Vertex Calculator

Quick Answer

Last updated: 10 November 2026 ·How we calculate this: the vertex formula h = −b ÷ 2a

How This Vertex Calculator Works

Enter a, b and c from the standard form y = ax² + bx + c. The calculator finds the x-coordinate of the vertex with h = −b ÷ 2a, then puts it back into the equation to find k. It writes the parabola in vertex form, gives the axis of symmetry, says whether the vertex is a maximum or minimum, and lists the roots, y-intercept, focus and directrix.

How to Use the Vertex Calculator

1. Write your equation in the form y = ax² + bx + c.
2. Enter a, b and c.
3. Press Calculate.
4. Read the vertex, vertex form and axis of symmetry.
5. Use the roots, focus and directrix if you need them.

Vertex Formulas

• Vertex x-coordinate: h = −b ÷ (2a)
• Vertex y-coordinate: k = f(h) = c − b² ÷ (4a)
• Vertex form: y = a(x − h)² + k
• Focus: (h, k + 1 ÷ (4a)); directrix: y = k − 1 ÷ (4a)

Key Terms

• Parabola: the U-shaped graph of a quadratic function.
• Vertex: the turning point of a parabola, its highest or lowest point.
• Axis of symmetry: the vertical line through the vertex that splits the parabola in half.
• Vertex form: y = a(x − h)² + k, where (h, k) is the vertex.
• Focus and directrix: the point and line that define a parabola: every point on it is the same distance from both.

Worked Example

A ball’s height in metres is y = −x² + 6x − 5. Find the highest point.

• a = −1, b = 6, c = −5
• h = −6 ÷ (2 × −1) = 3
• k = −(3)² + 6(3) − 5 = 4
• The vertex is (3, 4), a maximum because a is negative, and the vertex form is y = −(x − 3)² + 4.

Real-Life Examples

• Lowest point: y = 2x² − 8x + 3 has its minimum at (2, -5).
• Highest point: a ball on y = −x² + 6x − 5 peaks at (3, 4).
• No real roots: y = x² + 4x + 7 has vertex (-2, 3) above the x-axis and opens up, so it never crosses it.
• Profit: profit −2x² + 120x − 1,000 is greatest at x = 30, where it is $800.

Standard Form vs Vertex Form

• Standard form y = ax² + bx + c shows the y-intercept (c) straight away.
• Vertex form y = a(x − h)² + k shows the vertex (h, k) straight away.
• Factored form y = a(x − r₁)(x − r₂) shows the roots; the vertex is halfway between them.

Vertex Examples

Completing the Square

You can also reach vertex form by completing the square. For y = 2x² − 8x + 3:

• Take out a from the x terms: y = 2(x² − 4x) + 3
• Add and subtract (4 ÷ 2)² = 4 inside: y = 2(x² − 4x + 4 − 4) + 3
• Make the square: y = 2(x − 2)² − 8 + 3
• Simplify: y = 2(x − 2)² − 5, so the vertex is (2, −5).

Why the Vertex Matters

The vertex answers “highest” and “lowest” questions. For a thrown ball it gives the peak height and when it happens. For a business with profit −2x² + 120x − 1,000, the vertex at x = 30 gives the price or quantity with the most profit. In engineering, the focus of a parabola is where a satellite dish or a headlight reflector gathers or sends out signals.

The vertex also helps you sketch a graph quickly: plot the vertex, draw the axis of symmetry, add the y-intercept and its mirror image, and join them with a smooth U or ∩ shape.

Common Mistakes to Avoid

• Dropping the minus sign. h = −b ÷ 2a, so b = −8 gives h = +8 ÷ 2a.
• Reading vertex form backwards. In y = (x − 3)² + 4, h is +3, not −3.
• Forgetting to find k. Put h back into the original equation.
• Mixing up max and min. Positive a gives a minimum; negative a gives a maximum.
• Using a form that is not standard. Expand and collect terms before reading a, b and c.

Frequently Asked Questions

References & Sources