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与函数建模
The Purpose of this Lesson
::本课程的目的

In this lesson, you will explore linear and other types of functions represented using tables, equations, and graphs.
::在此教训中,您将使用表格、方程和图表来探索线性和其他类型的函数。

Introduction: Independent and Dependent Variables
::导言:独立和依赖变量

Functions help you understand the relationship between quantities in the world. You can decide which variable is independent (the x-value) and which is dependent (the y-value). Often, changes in the dependent variable are a direct consequence of changes in the independent variable. It is said that the dependent variable is a function of the independent variable. The number of fidget spinners a factory produces is a function of the number of workers. The price of tickets to see a basketball team play is a function of their win percentage. You can choose the most useful representation of the function for your purposes, and choose the best method for answering a question about the scenario.
::函数可以帮助您理解世界数量之间的关系。您可以决定哪个变量是独立的( x 值) ,哪个是独立的( y 值) 。通常, 依附变量的变化是独立变量变化的直接结果。据说, 依附变量是独立变量的函数。工厂生产的飞毛腿螺旋体数量是工人人数的函数。观看篮球队比赛的票价是其赢率的函数。您可以为您选择最有用的函数表达方式, 并选择回答有关情景问题的最佳方法。

The number of fidget spinners produced in a factory depends on the number of workers.

Activity 1: Finding a Useful Function
::活动1:寻找有用的函数

Example 1-1
::例1-1

Kiara lives in Sacramento. The high temperature today was $\begin{aligned} 70^{\circ} F \end{aligned}$ . Her friend Catarina lives in Lisbon, Portugal, where the high temperature was $\begin{aligned} 37^{\circ} C \end{aligned}$ . She has always struggled to quickly interpret temperatures given in Celsius degrees. It's time for her to find a function that converts Celsius to Fahrenheit. Luckily, she knows the freezing point and boiling point of water in both Celsius and Fahrenheit:
::她的朋友卡塔琳娜(Catarina)住在葡萄牙里斯本, 高温为37°C。她一直努力快速解析摄氏度的温度。现在是她找到将摄氏度转换为法赫伦海特的功能的时候了。幸运的是,她知道摄氏度和法赫伦海特的冰冷点和沸水点:

$\begin{aligned} \begin{array}{lcc} ^{\circ} C & ^{\circ} F \\ Freezing & 0 & 32 \\ Boiling & 100 & 212 \end{array} \end{aligned}$

::CF 冻结032 堆放100212

Which representation of temperature do you prefer, Celsius or Fahrenheit, and why? Would you prefer an equation which represents Fahrenheit as a function of Celsius, or the other way around? Why? Do you think the relationship between Fahrenheit and Celsius is linear? Explain.
::你喜欢哪种温度的表示, 摄氏度或华氏度, 以及为什么呢? 您更喜欢以摄氏度函数表示法氏度的方程式吗? 为什么? 您认为法氏度和摄氏度之间的关系是线性的吗? 请解释。

Solution: The relationship between Fahrenheit and Celsius is linear. If temperature increases by $\begin{aligned} 5^{\circ} C \end{aligned}$ , that corresponds to an increase of $\begin{aligned} 9^{\circ} F \end{aligned}$ , regardless of what the initial temperature was. Imagine a scale where this relationship wasn't linear!
::解决方案 : Fahrenheit 和摄氏之间的关系是线性。如果温度上升 5 °C, 则相当于增加 9 °F, 不论最初的温度是多少。想象一下这种关系不是线性的尺度 !

PLIX Interactive
::PLIX 交互式互动

Work it Out
::工作出来

Create a longer table which shows the conversion of values from degrees Celsius to Fahrenheit. Convert values from 0 to 50 in increments of 5. Which ordered pairs can you remember so that you have a quick reference when you hear a temperature in Celsius and want to get an estimate?
::创建一个较长的表格, 显示数值从摄氏度向华氏度的转换。将值从0转换为50, 递增值为5 。您可以记住哪对有顺序的对子, 这样当听到摄氏度的温度并想要获得估计值时可以快速引用 ?

Consider each of the following scenarios involving a relationship between two quantities . What is the independent variable and what is the dependent variable? Explain. Is the relationship linear or non-linear? Explain. Do you think that given enough information you could create a table, graph or equation for the function? Why or why not? What information would you need?

A child releases a balloon in a park. As time passes, the balloon rises.
::孩子在公园里放气球时间流逝气球就会升起

Sesame seeds are scattered on the ground. As more birds arrive, the number of seeds decrease.
::芝麻种子分散在地上,随着更多的鸟类到来,种子数量减少。

The number of bacteria in a petri dish are doubling every day.
::花生菜中的细菌数量每天都在翻倍

A bicycle passes the 120-kilometer mark in 3 hours, and the 200-kilometer mark in 7 hours.
::自行车在3小时内通过120公里的标记,在7小时内通过200公里的标记。

A factory produces 10 cars every 3 hours.
::工厂每3小时生产10辆汽车。

A gym membership costs $110 to open and $30 per month thereafter.
::健身馆会员费用110美元,之后每月30美元。

A child gets a giant bag of sweetened popcorn. He eats half of what's in the bag on the first day. On the second day, he eats half of what's left. And so on.
::一个孩子得到一大袋甜爆米花他第一天吃袋里一半的东西第二天吃剩下的一半

A rock is dropped from the top of a cliff into a reservoir. The height of the rock changes as time passes.
::岩石从悬崖的顶部掉入储油层。随着时间的流逝,岩石的高度会变化。

100 kilometers is about 62 miles.
::100公里约62英里

Jim and Todd are running a race. As Jim's heart rate increases, so does Todd's.
::吉姆和托德在赛跑,随着吉姆的心率上升,托德的心率也上升

::考虑下两种数量之间的关系的以下情景。独立变量和依附变量是什么? 解释。关系线性还是非线性? 解释 ; 解释。您认为给出了足够信息您可以为函数创建表格、图表或方程式吗? 为什么或不? 您需要什么信息? 儿童会在公园里释放气球。随着时间的流逝, 气球上升。芝麻种子散布在地面上。随着更多的鸟类的到来, 种子的数量会减少。花生盘中的细菌数量每天都会翻倍。自行车在3小时内通过120公里的标记, 而在7小时内通过200公里的标记。工厂每3小时生产10辆车; 为何不这样做? 您需要什么信息? 儿童会在公园里放气球? 随着时间的流逝世, 气球团会上升。当更多的鸟到达时, 他吃半个袋子里的东西。第二天, 他吃半个剩下的东西。然后, 岩石会从悬崖顶部掉到一个水库里。岩石的高度是每7小时10公里的高度。随着时间飞速变化, 。吉姆的心脏速度是62英里。

Activity 2: B eyond Linear Functions
::活动2:超越线性函数

As you saw in the last problem, n ot every relationship between two quantities is linear. But many of these relationships are still functions that can be modeled with equations, tables, and graphs. In this section you will begin by exploring a non-linear relationship.
::正如您在最后一个问题中看到的那样,并非两个数量之间的每一个关系都是线性关系。但其中许多关系仍然是可以用方程式、表格和图表建模的函数。在本节中,您将首先探索一种非线性关系。

Cheryl builds square-shaped kites

Work it Out
::工作出来

Cheryl builds square-shaped kites using lightweight wooden rods and fabric. The rods come in lengths that are in one-foot increments, so she can build a square kite with a side length of 1 foot, 2 feet, and so on. Sketch a few possible kites. Cheryl wants to know how the amount of fabric she needs for the kite changes as the side lengths increase. How can you measure the amount of fabric she needs? Make a table that expresses the relationship between the side length and the area of fabric required.
::Cheryl用轻巧的木棍和织物建造方形风筝。棒子的长度是一英尺长, 这样她就可以建造一平方风筝, 侧长1英尺、 2英尺等等。拖几只可能的风筝。 Cheryl想知道随着边长的增加, 她需要多少布料来改变风筝。您如何测量她需要的布料量? 制作一张表, 显示侧长与需要的织物面积之间的关系。

Do you prefer to express area as a function of the side length, or the other way around? Why? Find an equation which relates the two quantities.
::您是否更愿意用侧边长度的函数表示区域 ? 还是以另一边的函数表示区域 ? 为什么? 找到一个与两个数量相关的方程式。

Create a graph of this function. Is this function linear? How do you know?
::创建此函数的图形。这是线性函数吗? 你怎么知道?

In this scenario, does it make sense to discuss the area of fabric for a square with a side length of 7.5 feet? Why or why not? How about a side length of -2 feet? Why or why not? How about a side length of feet? Why or why not?
::在这种情形下,讨论长7.5英尺的广场的布料面积是否合理?为什么不行?侧长2英尺如何?为什么不行?侧长2英尺?为什么不行?侧长2英尺如何?侧长1英尺如何?为什么不行?为什么不行?侧长1英尺?为什么不行?为什么不行?为什么不行?

Does the area increase proportionally with the side length? Why or why not? Explain. As the side length doubles from 3 to 6, what happens to the area?
::区域是否随边长成比例地增加? 为什么或为什么不? 解释一下。当边长从3倍增加到6倍时,这个区域会怎样呢?

As the side length doubles from 5 to 10, what happens to the area? Why?
::当侧长从5乘5乘10乘10时,这个区域会怎样?为什么?

Comparing the expressions below gives you a way to prove the result described above. Explain the setup and each step in the process. Explain how the result proves the result you observed in the last problem. How is this different from the behavior of a linear function ?
::比较下面的表达式为您提供一个方法来证明上面描述的结果。解释设置和过程的每一个步骤。解释结果如何证明您在最后一个问题中观察到的结果。这与线性函数的行为有何不同?

$\begin{aligned} \begin{array}{ll} Input & Output \\ s & s^{2} \\ 2 s & (2 s)^{2} = (2 s) (2 s) = 2 \cdot 2 \cdot s \cdot s = 4 s^{2} \end{array} \end{aligned}$

::输入输出输出 22s( 2s) 2= ( 2s( 2s) = 222s= 4s2

Quadratic Functions
::二次曲线函数

The relationship between $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ in the equation $\begin{aligned} y = x^{2} \end{aligned}$ is quadratic . You may be familiar with the words quadruple, quadcopter, or quadrilateral. Each of those words is related to the idea of multiplying by 4, or creating a 4-sided figure. Quadratic equations will appear frequently throughout high school mathematics. Just like linear equations, they can be used to model a great many real-world scenarios, from square kites to any object with constant acceleration .

Activity 3: Linear and Non-linear Models
::活动3:线性和非线性模型

Work it Out
::工作出来

Below are scenarios which lead to functions relating two quantities. Decide if the functions are linear or non-linear. Explain your decisions. If you can give a name for the non-linear relationships, do so (for example, some of the relationships are quadratic).
::下面是导致两个数量相关的函数的情景。决定函数是线性还是非线性。请解释您的决定。如果您能给出非线性关系的名称, 请这样做( 例如, 有些关系是二次关系 ) 。

A car is driving at 50 miles per hour. The distance traveled is a function of the time that has passed.
::车速每小时50英里,行驶的距离是经过时间的函数。

The height of a falling baseball as a function of time is given by the equation $\begin{aligned} h (t) = - 4.9 t^{2} + 100. \end{aligned}$ .
::跌落的棒球作为时间函数的高度由等式 h(t)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\可以\\\\\\\\\\\\\\\\\\\\\\\\\\"\\\\\\"。

The relationship between a student's age and her height in decimal feet is expressed by the following ordered pairs: $\begin{aligned} (14, 4.9) (15, 5.0) (16, 5.4) \end{aligned}$
::学生年龄与其十进制脚的身高之间的关系由下列定购夫妇(14,4.9)(15,5.0)(16,5.4)表示。

The magnitude of earthquakes is measured using the Richter Scale. The table below represents the relationship between the amplitude of the shaking and the corresponding measurement of magnitude on Richter Scale.
::地震的大小是用“里氏比例尺”测量的。下表显示了震动的振幅与“里氏比例尺”相应测量的大小之间的关系。

$\begin{aligned} \begin{array}{cc} Amplitude & Richter Scale \\ 10 & 1 \\ 100 & 2 \\ 1000 & 3 \\ 10, 000 & 4 \\ 15, 000 & \approx 4.18 \end{array} \end{aligned}$

::振幅 - 摄氏101100210000310000010000415 000 4.18

A colony of penguins living in Antartica has doubled in size every year.
::生活在Antartica的企鹅群规模每年翻了一番。

A glider launched from the top of a cliff flies at an angle of depression of $\begin{aligned} 20^{\circ} \end{aligned}$ .
::从悬崖顶顶部发射的滑翔机以20°Z的压抑度为角

A glider launched from a cliff

Give five examples of scenarios which involve a relationship between two quantities: two that are linear and three that are non-linear. Determine the independent and dependent variable. Use your examples to summarize the differences and similarities between linear and non- and scenarios.
::举五个例子说明两种数量之间的关系的假设情景:两个是线性,三个是非线性。确定独立和依附的变量。用您的例子总结线性和非线性以及假设情景之间的差异和相似性。

The glider described above is launched from the top of a 1000 foot cliff, soaring into the flat plain below. Create a graph to represent this scenario. What do and represent? Use a protractor to measure the angle of depression from the horizontal, and graph the resulting line. Find the approximate from the graph. Write the equation for the line. What does each part of your equation represent? Use your equation to find where the glider lands (that is, the distance from the base of the cliff).
::上述滑翔机从1000英尺悬崖的顶部发射,飞向下面平原。创建一个图表来代表这个假设情景。什么是和代表的? 使用一个减压器从水平上测量低压角度, 并绘制所生成的线条。从图形中找到大约的长度。写入线的方程。您的方程的每个部分代表着什么? 使用您的方程查找滑翔场的位置( 即与悬崖底的距离 ) 。

Interactive
::交互式互动

A glider is launched from a clifftop, soaring into the flat plain below. Use the following interactive to vary the angle of depression and the height from which the glider is thrown. Describe the changes in the resulting flight path as the parameters are changed.
::滑翔机从悬崖顶上发射, 飞向下面平原。使用以下互动方式来改变压抑角度和滑翔机倾斜的高度。当参数发生变化时, 请描述由此而来的飞行路径的变化。

INTERACTIVE

Finding Functions

In the interactive, move the green point on the slider to change the line's angle of depression.
::在互动中,移动滑块上的绿点,以改变线条的压抑角度。

Move the red point on the slider to change the $y$ -intercept of the line.
::移动滑块上的红色点以更改线条的 Y 界面。

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Summary
::摘要

Functions represent relationships between different quantities in the world.
::功能代表世界不同数量之间的关系。

In order to find a function to represent a scenario, you decide which variable is dependent and which is independent.
::为了找到一个函数来代表一个假想,您可以决定哪个变量是依赖的,哪个变量是独立的。

You also assess whether the relationship is linear or non-linear.
::您还评估这种关系是线性关系还是非线性关系。

You can represent the relationship with a table, graph or equation, and if possible, find the function that relates the two quantities.
::您可以代表与表格、图表或方程式的关系,如果可能,可以找到与两个数量相关的函数。

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