# What is the vertex of my parabola?

Computes the vertex (h, k) of a parabola from y = ax² + bx + c or y = a(x − h)² + k, with the axis of symmetry, the minimum or maximum, the y-intercept, the real roots and both forms.

- Page: https://www.acalculator.org/math/vertex-calculator
- JSON spec: https://www.acalculator.org/math/vertex-calculator.json
- Version: 26717418820f

## Default answer

Example with the default inputs (My equation is ax² + bx + c, a 2, b -6, c 7): The vertex is (3/2, 5/2), on the axis x = 3/2.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| form | My equation is | Standard form y = ax² + bx + c, or vertex form y = a(x − h)² + k. |
| a | a | The coefficient of x² (or of the square in vertex form). It may not be 0. |
| b | b | The coefficient of x. |
| c | c | The constant term. |
| h | h | The x of the vertex in a(x − h)² + k. |
| k | k | The y of the vertex in a(x − h)² + k. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| vertex | Vertex (h, k) | The turning point of the parabola, as exact fractions. |
| h | h (x of the vertex) | h = −b ÷ (2a), as a decimal. |
| k | k (y of the vertex) | k = c − b² ÷ (4a), as a decimal. |
| axis | Axis of symmetry | The vertical line through the vertex, x = h. |
| opens | Opens | Up with a minimum at the vertex when a > 0; down with a maximum when a < 0. |
| standard | Standard form | y = ax² + bx + c. |
| vertexForm | Vertex form | y = a(x − h)² + k. |
| b | b | The coefficient of x: b = −2ah. |
| c | c (y-intercept) | The constant term c = ah² + k, where the parabola crosses the y-axis. |
| roots | Real roots (x-intercepts) | x = h ± √(−k ÷ a), when −k ÷ a ≥ 0. |

## Method

h = −b ÷ (2a), k = c − b² ÷ (4a); from vertex form, b = −2ah and c = ah² + k; roots x = h ± √(−k ÷ a).

## Assumptions

- a may not be 0 (then the graph is a line with no vertex).
- Coefficients are read as the exact decimals typed, so the vertex is an exact fraction.

## Worked examples

1. form = standard, a = 2, b = -6, c = 7 gives vertex = (3/2, 5/2), h = 1.5, k = 2.5, vertexForm = y = 2(x − 3/2)² + 5/2, opens = Up: the minimum is y = 5/2. Source: OpenStax, Algebra and Trigonometry 2e, §5.1 Quadratic Functions, Example 3 (vertex (3/2, 5/2); f(x) = 2(x − 3/2)² + 5/2), https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-1-quadratic-functions.
2. form = standard, a = 1, b = -4, c = 3 gives vertex = (2, −1), axis = x = 2, roots = x = 1 and x = 3, c = 3. Source: OpenStax, Algebra and Trigonometry 2e, §5.1 Quadratic Functions (h = −b ÷ (2a), k = f(h)), https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-1-quadratic-functions.
3. form = vertex, a = -0.5, h = -2, k = 8 gives standard = y = −(1/2)x² − 2x + 6, b = -2, c = 6, opens = Down: the maximum is y = 8, roots = x = −6 and x = 2. Source: OpenStax, Algebra and Trigonometry 2e, §5.1 Quadratic Functions (f(x) = a(x − h)² + k; a < 0 opens downward), https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-1-quadratic-functions.
4. form = standard, a = 1, b = 2, c = -1 gives vertex = (−1, −2), roots = x = −2.414213562 and x = 0.4142135624. Source: OpenStax, Algebra and Trigonometry 2e, §5.1 Quadratic Functions, https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-1-quadratic-functions.

## FAQ

### How do I find the vertex of a parabola?

For y = ax² + bx + c, the x of the vertex is h = −b ÷ (2a). Put h back into the equation for k. For y = 2x² − 6x + 7, h = 6 ÷ 4 = 3/2 and k = 2(9/4) − 9 + 7 = 5/2, so the vertex is (3/2, 5/2).

### Is there a formula for k?

Yes: k = c − b² ÷ (4a). For y = 2x² − 6x + 7, k = 7 − 36 ÷ 8 = 5/2, the same as putting h into the equation.

### What is the axis of symmetry?

The vertical line through the vertex, x = h. The parabola is a mirror image on each side of it.

### How do I tell a minimum from a maximum?

Look at the sign of a. When a > 0 the parabola opens up and the vertex is its lowest point (a minimum). When a < 0 it opens down and the vertex is its highest point (a maximum).

### How do I convert vertex form to standard form?

Expand a(x − h)² + k: b = −2ah and c = ah² + k. For y = −0.5(x + 2)² + 8, b = −2 × (−0.5) × (−2) = −2 and c = −0.5 × 4 + 8 = 6, so y = −0.5x² − 2x + 6.

### How are the roots found from the vertex?

Set y = 0: a(x − h)² = −k, so x = h ± √(−k ÷ a). When −k ÷ a is negative there is no real root; when it is 0, the vertex is the only root.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §5.1 Quadratic Functions (vertex h = −b/(2a), k = f(h); standard form f(x) = a(x − h)² + k; axis of symmetry x = −b/(2a); a > 0 opens upward; Example 3: f(x) = 2x² − 6x + 7 has vertex (3/2, 5/2)). https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-1-quadratic-functions (retrieved 2026-10-02)