Reflect a point over the x-axis calculator
Enter a point and get its mirror image across the x-axis: (3, 5) becomes (3, −5), because only the vertical coordinate changes sign. Points already on the axis stay put and are flagged as such. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- X
- 3
- Y
- 5
- Reflected x
- 3
- Reflected y
- -5
Reflecting a point over the x-axis is one of the core isometries in coordinate geometry: the image of P equals open parenthesis x comma y close parenthesis is the point with the same abscissa and the opposite ordinate, so P prime equals open parenthesis x comma negative y close parenthesis. That single sign flip of y is reflection across the horizontal axis of the plane, an involution, and the matrix diagonal of one and negative one applied to a column vector.
What reflection over the x-axis means and why people search for it
In Euclidean plane geometry the x-axis is the set of points whose ordinate is zero, the horizontal line y equals zero. Reflecting a point P equals open parenthesis x comma y close parenthesis across that axis produces the unique point P prime that lies on the opposite side of the line at the same perpendicular distance, with the segment PP prime perpendicular to the x-axis.
Algebraically the map is the ordinate sign flip P prime equals open parenthesis x comma negative y close parenthesis: the horizontal position is unchanged and vertical offsets reverse direction. Search intent clusters around phrases such as reflect a point over the x-axis, reflection over the x-axis calculator, find the image of a point across the horizontal axis, and coordinate geometry reflection (x,y) to (x,-y).
Where reflecting over the x-axis helps
Classrooms and tutoring platforms use a reflect-over-the-x-axis calculator so learners can check the image of arbitrary points without confusing reflection over the x-axis with reflection over the y-axis, the origin, or the line y equals x. Exam-prep generators need a deterministic oracle that rejects non-finite coordinates instead of silently coercing bad strings. Failed validation does not consume balance.
Coordinates may be any finite numbers within a large absolute bound suitable for ordinary classroom and CAD figures; the response shares the same unit system you already use. Keep requests small, pin the rounded fields in tests, and you get reproducible reflections for homework, LMS plugins, and edge workers without standing up symbolic math infrastructure of your own.
What you can do with it
Classroom and homework reflection drills
Students submit x and y for a point and verify the image across the x-axis as (x,-y), including rejection when a coordinate is missing or non-numeric.
Exam and LMS auto-grading
Mark coordinate-geometry items with a deterministic oracle that negates y correctly and never invents a reflection from invalid input.
FAQ
What formula does the calculator use?
Reflection over the x-axis is the ordinate sign flip (x,y) → (x,-y).
What is the reflection of (3, 5) over the x-axis?
It is (3, −5). The horizontal position stays the same and the vertical one changes sign, so the image sits on the other side of the axis at the same distance.
What do I get if I reflect the image again?
You get the original point back. Flipping the sign of the second coordinate twice undoes itself, so (3, −5) returns to (3, 5).
What if the point is already on the x-axis?
Then its image is the same point, because its second coordinate is zero and has no sign to flip. The calculator also marks that the point lies on the axis.
What happens if a coordinate is missing or is not a number?
The calculator stops with an error that names the coordinate at fault. It never turns a wrong entry into a guessed reflection.
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/reflect-point-x-axis \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"x":3,"y":5}'const res = await fetch("https://api.kit.forhosting.com/geom/reflect-point-x-axis", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"x": 3,
"y": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/reflect-point-x-axis",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"x": 3,
"y": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/reflect-point-x-axis", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"x":3,"y":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"x":3,"y":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/reflect-point-x-axis", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"x": 3,
"y": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.reflect_point_x_axis",
"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
This calculator accepts two finite numbers x and y, rejects missing or non-numeric coordinates with invalid_input so failed validation is never charged, and returns the reflected pair together with a flag when the original point already lies on the x-axis.
What reflection over the x-axis means and why people search for it
Delivering reflected_x and reflected_y as JSON makes the result easy to diff in continuous integration, feed into a larger pipeline, or copy into notes. Points already on the x-axis are fixed points of the isometry, and the response marks that case with on_axis true so clients can special-case UI without a second comparison.
Input contract, response shape, and validation behaviour
POST a JSON body with two required finite numbers: x, the abscissa of the point to reflect, and y, the ordinate. Each field may arrive as a JSON number or as a numeric string; missing fields, empty strings, NaN, and infinities fail with invalid_input naming the field so those failed calls are not charged. After parsing, the reflected point is computed as reflected_x equals x and reflected_y equals the negation of y. Results are rounded to a stable decimal precision so the same coordinates always yield the same JSON across JavaScript engines. The response echoes the input coordinates, returns reflected_x and reflected_y, sets on_axis to true when y equals zero, sets method to negate_y, and includes a short formula string restating the identity for display next to the answer. Example: x equals three and y equals five yields reflected_x equals three and reflected_y equals negative five because the x-axis keeps x and flips the sign of y. Example on the axis: x equals four and y equals zero yields the same point with on_axis true. There is no trigonometry, no matrix library, and no external geometry service—only a sign flip on a Worker isolate. Pair the response with your own plotting code if you need a sketch of the segment from P to P prime; this endpoint returns numbers only and does not emit plots or multi-step prose beyond the formula field.
Where an x-axis reflection API earns its keep, pricing, and limits
Engineering notebooks, CAD helpers, and game editors often reduce mirror-across-horizontal transforms and bilateral symmetry checks to this single map when validating that a figure is symmetric across the x-axis; calling a tiny parse API avoids pulling a full computer-algebra system for a two-number problem.
Integration use cases
CAD, mesh, and game horizontal symmetry checks
Tools that test symmetry across the horizontal axis or apply a mirror-over-x transform can recover the reflected point as structured JSON.
Technical questions
How do I reflect a point over the x-axis with this API?
Send a JSON body with finite numbers x and y. The response returns reflected_x = x and reflected_y = -y, the image of (x,y) across the x-axis.
What happens if a coordinate is missing or not a number?
The request fails with invalid_input because a finite number is required for every field. Those validation failures are not charged.
What if the point already lies on the x-axis?
The reflected point equals the original point and on_axis is true. That is a fixed point of the isometry, not an error.
What does the reflect-over-the-x-axis calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.