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・解释与预测基础
A jar is used to store marbles. The function \( f(x)=5x+270 \) gives the total mass, in grams, of the jar and the marbles inside, where x is the number of marbles in the jar. Which of the following is the best interpretation of \( f(0)=270 \) in this context?
参考答案
线性建模・解释与预测基础
The graph shown models the number of residents of a certain city x years after 2010. How many residents does this model estimate the city had in 2010?
参考答案
线性建模・解释与预测挑战
\( F(x)=\frac{9}{5}(x-273.15)+32 \)
The function F gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of x kelvins. If a temperature increased by 8.40 kelvins, by how much did the temperature increase, in degrees Fahrenheit?
The function F gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of x kelvins. If a temperature increased by 8.40 kelvins, by how much did the temperature increase, in degrees Fahrenheit?
参考答案
线性建模・解释与预测基础
The graph shown models the number of residents of a certain city x years after 2010. How many residents does this model estimate the city had in 2010?
参考答案
线性建模・解释与预测基础
The equation \( y=73x \) gives the estimated distance y, in kilometers, that a car traveled x hours after exiting a parking garage. What is the estimated distance, in kilometers, the car traveled 2 hours after exiting the parking garage?
参考答案
线性建模・解释与预测进阶
The graph in the xy-plane models the possible combinations of length x, in meters (m), and width y, in meters, for a rectangle with a perimeter of 24 m.
Which statement is the best interpretation of the point (10, 2) in this context?
参考答案
线性建模・解释与预测挑战
Kai used fabric measuring 4 yards in length to make each costume for a school play. The relationship between the number of costumes that Kai made, x, and the total length of fabric that he purchased y, in yards, is represented by the equation \( y-4x=3 \). What is the best interpretation of 3 in this context?
参考答案
线性建模・解释与预测进阶
Sarah used wax to make candles. The function \( f(x)=-4x+16 \) approximates the weight, in ounces, of wax Sarah 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?
参考答案
线性建模・解释与预测挑战
An office administrator spent a total of $161 to purchase x meals and y drinks for an office lunch. The equation \( 5x+3y=161 \) represents this situation. What is the best interpretation of 3y in this context?
参考答案
线性建模・解释与预测进阶
At a school carnival, students can win colored tickets that are worth a different number of points depending on the color. One student won G green tickets and R red tickets worth a total of 520 points. The equation 60G + 70R = 520 represents this situation. How many more points is a red ticket worth than a green ticket?
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 4 million in 1980 and increased by 0.041 million each year until 2007. 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
\( w\left(t\right)=300-4t \)
The function w models the volume of liquid, 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
