# Vertex Form Calculator

Convert standard form y = ax² + bx + c to vertex form y = a(x-h)² + k. Find the vertex, axis of symmetry, and parabola direction instantly.

## What this calculates

Convert any quadratic equation from standard form to vertex form. Enter the coefficients a, b, and c to find the vertex, axis of symmetry, and whether the parabola opens up or down.

## Inputs

- **Coefficient a (x² term)** — The coefficient of x². Must not be zero.
- **Coefficient b (x term)** — The coefficient of x.
- **Coefficient c (constant)** — The constant term.

## Outputs

- **Vertex Form** — formatted as text — The equation in vertex form y = a(x - h)² + k.
- **Vertex (h, k)** — formatted as text — The vertex of the parabola.
- **Axis of Symmetry** — formatted as text — The vertical line through the vertex.
- **Direction** — formatted as text — Whether the parabola opens upward or downward.
- **Y-Intercept** — Where the parabola crosses the y-axis (x = 0).
- **Standard Form** — formatted as text — The original equation y = ax² + bx + c.

## Details

The vertex form of a quadratic equation reveals the location of the vertex at a glance. Every quadratic y = ax² + bx + c can be rewritten as y = a(x - h)² + k.

**Finding h and k:**

h = -b / (2a)
k = c - b² / (4a)

Alternatively, k = f(h) -- just plug h back into the original equation.

**Worked Example:**

Convert y = 2x² - 12x + 22 to vertex form.

1. h = -(-12) / (2 x 2) = 12/4 = 3
2. k = 22 - (-12)² / (4 x 2) = 22 - 144/8 = 22 - 18 = 4
3. Vertex form: y = 2(x - 3)² + 4
4. Vertex: (3, 4)
5. Axis of symmetry: x = 3
6. Opens upward because a = 2 > 0

**Why vertex form matters:**

- The vertex (h, k) is immediately visible.
- The axis of symmetry is x = h.
- If a > 0, the vertex is the minimum. If a < 0, it is the maximum.
- Graphing is easy: plot the vertex, then use a to determine the width and direction.

## Frequently Asked Questions

**Q: What is vertex form?**

A: Vertex form is y = a(x - h)² + k, where (h, k) is the vertex of the parabola and a controls the width and direction. It makes it easy to identify the highest or lowest point of the parabola without any additional computation.

**Q: How do I convert standard form to vertex form?**

A: Calculate h = -b/(2a) and k = c - b²/(4a). For y = x² + 8x + 12: h = -8/2 = -4, k = 12 - 64/4 = 12 - 16 = -4. So vertex form is y = (x + 4)² - 4. You can also complete the square to arrive at the same result.

**Q: What is the axis of symmetry?**

A: The axis of symmetry is the vertical line x = h that passes through the vertex. Every parabola is symmetric about this line, meaning for any point on one side, there is a mirror-image point on the other. For y = 3(x - 5)² + 2, the axis of symmetry is x = 5.

**Q: How do I tell if a parabola opens up or down?**

A: Look at the coefficient a. If a > 0, the parabola opens upward and the vertex is the minimum point. If a < 0, it opens downward and the vertex is the maximum. The larger the absolute value of a, the narrower the parabola.

**Q: Is vertex form the same as completing the square?**

A: Completing the square is the algebraic process used to derive vertex form. The end result is the same equation y = a(x - h)² + k. This calculator uses the direct formulas h = -b/(2a) and k = c - b²/(4a), which are equivalent to completing the square but faster to compute.

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Source: https://vastcalc.com/calculators/math/vertex-form
Category: Math
Last updated: 2026-04-08