🎢 二次函数：抛物线的一生🎢 Quadratic Functions: The Life of a Parabola
上一片里，函数画出来都是笔直的一条线。这一片让 x 乘上它自己——线马上弯成一条抛物线：喷泉、投篮、卫星锅，都是它的舞台。Last leaf, every function drew a straight line. Now let x multiply itself — the line instantly bends into a parabola. Fountains, basketball shots and satellite dishes all take its stage.
第 1 步：一束水柱，一条抛物线Step 1: One water jet, one parabola
上一片的直线只有 x 的一次项，笔直笔直；那让 x 乘它自己，形状会变成什么样？The last leaf's line had x to the first power — dead straight. So what happens when x multiplies itself?
喷泉喷出的水柱在空中画出的就是抛物线，它的式子长这样：y=ax²+bx+c，其中 a 不能为 0（否则就没有 x² 了）。a 是开口的“方向盘”：a 大于 0，开口向上；a 小于 0，开口向下。|a| 越大开口越窄（越瘦），|a| 越小开口越宽（越胖）。小算盘机器人在调形台上一推滑块，水柱的形状马上跟着变。A fountain's water jet traces a parabola, written as y=ax²+bx+c, where a cannot be 0 (otherwise there is no x²). The number a steers the opening: a above 0 opens upward, a below 0 opens downward. The bigger |a| is, the narrower the opening; the smaller |a| is, the wider it gets. The abacus robot slides one knob on the tuner and the jet immediately changes shape.
水柱先升后落，它最高的一点藏在哪里？The jet rises, then falls — where is its highest point hiding?
第 2 步：顶点是最高点Step 2: The vertex is the peak
抛物线的最高点（或最低点）叫顶点——它是整条曲线的“分水岭”。水柱到达顶点时不再上升，接下来一路下落。开口向上时，顶点是最低点，像碗底；开口向下时，顶点是最高点，像山顶。小算盘机器人坐的滑梯就是从顶点出发的：两边一样陡，滑下去的速度感也一模一样。The highest (or lowest) point of a parabola is the vertex — the curve's watershed. When the jet reaches the vertex it stops rising and heads down. Opening upward, the vertex is the lowest point, like a bowl's bottom; opening downward, it is the peak, like a hilltop. The slide the abacus robot sits on starts at the vertex: both slopes are equally steep, and the ride feels exactly the same on either side.
机器人滑下去时发现：左边和右边的坡一模一样——这是巧合吗？Sliding down, the robot notices both slopes are identical — a coincidence?
第 3 步：对称轴像一面镜子Step 3: The axis of symmetry is a mirror
不是巧合。抛物线左右两边完全对称，像照镜子；穿过顶点的那条竖线叫对称轴（axis of symmetry），它正好在正中间。卫星锅正是利用了这一点：锅面是抛物线绕对称轴转一圈得到的，天上各个方向来的信号经锅面一反射，全都聚到同一个点（焦点）上，锅再小也能听清远方。篮球的投篮弧线、喷泉的水柱，同样沿对称轴左右对称。No coincidence. A parabola mirrors itself left and right; the vertical line through the vertex is the axis of symmetry, sitting exactly in the middle. A satellite dish exploits this: its surface is a parabola spun around that axis, so signals arriving from any direction bounce off and gather at one single point — the focus. Even a small dish hears faraway signals clearly. A basketball arc and a fountain jet are symmetric about the same axis.
🎮 你来当调形师（1 分钟）🎮 Your turn: the shape tuner (1 minute)
道理讲完了。拖动 a 滑块，让抛物线开口向上、向下、变胖、变瘦——完成 4 个任务，就是调形师。Theory done. Drag the a slider to make the parabola open up, open down, grow wide, and grow narrow — finish 4 tasks to become the shaper.
一句话记住它：二次函数 = 一条抛物线；a 管开口方向和胖瘦，顶点是最高/最低点，对称轴两边像照镜子。Remember it in one line: a quadratic function is a parabola — a sets the direction and width, the vertex is the peak or the pit, and the curve mirrors itself across the axis of symmetry.
y=ax²+bx+c（a≠0）画出来是抛物线，最高次数是 2y=ax²+bx+c (a≠0) draws a parabola — degree 2a>0 开口向上，a<0 开口向下；|a| 越大越瘦、越小越胖a>0 opens up, a<0 opens down; bigger |a| = narrower, smaller |a| = wider顶点是最高/最低点，对称轴穿过顶点，两边完全对称The vertex is the peak or pit; the axis of symmetry runs through it, mirroring both sides
内容参考 OpenStax 等公开教材，多来源核对 · AI 生成、人工审核 · 发现错误欢迎指正，帮这片叶子长得更好。Based on OpenStax and other open textbooks, cross-checked across sources · AI-generated, human-reviewed · Spotted a mistake? Tell us — help this leaf grow.
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