这部分需要会什么？
- 结合单位解释线性模型中的系数
- 利用模型预测并理解交点
- 区分模型的代数范围与实际适用范围
核心概念与公式
在 \(y=mx+b\) 中，\(m\) 表示 \(x\) 每增加 1，预测的 \(y\) 增加 \(m\)；若 \(m<0\)，表示减少。\(b\) 是输入为零时的模型值，只有 \(x=0\) 在情境中有意义时，才能解释为实际初始值。
两个费用模型的交点表示费用相等的输入；零点表示模型输出降到零的输入。利用观测范围内的输入作预测叫插值；超出范围的外推需要格外注意模型是否仍然适用。
做题时怎样使用
- 先读清变量定义，例如 \(t\) 是某年之后的年数还是具体年份
- 把斜率翻译成带有“每”和单位的句子
- 代入预测输入，并检查是否落在合理的时间、数量或测量范围内
容易混淆的地方
模型中的负斜率表示每单位减少固定量，不是减少固定百分比。若时间单位从年变为月，系数也要随之改变。
用一个例题串起方法
知识点教学例题水箱余水量满足 \(V=120-8t\)，其中 \(V\) 以升计、\(t\) 以分钟计。解释系数并求水箱何时排空。
展开解题思路 ＋
初始有 120 升水，每分钟减少 8 升。令 \(V=0\)，得 \(t=15\)。排空前模型的实际时间范围为 \(0\le t\le15\)，不能把负水量当作真实结果。
本知识点题库
先独立作答，再查看答案与解析。
线性建模・解释与预测挑战
\( \frac{x}{7}+\frac{y}{5}=\frac{43}{35} \)
An engineer connects resistors in series, where the resistors in the series have a total resistance of \( \frac{43}{35} \) ohms. In this series, there are x resistors of type A, which each have a resistance of a ohms, and y resistors of type B, which each have a resistance of b ohms. The given equation represents this situation. According to this equation, what is the positive difference between the value of a and the value of b?
参考答案
线性建模・解释与预测挑战
The graph shows the estimated boiling point y in degrees Celsius, of a straight-chain alkane with a molecular weight of x grams per mole, where \( 1\le x\le280 \). Which statement is the best interpretation of the point (169.39, 219.86)?
参考答案
线性建模・解释与预测挑战
A museum exhibit displays 73 artifacts on day 1. On each following day, 3 artifacts are added to the exhibit. The function \( f(x)=3(x-1)+73 \) represents this situation, where \( f(x) \) is the total number of artifacts exhibited at the end of day x. According to the function, what is the number of artifacts that have been added to the exhibit by the end of day x?
参考答案
线性建模・解释与预测进阶
\( w(w+89) \)
The given expression represents the area, in square centimeters, of a rectangle, where \( w \) represents the width, in centimeters (cm), of the rectangle. What is the best interpretation of \( w+89 \) in this context?
参考答案
线性建模・解释与预测进阶
The equation \( d=17w+100 \) gives the total amount of money d, in dollars, that James plans to save w weeks after he starts to save. What is the total amount of money, in dollars, he plans to save in 10 weeks?
参考答案
线性建模・解释与预测进阶
The equation \( d=25w+100 \) gives the total amount of money d, in dollars, that Jerome plans to save w weeks after he starts to save. What is the total amount of money, in dollars, he plans to save in 10 weeks?
参考答案
线性建模・解释与预测挑战
A museum exhibit displays 69 artifacts on day 1. On each following day, 7 artifacts are added to the exhibit. The function \( f(x)=7(x-1)+69 \)represents this situation, where \( f(x) \) is the total number of artifacts exhibited at the end of day x. According to the function, what is the number of artifacts that have been added to the exhibit by the end of day x ?
参考答案
线性建模・解释与预测进阶
The function \( f(x) \) is defined as 34 more than 5 times a number x. If \( y=f(x) \) is graphed in the xy-plane, what is the best interpretation of the x-intercept?
参考答案
线性建模・解释与预测进阶
A model estimates that the population of a state was 4 million in 1980 and increased by 0.041 million each year until 2007.
According to the model, what was the estimated population of this state, in millions, 10 years after 1980?
According to the model, what was the estimated population of this state, in millions, 10 years after 1980?
参考答案
线性建模・解释与预测进阶
The distance d, in feet, from the ground to a flag on a flagpole t seconds after a person begins lowering it is given by the function \( d(t)=26-4t \). In this context, what does 26 represent?
参考答案
线性建模・解释与预测挑战
The graph of the function \( f \), where \( y=f(x), \) gives the total cost y, in dollars, for a certain video game system and x games. What is the best interpretation of the slope of the graph in this context?
参考答案
线性建模・解释与预测挑战
\( F(x)=\frac{9}{5}(x-273.15)+32 \)
The function F gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of x kelvins. If a temperature increased by 6.50 kelvins, by how much did the temperature increase, in degrees Fahrenheit?
The function F gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of x kelvins. If a temperature increased by 6.50 kelvins, by how much did the temperature increase, in degrees Fahrenheit?
参考答案
