Parabola vertex form calculator
Enter 1, −4 and 3 and y = x² − 4x + 3 becomes y = (x − 2)² − 1, with its vertex at (2, −1). Check a completed-square answer without flipping a sign in the linear term. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- A
- 1
- B
- -4
- C
- 3
- H
- 2
- K
- -1
- Vertex form
- (x - 2)^2 - 1
- Standard form
- x^2 - 4x + 3
Standard form y equals ax squared plus bx plus c hides the vertex of a parabola inside the linear and constant terms, so students, tutors, and automation scripts often need a reliable conversion into vertex form a times the square of x minus h, plus k. This parabola vertex form calculator accepts the three finite coefficients a, b, and c, applies the closed identities h equals negative b over two a and k equals c minus b squared over four a, and returns the readable vertex-form equation together with the vertex coordinates.
From standard form to vertex form in one identity
A real parabola opening vertically is completely described by three coefficients in the standard polynomial y equals a x squared plus b x plus c, provided a is not zero. Completing the square algebraically rewrites that polynomial as a times the square of the quantity x minus h, plus a constant k. The abscissa of the vertex is always h equals negative b divided by two a, because that is the unique critical point of a quadratic: the derivative two a x plus b vanishes exactly there.
Substituting x equals h into the original polynomial, or using the compact identity k equals c minus b squared over four a, yields the ordinate of the vertex. The sign of a decides orientation: positive a opens upward so the vertex is a minimum; negative a opens downward so the vertex is a maximum.
Once h and k are known, the vertex form y equals a of x minus h squared plus k makes the axis of symmetry x equals h and the extreme value y equals k obvious at a glance, which is why textbooks, graphing calculators, and exam rubrics prefer that presentation for sketching and for comparing shifts of the same parent parabola.
Where a vertex-form conversion earns its keep
Classrooms and tutoring platforms use a parabola vertex form calculator so learners can check that their hand rewrite of ax squared plus bx plus c matches the canonical a of x minus h squared plus k without flipping a sign in the linear term. Exam generators and LMS auto-graders need a deterministic oracle that rejects a equals zero as invalid input and returns stable rounded h and k for arbitrary finite coefficients.
Content sites that rank for parabola vertex form calculator benefit from a live free browser path whose results match the paid automation path byte for byte on the same pure module. Keep requests small, pin the rounded vertex in tests, and you get reproducible form conversion for homework, LMS plugins, and edge workers without standing up symbolic math infrastructure of your own.
What you can do with it
Classroom checks of vertex form
Let students enter a, b, and c for y=ax²+bx+c and verify the rewritten a(x−h)²+k and the vertex against textbook completing-the-square work.
Notebooks and optimum of a quadratic model
Convert fitted standard-form quadratics into vertex form so scripts can read the extreme point (h, k) without a computer-algebra dependency.
FAQ
How do I convert a parabola from standard form to vertex form?
Send finite numbers a (non-zero), b, and c for y=ax²+bx+c.
Does this calculator also return focus and directrix?
No. This capability returns vertex form and the vertex only. For focus, directrix, and axis of symmetry, use a dedicated parabola geometry tool.
How do I know if the vertex is a minimum or a maximum?
A positive a makes the vertex the lowest point of the curve and a negative a makes it the highest. In vertex form that extreme value is simply k, the number added at the end.
Why do textbooks prefer vertex form?
Because the axis of symmetry and the extreme value can be read at a glance: the axis is the vertical line through h and the extreme value is k. That makes sketching and comparing shifted parabolas quick.
Why is a coefficient a of zero rejected?
Without a squared term there is nothing to complete into a square: the graph is a straight line with no vertex. The calculator says so instead of inventing a vertex.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/geom/parabola-vertex-form \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":1,"b":-4,"c":3}'const res = await fetch("https://api.kit.forhosting.com/geom/parabola-vertex-form", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 1,
"b": -4,
"c": 3
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/parabola-vertex-form",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 1,
"b": -4,
"c": 3
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/parabola-vertex-form", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":1,"b":-4,"c":3}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":1,"b":-4,"c":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/parabola-vertex-form", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"a": 1,
"b": -4,
"c": 3
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.parabola_vertex_form",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
When the leading coefficient a is zero the graph is a line rather than a parabola, so the task fails with invalid_input instead of inventing a vertex.
From standard form to vertex form in one identity
This endpoint encodes exactly that rewrite: it does not approximate roots, does not sample the curve, and does not call a symbolic engine—it evaluates the two identities, rounds coefficients to twelve decimal places for stable machine comparison, and builds a compact display string that collapses unit coefficients and zero terms the way a careful human would write the answer on paper.
Input contract, response fields, and failure modes
Example: a equals one, b equals negative four, and c equals three produces h equals two, k equals negative one, vertex at two comma negative one, and the equation y equals open parenthesis x minus two close parenthesis squared minus one. Failed validation is not billed. Pair the response with your own plotter if you need a sketch; this capability converts forms and reports the vertex only. POST a JSON object with three required finite numbers named a, b, and c. Each field may arrive as a JSON number or as a decimal numeric string; missing fields, empty strings, non-numeric text, NaN, and Infinity all fail with invalid_input and name the offending field. On success the body echoes the rounded coefficients a, b, and c; reports h and k separately; packages the vertex as an object with x and y; supplies vertex_form as the polynomial body without the leading y equals, standard_form as the original polynomial body, and equation as the full y equals vertex-form line ready for display; labels opens as up or down from the sign of a; and includes a short formula string restating the identities for classroom or log use.
Where a vertex-form conversion earns its keep
Engineering notebooks that shift and scale quadratic cost models often store parameters in standard form for regression and then need vertex form to read off the optimum operating point; a tiny parse API avoids pulling a full computer-algebra library for three coefficients.
Integration use cases
LMS and exam auto-grading
Mark convert-to-vertex-form items with a deterministic oracle that rejects a=0 and returns stable rounded h and k for arbitrary finite coefficients.
Technical questions
What happens if the leading coefficient a is zero?
The task fails with invalid_input. When a=0 the graph is a line y=bx+c, not a parabola, so there is no vertex form of the type a(x−h)²+k.
What is the vertex of y=ax²+bx+c?
The vertex is the point (h,k) with h=-b/(2a) and k=c-b²/(4a). It is the minimum when a>0 and the maximum when a<0.
What does the parabola vertex form calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.