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
The equation \( d=8t+210 \) gives the distance d, in feet, from a reference point on the surface of Mars to a Mars rover, where t is the number of minutes after the start of a period of observation. By how many feet does the Mars rover’s distance from the reference point increase over a period of 20 minutes?
原库参考解析
线性建模・解释与预测挑战
\( f(x)=60x+8 \)
The given function \( f \) estimates the distance a train has traveled, in kilometers, from a station in a certain city \( x \) hours after crossing the city border. What is the best interpretation of 8 in this context?
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・解释与预测挑战
On January 1,2000, the population of a town was 26,255 and on January 1,2010, the population was 26,950. The equation \( 10x+26,255=26,950 \) describes this situation. Which of the following is the best interpretation of x in this context?
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 \( y=f(x) \), 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・解释与预测挑战
At the beginning of an experiment, the temperature of a liquid is 23 degrees Celsius (°C). During the first 4.0 minutes of the experiment, the temperature of the liquid increases at an average rate of 6.5°C per minute. Then, the temperature of the liquid increases at a constant rate of 2.4°C per minute. If the temperature of the liquid reaches \( 63^{\circ}\text{C} \) x minutes after the beginning of the experiment, where \( x>4.0 \), which equation represents this situation?
原库参考解析
线性建模・解释与预测挑战
\( \frac{x}{5}+\frac{y}{9}=\frac{47}{45} \)
An engineer connects resistors in series, where the resistors in the series have a total resistance of \( \frac{47}{45} \) 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?
原库参考解析
线性建模・解释与预测进阶
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?
原库参考解析
线性建模・解释与预测挑战
\( \frac{x}{7}+\frac{y}{5}=\frac{41}{35} \)
An engineer connects resistors in series, where the resistors in the series have a total resistance of \( \frac{41}{35} \) 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?
原库参考解析
线性建模・解释与预测挑战
The equation \( d=20w+100 \) gives the total amount of money d, in dollars, that Janna plans to save w weeks after she starts to save. What is the total amount of money, in dollars, she plans to save in 10 weeks?
原库参考解析
线性建模・解释与预测进阶
To purchase a used car at a total price of 13,000 dollars, a onetime down payment is required, and then fixed monthly payments are made for the remaining amount owed for the car. The equation \( 13,000=2,600+260t \) represents this situation, where t is the number of fixed monthly payments that are made. Which of the following is the best interpretation of 260 in this context?
原库参考解析
线性建模・解释与预测挑战
\( H=2.419L+22.83 \)
A group of biology students conducted an experiment to study the relationship between a person's femur length and the person's height. The given equation describes the relationship between the length L, in inches, of a student's femur and the student's estimated height H, in inches, for the students in the group. Which of the following is the best interpretation of 2.419 in this context?
原库参考解析
