parabola 顶点式计算器（一般式 ↔ 顶点式，顶点、对称轴、根）Vertex Form Calculator (Standard ↔ Vertex Form, Vertex, Axis, Roots)

使用说明与常见问题How-to & FAQ

「顶点式计算器（一般式 ↔ 顶点式，顶点、对称轴、根）」是什么？What is the Vertex Form Calculator (Standard ↔ Vertex Form, Vertex, Axis, Roots) tool?

把 y = ax² + bx + c 化成顶点式 y = a(x − h)² + k（整数系数时给出精确分数），或由顶点式展开成一般式；同时给出顶点、对称轴、开口方向、最值、与 x 轴的交点和 y 截距。本地处理。Turn y = ax² + bx + c into vertex form y = a(x − h)² + k (exact fractions for integer coefficients), or expand vertex form back to standard form. Also shows the vertex, axis of symmetry, direction of opening, minimum or maximum, x-intercepts and y-intercept. Processed locally.

如何使用「顶点式计算器（一般式 ↔ 顶点式，顶点、对称轴、根）」？How do I use the Vertex Form Calculator (Standard ↔ Vertex Form, Vertex, Axis, Roots) tool?

在「顶点式计算器（一般式 ↔ 顶点式，顶点、对称轴、根）」中，填写「方向」、「a」、「b」、「c」、「h」等参数，结果会随输入实时计算并显示在下方。In the Vertex Form Calculator (Standard ↔ Vertex Form, Vertex, Axis, Roots): Fill in "Direction", "a", "b", "c", "h"; results are recalculated instantly as you type.

怎样把一般式化成顶点式？How do I convert standard form to vertex form?

配方：h = −b/(2a)，k = c − b²/(4a)，于是 y = a(x − h)² + k。例如 y = 2x² − 6x + 1：h = 3/2，k = 1 − 36/8 = −7/2，所以 y = 2(x − 3/2)² − 7/2。Complete the square: h = −b/(2a) and k = c − b²/(4a), so y = a(x − h)² + k. For y = 2x² − 6x + 1: h = 3/2 and k = 1 − 36/8 = −7/2, giving y = 2(x − 3/2)² − 7/2.

h 为负数时怎么写？How is a negative h written?

h = −2 时写成 (x + 2)²，不会出现 (x − (−2))² 这种写法；系数为 1 时也不写出 1。For h = −2 it is written (x + 2)², never (x − (−2))²; a coefficient of 1 is not written either.

能举个计算实例吗？Can you show a worked example?

例如在「顶点式计算器（一般式 ↔ 顶点式，顶点、对称轴、根）」中填 方向 = 一般式 → 顶点式，a = 2，b = -6，c = 1，结果区显示：顶点式：y = 2(x − 3/2)² − 7/2；一般式：y = 2x² − 6x + 1；顶点 (h, k)：(3/2, −7/2)；对称轴：x = 3/2；开口：向上，最小值 −7/2；与 x 轴交点：x = 0.177124, 2.822876；y 截距：1For example, entering Direction = Standard → vertex, a = 2, b = -6, c = 1 in the Vertex Form Calculator (Standard ↔ Vertex Form, Vertex, Axis, Roots) gives: Vertex form: y = 2(x − 3/2)² − 7/2; Standard form: y = 2x² − 6x + 1; Vertex (h, k): (3/2, −7/2); Axis of symmetry: x = 3/2; Opens: upward, minimum −7/2; x-intercepts: x = 0.177124, 2.822876; y-intercept: 1

哪些输入会被检查？Which inputs are validated?

「顶点式计算器（一般式 ↔ 顶点式，顶点、对称轴、根）」会检查这些输入——「a」：必须填写，须为数字；「b」：须为数字；「c」：须为数字；「h」：须为数字；「k」：须为数字；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Vertex Form Calculator (Standard ↔ Vertex Form, Vertex, Axis, Roots) checks its inputs — "a": required, must be a number; "b": must be a number; "c": must be a number; "h": must be a number; "k": must be a number; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.

有哪些选项可以调整？What options can I adjust?

「顶点式计算器（一般式 ↔ 顶点式，顶点、对称轴、根）」的设置：「方向」可选 一般式 → 顶点式、顶点式 → 一般式（默认 一般式 → 顶点式）；「a」默认 2；「b」默认 -6；「c」默认 1；「h」默认 3；「k」默认 -4。每次生成结果都按当前设置处理。Settings in the Vertex Form Calculator (Standard ↔ Vertex Form, Vertex, Axis, Roots) — "Direction": Standard → vertex, Vertex → standard (default Standard → vertex); "a" default 2; "b" default -6; "c" default 1; "h" default 3; "k" default -4. The result always uses the current settings.

计算精度如何？How precise are the results?

「顶点式计算器（一般式 ↔ 顶点式，顶点、对称轴、根）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Vertex Form Calculator (Standard ↔ Vertex Form, Vertex, Axis, Roots) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.

「顶点式计算器（一般式 ↔ 顶点式，顶点、对称轴、根）」免费吗？需要注册吗？Is the Vertex Form Calculator (Standard ↔ Vertex Form, Vertex, Axis, Roots) free? Do I need an account?

「顶点式计算器（一般式 ↔ 顶点式，顶点、对称轴、根）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Vertex Form Calculator (Standard ↔ Vertex Form, Vertex, Axis, Roots) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.