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
At a state fair, attendees can win tokens that are worth a different number of points depending on the shape. One attendee won S square tokens and C circle tokens worth a total of 1,120 points. The equation \( 80S+90C=1,120 \) represents this situation. How many more points is a circle token worth than a square token?
原库参考解析
线性建模・解释与预测进阶
A certain township consists of a 5-hectare industrial park and a 24-hectare neighborhood. The total number of trees in the township is 4,529. The equation \( 5x+24y=4,529 \) represents this situation. Which of the following is the best interpretation of x in this context?
原库参考解析
线性建模・解释与预测基础
Keenan made 32 cups of vegetable broth. Keenan then filled x small jars and y large jars with all the vegetable broth he made. The equation \( 3x+5y=32 \) represents this situation. Which is the best interpretation of 5y in this context?
原库参考解析
线性建模・解释与预测挑战
5G +45R =380
At a school fair, students can win colored tokens that are worth a different number of points depending on the color. One student won G green tokens and R red tokens worth a total of 380 points. The given equation represents this situation. How many more points is a red token worth than a green token?
原库参考解析
线性建模・解释与预测挑战
Keenan made 32 cups of vegetable broth. Keenan then filled x small jars and y large jars with all the vegetable broth he made. The equation 3x +5y =32 represents this situation. Which is the best interpretation of 5y 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 2.10 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
An object hangs from a spring. The formula l =30+2w relates the length l , in centimeters, of the spring to the weight w, in newtons, of the object. Which of the following describes the meaning of the 2 in this context?
原库参考解析
线性建模・解释与预测挑战
An economist modeled the demand Q for a certain product as a linear function of the selling price P. The demand was 20,000 units when the selling price was $40 per unit, and the demand was 15,000 units when the selling price was $60 per unit. Based on the model, what is the demand, in units, when the selling price is $55 per unit?
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 9.10 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
According to data provided by the US Department of Energy, the average price per gallon of regular gasoline in the United States from September 1, 2014, to December 1, 2014, is modeled by the function F defined below, where F(x) is the average price per gallon x months after September 1.
F(x) =2.74 -0.19(x -3)
The constant 2.74 in this function estimates which of the following?
原库参考解析
线性建模・解释与预测进阶
The graph shown models the number of candy bars a certain machine wraps with a label in x seconds.
According to the graph, what is the estimated number of candy bars the machine wraps with a label per second?
原库参考解析
线性建模・解释与预测进阶
d =16tThe given equation represents the distance d, in inches, where t represents the number of seconds since an object started moving. Which of the following is the best interpretation of 16 in this context?
原库参考解析
