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
\( W\left(x\right)=4x+80 \)
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
A bank account was opened with an initial deposit. Over the next several months, regular deposits were made into this account, and there were no withdrawals made during this time. The graph of the function \( f \) shown, where \( y=f\left(x\right) \), estimates the account balance, in dollars, in this bank account x months since the initial deposit. To the nearest whole dollar, what is the amount of the initial deposit estimated by the graph?
原库参考解析
线性建模・解释与预测进阶
The function \( F\left(m\right)=7m \), models the force, in newtons, acting on each object in a certain field as a function of the object's mass m, in kilograms. Which statement is the best interpretation of \( F\left(4\right)=28 \) in this context?
原库参考解析
线性建模・解释与预测进阶
A school swim team sold shirts to raise money. The team raised $13 from each short-sleeved shirt sold and raised $14 from each long-sleeved shirt sold. The team raised a total of $423 from the sale of these shirts. The equation \( 13x+14y=423 \) represents this situation. Which of the following is the best interpretation of y in this context?
原库参考解析
线性建模・解释与预测进阶
A linear model estimates the population of a city from 1995 to 2019. The model estimates the population was 55 thousand in 1995, 217 thousand in 2015, and x thousand in 2019. To the nearest whole number, what is the value of x?
原库参考解析
线性建模・解释与预测进阶
A family has money in an account for renting movies online. Each time the family rents a movie, the cost of the rental is withdrawn from the account, and each rental costs the same amount of money. The function \( f\left(m\right)=21-3m \) gives the amount of money, in dollars, in the account after the family has rented m movies. Which of the following represents the amount of money, in dollars, withdrawn from the account each time the family rents a movie?
原库参考解析
线性建模・解释与预测进阶
The graph models the number of active projects a company was working on x months after the end of November 2011, where \( 0\le x\le6 \). According to the model, what is the predicted number of active projects the company was working on at the end of November 2011?
原库参考解析
线性建模・解释与预测进阶
\( d=15t \)
The given equation relates the distance d, in inches, of an object from its starting position to the time t, in seconds, since the object started moving. What is the rate of change of the object's distance from its starting position over time?
原库参考解析
线性建模・解释与预测进阶
A company that manufactures staplers calculates its monthly profit, in dollars, by subtracting its fixed monthly costs, in dollars, from its monthly sales revenue, in dollars. The equation \( 11,000=2.00x-6,500 \) represents this situation for a month where x staplers are manufactured and sold. Which statement is the best interpretation of \( 2.00x \) 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 24 times its width. The function y=(24w)(w) 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 24w in this context?
原库参考解析
线性建模・解释与预测进阶
A linear model estimates the population of a city from 1992 to 2016. The model estimates the population was 61 thousand in 1992, 217 thousand in 2012, and x thousand in 2016. To the nearest whole number what is the value of x?
原库参考解析
线性建模・解释与预测挑战
Kenny engages in up to 3 types of exercise each week for a total of 6 hours while training for a triathlon. Kenny swims for the same number of minutes each week. The equation \( y=360-x-170 \) represents the situation where Kenny bikes for x minutes during a week and runs for any remaining training time y, in minutes. If this equation is graphed in the xy-plane, which statement is the best interpretation of the x-intercept of the graph?
原库参考解析
