函数、坐标与参数Functions, Coordinates & Parameters
同一条规则里的参数改变时，输入与输出的关系怎样变化？How does changing a parameter change the relationship between input and output?
前置知识：Prerequisites: 高中代数与平面坐标；熟悉的读者可先做文末小检查。High-school algebra and Cartesian coordinates. If these are familiar, try the check at the end first.
学完后你能：After this lesson you can: 读出坐标，区分输入和参数，并预测斜率或截距变化的效果。Read coordinates, distinguish inputs from parameters, and predict the effect of slope or intercept changes.
动手观察Explore
先预测：让截距增加，直线会如何移动？再调节两个参数，核对图与表。Predict: what happens when the intercept increases? Adjust both parameters and compare the graph with the table.
从观察到解释From observation to explanation
先观察一个输入Observe one input
横轴表示输入 x，纵轴表示输出 y。点 (2, 5) 的意思是：输入 2，规则给出输出 5。改变输入是在同一条图线上寻找另一个点。The horizontal axis is input x and the vertical axis is output y. The point (2, 5) means the rule outputs 5 for input 2. Changing the input selects a different point on the same graph.
再观察整条规则Now change the rule
先固定 a = 2、b = 1：x = −1、0、1 对应 y = −1、1、3。把 b 增加 1，所有输出都增加 1，直线平行上移；把 a 增加，则每向右走一步时 y 的增量改变。First fix a = 2 and b = 1: inputs −1, 0, 1 give outputs −1, 1, 3. Increasing b by 1 adds 1 to every output and shifts the line upward. Increasing a changes the rise for each step to the right.
用在机器学习里Connection to machine learning
一个简单模型也可以写成 y = ax + b：输入是特征，参数控制预测规则。训练会调整参数；这里先只关注参数变化如何影响图像。A simple model can also be written y = ax + b: the input is a feature and the parameters control the prediction rule. Training adjusts these parameters; here we focus only on how they affect the graph.
公式与符号Formula and symbols
x 是输入，y 或 f(x) 是输出；a 是斜率，b 是纵轴截距。a、b 是可以固定或调整的参数。x is the input and y or f(x) is the output; a is the slope and b the vertical intercept. The parameters a and b can be held fixed or adjusted.
留意这个区别Watch for this
区分改变一个输入与改变参数：前者沿同一图线移动，后者改变整条规则。斜率为负时，x 增大反而使 y 减小。Distinguish changing an input from changing a parameter: the former moves along one graph, while the latter changes the rule. A negative slope means y decreases as x increases.
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