📏 一次函数：y = kx + b 的直线📏 Linear Functions: the line y = kx + b
上一片给机器装上了“倒挡”——反着开，就能把输出还原成输入。这一片来看看机器铺里最忙的一台：出租车计价器。它算出来的账单，画在纸上居然是一条笔直的线。Last leaf gave the machine a reverse gear — run it backwards and an output becomes an input again. This leaf visits the busiest machine in the shop: a taxi meter. Its bill, drawn on paper, turns out to be a perfectly straight line.
第 1 步：起步价 + 每公里单价Step 1: start fare + price per km
小算盘机器人打了一辆车，先看看车上的计价牌。The little abacus robot hails a taxi — let's read the fare board.
计价牌上写着：起步价 8 元，每公里 2 元。坐 x 公里，车费就是 y = 2x + 8。这类“只乘一次 x 再加”的函数叫一次函数，样子统一是 y = kx + b：k = 2 是每公里加的钱，b = 8 是起步价。The board reads: start fare 8 yuan, plus 2 yuan per km. Ride x kilometers and the fare is y = 2x + 8. A function that multiplies x once and adds a constant is a linear function, always shaped y = kx + b: k = 2 is the money added per km, b = 8 is the start fare.
为什么这张账单画出来一定是直线？Why is this bill always a straight line when drawn?
第 2 步：两个旋钮，一条直线Step 2: two knobs, one straight line
每多坐 1 公里都多加 2 元——上升量固定不变，点就连成一条直线。k 管陡缓：k 越大线越陡，车费涨得越快；k 为负，线就向下走。b 管起点：x = 0 时的值，也就是起步价。调这两个旋钮，直线能穿过任何指定的点；而钉住两个点，k 和 b 就被唯一确定了。Every extra kilometer adds exactly 2 yuan — a fixed rise, so the points join into a straight line. k sets the steepness: bigger k, steeper line, faster-growing fare; negative k slopes down. b sets the start: the value at x = 0, the start fare. Tune the two knobs and the line can pass through any chosen points; pin two points down and k and b are locked in.
一次函数藏在生活里的什么地方？Where does a linear function hide in daily life?
第 3 步：楼梯扶手就是一条直线Step 3: a staircase handrail is a straight line
楼梯每级升高一样、前进一样，所以扶手笔直。k 就是“每前进 1 步升高多少”，b 就是第一级的高度。出租车费、水费、打印费……凡是“固定起步 + 按量计费”的账单，画的都是同一条直线。Every stair rises the same and runs the same, so the handrail is straight. k is “how much you rise per step forward”, and b is the height of the first step. Taxis, water bills, printing fees — any bill that is “fixed start + pay per unit” draws the very same straight line.
🎮 你来开出租车（30 秒）🎮 Your turn: drive the taxi (30 seconds)
拖动 k 和 b 两个旋钮，让直线同时穿过两个金色靶点。4 关全过就通关。Drag the k and b knobs until the line passes through both golden targets. Clear all 4 levels to win.
一句话记住它：一次函数 y = kx + b 的图像是一条直线——b 管起点（x = 0 时的值），k 管陡缓（每多 1 个 x，y 增加多少）。Remember it in one line: y = kx + b graphs as a straight line — b sets the start (the value at x = 0), k sets the steepness (how much y grows per extra x).
起步价 8 元 + 每公里 2 元 → y = 2x + 8：b = 起步价，k = 单价Start fare 8 + 2 per km → y = 2x + 8: b is the start fare, k the price per kmk 越大线越陡，k 为负线向下；b 是 x = 0 时的起点Bigger k, steeper line; negative k slopes down; b is the height at x = 0“固定起步 + 按量计费”的账单都是直线；两点就能定下 k 和 bAny “fixed start + per-unit” bill is a line; two points pin down k and b
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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