Vertex and Axis of Symmetry Calculator
Find the vertex, axis of symmetry, opening direction, and minimum/maximum classification of a quadratic ax² + bx + c, plus its real roots and a step-by-step derivation.
Input
Coefficient of x². Must not be zero.
Coefficient of x.
Constant term.
Output
More ways to use this tool
REST API
curl -X POST https://api.iotools.cloud/v1/tool/vertex-axis-calculator \
-H "Authorization: Bearer YOUR_API_KEY" \
-H "Content-Type: application/json" \
-d '{
"coeffA": "1",
"coeffB": "-4",
"coeffC": "3"
}'Swap in your own key from your account. The tool's fields are the body — no wrapper.
Ask an AI agent
Use the IOTools `vertex-axis-calculator` tool (Vertex and Axis of Symmetry Calculator) on this input:
YOUR_INPUT_HEREPaste this at any agent connected to the IOTools MCP server, then add your input.
Embed widget
<iframe
src="https://iotools.cloud/embed/vertex-axis-calculator/"
width="100%" height="520" frameborder="0" scrolling="no" loading="lazy"
title="Vertex and Axis of Symmetry Calculator — iotools.cloud"
sandbox="allow-scripts allow-forms allow-same-origin allow-downloads allow-popups allow-popups-to-escape-sandbox"
allow="clipboard-write"
style="width:100%;border:1px solid #e5e7eb;border-radius:12px;overflow:hidden"></iframe>
<script src="https://iotools.cloud/embed.js" async></script>Drop this into your own page — free, no key required, just a link back.
| Cost per API/MCP call | From 5 credits |
|---|---|
| Need more credits? | View pricing |
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Guides
The Vertex and Axis of Symmetry Calculator analyzes a quadratic in the form ax² + bx + c and returns everything that describes its parabola's shape: the vertex point, the axis of symmetry, which way it opens, and whether the vertex is a minimum or a maximum — plus the real roots and a full step-by-step derivation.
It's built for algebra students graphing parabolas by hand, teachers checking worked examples, and anyone who needs a quadratic's vertex and axis of symmetry without redoing the completing-the-square algebra.
How to use it
- Enter the coefficient a (the number multiplying x²). It must not be zero.
- Enter the coefficient b (the number multiplying x).
- Enter the coefficient c (the constant term).
- Read Vertex (h, k), Axis of Symmetry, Opens (upward/downward), and Vertex Is a (minimum/maximum) for the parabola's shape, Real Roots for its x-intercepts, and Step-by-Step Solution for the full derivation.
Results update automatically as you type. For example, entering a = 1, b = -4, c = 3 (for x² − 4x + 3) returns vertex (2, -1), axis of symmetry x = 2, opens Upward, vertex is a Minimum, and real roots x₁ = 3, x₂ = 1.
What is the axis of symmetry?
The axis of symmetry is the vertical line that splits a parabola into two mirror-image halves — it always passes through the vertex. For y = ax² + bx + c, it's:
x = h = −b / (2a)Every point on the parabola has a mirror point the same distance on the other side of this line, at the same height.
How is the vertex found?
The vertex (h, k) is found the same way as completing the square — factor a out of the x² and x terms, then complete the square:
h = −b / (2a)
k = c − b² / (4a)For the Completing the Square Calculator's full step-by-step version of this derivation (with the intermediate algebra shown), see that tool directly.
How do I know if the vertex is a minimum or maximum?
The sign of a decides it:
- a > 0 — the parabola opens upward, and the vertex is the lowest point: a minimum.
- a < 0 — the parabola opens downward, and the vertex is the highest point: a maximum.
Why does "Real Roots" sometimes say there are none?
The real roots are the parabola's x-intercepts — where it crosses the x-axis. That only happens when the discriminant Δ = b² − 4ac is zero or positive:
- Δ > 0 — two distinct real roots (two x-intercepts).
- Δ = 0 — one repeated real root, exactly at the vertex (the parabola just touches the x-axis).
- Δ < 0 — no real roots at all: the whole parabola sits above or below the x-axis without ever touching it. To see the complex conjugate pair instead, use the Quadratic Formula Solver.
Privacy
This calculator runs entirely in your browser. Your coefficients are never uploaded, logged, or stored — the computation happens locally on your device.
Related tools
For the vertex form derivation shown step by step, see the Completing the Square Calculator. For roots via the discriminant (including complex roots), see the Quadratic Formula Solver.