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
A bookstore is giving away free books every day. The equation \( f(x)=162-9x \) gives the number of free books, \( f(x) \), that remain to be given away after x days. Which of the following is the best interpretation of \( f(18)=0 \) in this context?
原库参考解析
线性建模・解释与预测挑战
As part of a science experiment on evaporation, Ella measured the height of water in a glass over a period of time. The function \( f(x)=32-0.17x \) gives the estimated height, in centimeters (cm), of the water in the glass x days after the start of the experiment. Which of the following is the best interpretation of 32 in this context?
原库参考解析
线性建模・解释与预测进阶
On January 1, 2000, the population of a town was 26,257 and on January 1, 2010, the population was 26,858. The equation \( 10x+26,257=26,858 \) 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
The function w models the weight, in ounces, of the remaining solid wax of a burning candle as a function of the time x , in hours, after the candle was lit. The graph of \( y=w(x) \) is shown. According to the model, what was the weight, in ounces, of the candle's remaining solid wax 12 hours after the candle was lit?
原库参考解析
线性建模・解释与预测挑战
A cable company charges new customers a onetime installation fee and a monthly service fee for cable service. The equation \( y=50x+120 \) gives the total amount y, in dollars, the cable company charges new customers for x months of cable service. What is the amount, in dollars, of the onetime installation fee the cable company charges new customers?
原库参考解析
线性建模・解释与预测挑战
A bookstore is giving away free books every day. The equation \( f(x)=200-10x \) gives the number of free books, \( f(x) \), that remain to be given away after x days. Which of the following is the best interpretation of \( f(20)=0 \) in this context?
原库参考解析
线性建模・解释与预测挑战
Sunspots are temporary dark spots on the surface of the Sun. The monthly mean number of sunspots for a calendar month is the average of the number of sunspots observed each day during the month. The linear function s models the monthly mean number of sunspots as a function of the number of months, x, since December 2013, where \( 0\le x\le30 \) . The graph of \( y=sx \) is shown.
Which of the following is the best estimate for the monthly mean number of sunspots in December 2014?
原库参考解析
线性建模・解释与预测进阶
A rectangle has a length that is 70 times its width. The function \( y=\left(70w\right)\left(w\right) \) 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 70w in this context?
原库参考解析
线性建模・解释与预测进阶
A certain gas is placed inside a container with a constant volume. The graph shows the estimated pressure y, in pounds per square inch (psi), of the gas when its temperature is x kelvins.
What is the estimated pressure of the gas, in psi, when the temperature is 500 kelvins?
原库参考解析
线性建模・解释与预测进阶
The graph shows the estimated balance of a bank account, in dollars, x years after an initial deposit is made. According to the graph, which of the following is closest to the number of years it took for the estimated balance to increase from the initial deposit to 125 dollars?
原库参考解析
线性建模・解释与预测挑战
The function \( f \) models the resistivity of copper, in nanoohm · meters, as a function of the temperature x, in kelvins, of the copper. The graph of \( y=f(x) \) is shown.
According to the model, which of the following statements about the estimated resistivity of copper is true?
原库参考解析
线性建模・解释与预测挑战
\( p(x)=132-4x \)
The function p shown gives the number of blank sheets of paper, \( p(x) \), remaining in a printer x days after the printer had been loaded with paper. Which statement is the best interpretation of \( p(6)=108 \)?
参考答案
