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
\( T\left(x\right)=67-8x \)
The function T estimates the temperature, in degrees Fahrenheit (°F), in a certain city one day in January, where x is the number of hours after 10 a.m. and \( 1\le x\le5 \) . According to the function, what was the estimated decrease in temperature each hour, in °F in this city during this time period?
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 $900 by working at her regular job and at a second job doing part-time work. The equation \( 21h+10c=900 \) 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 21 in this context?
原库参考解析
线性建模・解释与预测进阶
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 (141.32, 172.26)?
原库参考解析
线性建模・解释与预测进阶
\( 20.5x+25.75y=411 \)
Odalys ordered mulch and river rock, which cost a total of $411, for her home. The given equation represents the relationship between the number of cubic yards of mulch, x, and the number of tons of river rock, y, Odalys ordered. How much more, in dollars, did a ton of river rock cost Odalys than a cubic yard of mulch?
原库参考解析
线性建模・解释与预测进阶
The graph models the number of online newsletter subscribers at the end of every six-month period, where x is the number of six-month periods since the end of January 1992 and \( 0\le x\le4 \) . Which statement is the best interpretation of the point (1, 750) in this context?
原库参考解析
线性建模・解释与预测进阶
A marine biologist uses a linear model to estimate the weight of a blue whale after it is born. The model estimates that a certain blue whale weighs 5,910 pounds at birth and gains 10.0 pounds per hour, for 120 hours, after it is born. Based on this model, what is the estimated weight, in pounds, of this blue whale 7 hours after it is born?
原库参考解析
线性建模・解释与预测进阶
\( s=10+4t \)
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 5 seconds after it began to accelerate?
原库参考解析
线性建模・解释与预测进阶
Antwon's hair will grow at a constant rate from a length of 3 inches to a length of 5 inches. The equation \( 0.3t+3=5 \) describes this situation, where t is the number of months it will take for Antwon's hair to reach a length of 5 inches. Which of the following is the best interpretation of 0.3 in this context?
原库参考解析
线性建模・解释与预测挑战
While walking on a path, Maksim stopped at a water fountain. The graph shows the total distance y, in meters, Maksim had walked on the path x minutes after leaving the water fountain.
What distance, in meters, did Maksim walk on the path in the 10 minutes after he left the water fountain?
原库参考解析
线性建模・解释与预测进阶
The graph shows the total amount of rainfall y, in centimeters, from the start of a 10-hour period, where x is the number of hours after the start of the period. Which of the following statements about the rainfall during this time period is true?
原库参考解析
线性建模・解释与预测挑战
The function t gives the estimated melting temperature, in degrees Celsius, of a DNA molecule as a function of the percent of guanine-cytosine (GC) content, x, in the DNA molecule. The graph of \( y=t(x) \) is shown. According to the graph, which of the following statements is NOT true?
原库参考解析
线性建模・解释与预测挑战
Scientists collected fallen acorns that each housed a colony of the ant species P. ohioensis and analyzed each colony's structure. For any of these colonies, if the colony has x worker ants, the equation \( y=0.67x+2.6 \) , where \( 20\le x\le110 \) , gives the predicted number of larvae, y, in the colony. If one of these colonies has 39 worker ants, which of the following is closest to the predicted number of larvae in the colony?
原库参考解析
