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
The graph gives the estimated population y, in thousands, of a town x years since 2001, where 0≤ x ≤5 . which of the following best describes the increase in the estimated population from x = 0 to x = 1?
原库参考解析
线性建模・解释与预测进阶
If a circuit consists of resistors connected in series, its resistance, in ohms, is equal to the sum of the resistances, in ohms, of all these resistors. A particular circuit consists of x identical resistors of type X and y identical resistors of type Y, which are connected in series. The resistance R, in ohms, of this circuit is given by the equation R = 13x +15y . Which of the following is the best interpretation of 13 in this context?
原库参考解析
线性建模・解释与预测进阶
s =50 +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 2 seconds after it began to accelerate?
原库参考解析
线性建模・解释与预测进阶
f(x)= 3x+500
The function shown gives the total cost f(x), in dollars, for making x units of a certain product. What is the total cost, in dollars, for making 100 units of this product?
原库参考解析
线性建模・解释与预测进阶
A model estimates that the population of a state was 3 million in 1980 and increased by 0.043 million each year until 2008. According to the model, what was the estimated population of this state, in millions, 10 years after 1980?
原库参考解析
线性建模・解释与预测进阶
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 a worker ants, the equation y = 0.67x + 2.6, where \( 20\le x\le110 \) , gives the predicted number of larvae, y, in the colony. lf one of these colonies has 35 worker ants, which of the following is closest to the predicted number of larvae in the colony?
原库参考解析
线性建模・解释与预测进阶
19.5x+24.25y=583
Odalys ordered mulch and river rock, which cost a total of $583, 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
f(x)=7x+3
The function f gives the estimated height, in feet, of a willow tree x years after its height was first measured. Which statement is the best interpretation of 3 in this context?
原库参考解析
线性建模・解释与预测进阶
Harper deposited a fixed amount into her bank account each month. The function f(t)= 250+30t gives the amount, in dollars, in Harper's bank account after t monthly deposits. What is the best interpretation of 30 in this context?
原库参考解析
线性建模・解释与预测挑战
A school swim team sold shirts to raise money.The team raised $13 from each short-sleeved shirt sold and raised $15 from each long-sleeved shirt sold. The team raised a total of $474 from the sale of these shirts. The equation 13x + 15y = 474 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・解释与预测挑战
w(t)=600-3t
The function w models the volume of water, in milliliters, in a container t seconds after it begins draining from a hole at the bottom, According to the model, what is the predicted volume, in milliliters, draining from the container each second?
原库参考解析
线性建模・解释与预测挑战
A linear model estimates the population of a city from 1992 to 2016. The model estimates the population was 56 thousand in 1992, 218 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
