Why does it matter? Whenever something follows a parabola, the vertex answers "what is the most (or least), and when?" A ball thrown upwards with height h = 20t − 5t² metres reaches its vertex at t = −20 ÷ (2 × −5) = 2 seconds, when h = 40 − 20 = 20 m. So it rises to 20 m after 2 seconds, then falls. The vertex is also the middle of the two solutions of a quadratic equation, which is exactly why the quadratic formula starts with −b ÷ 2a.

The axis of symmetry is then the line x = (your x-coordinate). When you sketch the graph, plot the vertex first and work outwards in both directions.

What is the vertex formula?#

For y = ax² + bx + c, the vertex is at x = −b ÷ 2a. Substitute that into the equation to get y, or use y = c − b² ÷ 4a. The vertex is the point (−b ÷ 2a, c − b² ÷ 4a).

What is vertex form?#

Vertex form is y = a(x − h)² + k. The vertex is (h, k), so you can read it straight off. y = 2(x − 3)² + 1 has its vertex at (3, 1).

How do you find the vertex from two x-intercepts?#

The vertex is halfway between them. Add the two x-intercepts and divide by 2 to get the x-coordinate, then substitute to find y. This works for factored (intercept) form, y = a(x − p)(x − q).

How do you find the vertex from a graph?#

Look for the point where the curve turns. If it is not on a grid line, find two points on the curve at the same height: the vertex is directly halfway between them.

How do you find the vertex by completing the square?#

Rewrite y = x² + bx + c as y = (x + b/2)² − (b/2)² + c. For y = x² − 6x + 5 that is (x − 3)² − 4, so the vertex is (3, −4).

What is the focus of a parabola?#

The focus is a different point, inside the curve. Every point on a parabola is the same distance from the focus as from a line called the directrix, and the vertex sits exactly halfway between them. For y = ax², the focus is at (0, 1 ÷ 4a).

Can a parabola have more than one vertex?#

No. A parabola has exactly one vertex and one axis of symmetry. (Ellipses and hyperbolas have two vertices on their main axis, which is where the plural “vertices” comes from.)