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・解释与预测进阶
\( W\left(x\right)=2x+30 \)
The function W gives the total time, in minutes, it takes for a gardener to water x plants and mow the grass. According to the function, how many minutes does it take the gardener to mow the grass?
原库参考解析
线性建模・解释与预测进阶
As part of a science experiment on evaporation, Ella measured the height of water in a glass over a period of time. The function \( f\left(x\right)=39-0.17x \) gives the estimated height, in centimeters (cm), of the water in the glass x days after the start of the experiment. What is the best interpretation of 0.17 in this context?
原库参考解析
线性建模・解释与预测进阶
A total of 263 toothpicks of equal length were used to construct two types of figures: triangles and squares. The triangles and squares were constructed so that no two figures had a common side. The equation \( 3x+4y=263 \) represents this situation, where x is the number of triangles constructed and y is the number of squares constructed. What is the best interpretation of \( \left(x,\ y\right)=\left(21,\ 50\right) \)in this context?
原库参考解析
线性建模・解释与预测挑战
\( F\left(x\right)=\frac{9}{5}\left(x-273.15\right)+32 \)
The function F gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of x kelvins. If a temperature increased by 7.70 kelvins. By how much did the temperature increase, in degrees Fahrenheit?
原库参考解析
线性建模・解释与预测挑战
Audrey's hair will grow at a constant rate from a length of 7 inches to a length of 11 inches. The equation \( 0.6t+7=11 \) describes this situation, where t is the number of months it will take for Audrey's hair to reach a length of 11 inches. Which of the following is the best interpretation of 0.6 in this context?
原库参考解析
线性建模・解释与预测挑战
A rectangle has a length that is 31 times its width. The function \( y=\left(31w\right)\left(w\right) \) represents this situation, where y is the area, in square feet, of the rectangle and \( y>0 \) . Which of the following is the best interpretation of 31w in this context?
原库参考解析
线性建模・解释与预测进阶
A rectangle has a length that is 72 times its width. The function \( y=\left(72w\right)\left(w\right) \) represents this situation, where y is the area, in square feet, of the rectangle and \( y> 0 \). Which of the following is the best interpretation of 72w in this context?
原库参考解析
线性建模・解释与预测进阶
\( s=40+2t \)
The equation gives the speed s, in miles per hour, of a certain car t seconds after it began to accelerate. What is the speed, in miles per hour, of the car 4 seconds after it began to accelerate?
原库参考解析
线性建模・解释与预测挑战
In one week in 2017, a technician earned a total of $700 by working at her regular job and at a second job doing part-time work. The equation \( 16h+13c=700 \) represents this situation, where h is the number of hours worked at her regular job and c is the number of hours worked at her second job. Which of the following is the best interpretation of 16 in this context?
原库参考解析
线性建模・解释与预测进阶
\( s=40+2t \)
The equation gives the speed s, in miles per hour, of a certain car t seconds after it began to accelerate. What is the speed, in miles per hour, of the car 3 seconds after it began to accelerate?
原库参考解析
线性建模・解释与预测进阶
In one week in 2017, a technician earned a total of $1,055 by working at her regular job and at a second job doing part-time work. The equation \( 20h+17c=1,055 \) represents this situation, where h is the number of hours worked at her regular job and c is the number of hours worked at her second job. Which of the following is the best interpretation of 20 in this context?
原库参考解析
线性建模・解释与预测挑战
A rectangle has a length that is 92 times its width. The function \( y=\left(92w\right)\left(w\right) \) represents this situation, where y is the area, in square feet, of the rectangle and \( y>0 \) . Which of the following is the best interpretation of 92w in this context?
原库参考解析
