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
22.5x+ 26.75y=242
Odalys ordered mulch and river rock, which cost a total of $242, 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
g(m)= -0.05m+12.4
The given function g models the number of gallons of gasoline that remains from a full gas tank in a car after driving m miles, According to the model, about how many gallons of gasoline are used to drive each mile?
原库参考解析
线性建模・解释与预测进阶
The graph models the population, in thousands, of a town x years since 2004, where 0 < x < 15. Which statement is the best interpretation of the point (1, 3.21)?
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 Each time the family rents a movie, the cost of the rental is withdrawn from the account, and each rental costs the same amount of money.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 (m)= 32- 4m 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 Which of the following represents the amount of money, in dollars, withdrawn from the account each time the family rents a movie?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・解释与预测挑战
Elise used x packs of red beads and y packs of green beads to make 13 bracelets. She used a total of 260 red and green beads. The equation 45x + 25y = 260 represents this situation. Which is the best interpretation of 25y in this context?
原库参考解析
线性建模・解释与预测挑战
The function f(x) = 7x + 287 predicts the total number of brown sparrows in a certain region x months after the initial observation. Which statement best interprets the y-intercept of the graph of y = f(x) in the xy-plane in this context?
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 x worker ants, the equation y = 0.67x ＋2.6, where 20≤ x ≤110, gives the predicted number of larvae, y, in the colony. If one of these colonies has 46 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
In 2000, a census was taken to determine the population of a certain town. The graph gives the estimated population y, in thousands, of this town x years after the 2000 census was taken, where 0 ≤ x ≤ 5. Based on the graph, which of the following is the closest to the estimated population 2 years after the 2000 census was taken?
原库参考解析
线性建模・解释与预测挑战
A certain open star cluster contains M-type stars and K-type stars. The estimated total mass of M-type and K-type stars in this open star cluster is 132,778 quettagrams. The graph shown models the possible combinations of the number of M-type stars, x, and K-type stars, y, that could be in this open star cluster if all the M-type stars have the same estimated mass and all the K-type stars have the same estimated mass.Based on the graph, which of the following is closest to the estimated mass, in quettagrams, of each M-type star in this cluster?
原库参考解析
线性建模・解释与预测挑战
At a depth of h meters below the surface, the estimated total pressure P1, in kilopascals (kPa), a diver experienced is given by P1 =10h ＋101, which is a combination of estimated water pressure and estimated atmospheric pressure. After maintaining a certain depth, the diver descended d additional meters. The estimated total pressure P2, in kPa, the diver experienced after descending these d additional meters is given by P2 = 10d ＋311. Which of the following is the best interpretation of 311 in this context?
原库参考解析
线性建模・解释与预测进阶
T(x) =61 - 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≤ x ≤7. Which statement is the best interpretation of T(5)= 21 in this context?
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 5 inches to a length of 9 inches. The equation 0.4t＋5= 9 describes this situation, where t is the number of months it will take for Antwon's hair to reach a length of 9 inches. Which of the following is the best interpretation of 0.4 in this context?
原库参考解析
