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 shown gives the estimated value, in dollars, of a mountain bike as a function of the number of months since it was purchased. What is the best interpretation of the y-intercept of the graph 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
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 120,134 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 +461. Which of the following is the best interpretation of 461 in this context?
原库参考解析
线性建模・解释与预测挑战
\( F\left(x\right)=\frac{9}{5}\left(x-273.15\right)+32 \)
The function F gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of x kelvins. If a temperature increased by 13.30 kelvins, by how much did the temperature increase, in degrees Fahrenheit?
原库参考解析
线性建模・解释与预测挑战
Eli used wax to make candles. The function f (x)= - 9x + 54 approximates the weight, in ounces, of wax Eli had remaining after making x candles. Which statement is the best interpretation of 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・解释与预测挑战
\( f\left(x\right)=50x+12 \)
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 12 in this context?
原库参考解析
线性建模・解释与预测挑战
Two students are playing a game. In the first round, Player 1 answers 49 questions. If an answer is correct, Player 1 earns 1 point. If an answer is incorrect, Player 2 will earn 1 point instead. The graph shows \( y=f\left(x\right) \) , where y is the number of points Player 2 will earn when x is the number of points Player 1 earns. Which of the following is the best interpretation of the point (49, 0) in this context?
原库参考解析
线性建模・解释与预测挑战
The function f(x) is defined as 17 more than 3 times a number x. If y = f(x) is graphed in the xy-plane, what is the best interpretation of the x-intercept?
原库参考解析
线性建模・解释与预测挑战
The graph gives the estimated population y, in thousands, of a town x years since 2007, 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・解释与预测挑战
T(x) = 62 - 8x
The function T estimates the temperature, in degrees Fahrenheit (ºF), in a certain city one day in January, where x is the number of hours after 10 a.m.and 1≤ x ≤5. According to the function, what was the estimated decrease in temperature each hour, in ºF, in this city during this time period?
原库参考解析
线性建模・解释与预测进阶
\( s=10+2t \)
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 3 seconds after it began to accelerate?
原库参考解析
