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比较函数类型
The Purpose of This Lesson
::本课程的目的

In this lesson you will compare and contrast the different types of functions you explored in this chapter. You'll consider the scenarios they model, the differences in their , the structure and parameters of their equations, and the patterns visible in their tables and graphs.
::在此教训中, 您将比较和对比您在本章中探讨的不同类型函数。您将考虑他们模型的情景、它们的方程差异、它们的方程结构和参数, 以及它们在表格和图表中可见的模式。

Introduction: Types of Functions
::导言:职能类型

Th ere are several function types, including linear, exponential, quadratic, and rational. Each type has particular identifying traits and specific applications and uses. Knowing how to identify and apply different function types is vital to making them useful in applied mathematics.
::有几种功能类型, 包括线性、指数性、二次曲线和理性性。每种类型都有特定的识别特性和具体应用及用途。了解如何识别和应用不同的功能类型对于使其在应用数学中发挥作用至关重要。

Activity 1: Comparing Functions
::活动1:比较职能

Example 1-1
::例1-1

Identify each of the following equations as linear, exponential, quadratic, rational, or none of those.
::将以下方程式中的每一个方程式识别为线性、指数性、二次方程式、理性方程式或非线性、指数性、二次方程式、理性方程式。

$\begin{aligned} \begin{array}{l} y = 5 x^{2} \\ y = \sqrt{x} \\ y = 2^{x} \\ y = 3 x + 4 \\ y = 7 x^{3} \\ y = (\frac{10}{x}) \\ y = (\frac{10}{x^{2}}) \\ y = 7 (3)^{x} \\ y = | x | \end{array} \end{aligned}$
$\begin{align*}\begin{array}{l} y=5x^2\\ y=\sqrt{x}\\ y=2^x\\ y=3x+4\\ y=7x^3\\ y=\left( \frac{10}{x} \right)\\ y=\left( \frac{10}{x^2} \right)\\ y=7(3)^x\\ y=\left| x \right| \end{array}\end{align*}$
::y= (10x)y= (10x2)y= (10x2)y= 7(3)xy= 7(3)xy*x*x*}= (10x)y= (10x)y= (7x2y=)

Solution:

$\begin{align*}\begin{array}{l|l|l} \text{Function} & \text{Type} & \text{Explanation}\\ \hline y=5x^2 & \text{quadratic} & \text{The structure of a quadratic is}~y=ax^2.\\ y=\sqrt{x} & \text{square root} & \text{This type not explored in this chapter, you will see it later.}\\ y=2^x & \text{exponential} & \text{Exponential function structure is}~y=a(b)^x.~\text{In this case}~a=1.\\ y=3x+4 & \text{linear} & \text{Linear functions in slope-intercept form look like}~y=mx+b.\\ y=7x^3 & \text{cubic} & \text{Not explored in this chapter, though you will see it later.}\\ y=\frac{10}{x} & \text{rational} & \text{Simple rational functions have the structure}~y=\frac{a}{x}\\ y=\frac{10}{x^2} & \text{rational} & \text{A more complex rational function.}\\ y=7(3)^x & \text{exponential} & \text{Fits the structure described above.}\\ y=\left| x \right| & \text{absolute value} & \text{Not explored in the chapter, you'll see it later.} \end{array} \end{align*}$
::解析函数TypeExplainationy= 5x2quatratic 。二次方形结构为 y= ax2. y=xquare root 。本章未探讨的类型。您可以稍后看到它。 y= 2xexexexitive 函数结构为 y= a(b)x。在此情况下, 斜度- intervict 形态中a= 1. y= 3x+ 4linearLinear 函数看起来像 y= mx+b.y= 7x3cubic 。虽然您可以看到它稍后会看到它。 y= 10xationalSempty 理性函数结构为 y= = 10x2LationalA 更复杂的理性函数。 y= 7= 3xexexexponitialFits 上述结构没有在章节中进行探讨, 您以后会看到它。

A rocket accelerates at 20 meters per second, per second

Example 1-2
::例1-2

Each of the following scenarios can be modeled with a linear, exponential, quadratic, or rational function . Determine which type would work best. Complete the following statement for each: I could write an equation for ________ as a function of _________.
::以下每种情景都可以以线性、指数性、二次函数或理性函数为模型。确定哪种类型最有效。填写以下每种语句:我可以以 __ 的函数写出 __ 的方程式。

Identify the most appropriate type of model for each situation:
::为每一种情况确定最适当的模式类型:

A gym membership costs $45 to start and $30 per month thereafter.
::健身馆会员费用45美元,此后每月30美元。

A rocket accelerates at 20 meters per second, per second.
::一枚火箭加速速度为每秒20米,每秒20米。

The population of deer in a forest doubles every year.
::森林中鹿的人口每年翻一番。

A car is traveling at 45 miles per hour.
::一辆汽车每小时45英里行驶。

A car accelerates at 45 miles per hour, per hour.
::车速每小时45英里,每小时45英里。

A bicycle factory starts out with 50 completed bikes on site. It produces 35 every hour.
::一家自行车厂开始时,现场有50辆完成的自行车,每小时生产35辆。

The density of the gas as a function of volume . For instance, the volume of a balloon increases, while the mass of the gas inside stays constant .
::气体的密度随体积的函数而变化。例如,气球的体积增加,而气球内部的气体质量则保持不变。

A plane descends from 30,000 feet, dropping 200 feet for every 500 feet of horizontal movement.
::飞机从3万英尺下坠,每水平移动500英尺,降下200英尺。

100,000 bacteria are in a petri dish. They are halving in number every day.
::每天有10万种细菌,数量减半。

Solution: The most appropriate type of function is identified, consider why each answer is correct.
::解决办法:确定最合适的职能类型,考虑为什么每个答案正确。

linear
::线

quadratic
::二次

exponential
::指数指数

linear
::线

quadratic
::二次

linear
::线

rational
::合理合理

linear
::线

exponential
::指数指数

Work It Out
::F. 工作外外

Several functions were introduced in this chapter using different scenarios. For many of these scenarios, the $\begin{align*}x\end{align*}$ -value was time, measured in seconds, hours, days, or other units. For these scenarios, it didn't make sense to consider the results of substituting negative values for $\begin{align*}x\end{align*}$ . You will do this now, to get more complete pictures of the behavior of these functions.
::本章采用不同的假设情景引入了多个函数。在很多这些假设情景中, x 值是时间, 以秒、小时、天数或其他单位衡量。对于这些假设情景, 考虑以负值代替 x 的结果是没有意义的。您现在要这样做, 才能获得这些函数行为的更完整的图片。

Consider the quadratic function , $\begin{align*}y=x^2\end{align*}$ . Complete the table below and graph the function. What characteristics of quadratic functions can you observe now that you have graphed results for negative $\begin{align*}x\end{align*}$ values? Find the first differences (change in $\begin{align*}y\end{align*}$ ) and second differences (change in the differences) between consecutive $\begin{align*}y\end{align*}$ -values. What do you notice?

$\begin{aligned} \begin{array}{cc} x & y \\ - 5 \\ - 4 \\ - 3 \\ - 2 \\ - 1 \\ 0 \\ 1 \\ 2 \\ 3 \\ 4 \\ 5 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{c|c} x & y\\ \hline -5 &\\ -4 &\\ -3 &\\ -2 &\\ -1 &\\ 0 &\\ 1 &\\ 2 &\\ 3 &\\ 4 &\\ 5 &\\ \end{array}\end{align*}$
::y=x2. 考虑二次函数, y=x2. 完成下面的表格, 并绘制函数图。当您已经为负 x 值绘制了图表结果时, 您可以看到二次函数的什么特性? 查找连续的 y 值之间的第一个差异( y) 和第二个差异( y 变化) 。您注意到什么 ? xy - 5 - 4 - 3 - 2 - 1012345

Quadratic Functions in Tables
::表格中的二次曲线函数

A table can be represented by a quadratic function if the second differences of consecutive $\begin{align*}y\end{align*}$ -values are constant:
::如果连续 y 值的第二个差异不变,表格可以用二次函数表示 :

Since the change in $\begin{align*}y\end{align*}$ increases by 4 each time, the second differences are constant and the table represents a quadratic function.
::由于y的变动每次增加4个,第二个差异不变,表格代表二次函数。

The table for an exponential function is shown below. Do the same patterns you've already observed in hold true here? Explain. What additional patterns do you observe? Explain. Graph this exponential function. Does it appear that the graph will ever cross the $\begin{align*}x\end{align*}$ -axis? W hy might this be the case?

$\begin{aligned} \begin{array}{cc} x & y \\ - 5 & \frac{1}{32} \\ - 4 & \frac{1}{16} \\ - 3 & \frac{1}{8} \\ - 2 & \frac{1}{4} \\ - 1 & \frac{1}{2} \\ 0 & 1 \\ 1 & 2 \\ 2 & 4 \\ 3 & 8 \\ 4 & 16 \\ 5 & 32 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{c|c} x & y\\ \hline -5 & \frac{1}{32}\\ -4 & \frac{1}{16}\\ -3 & \frac{1}{8}\\ -2 & \frac{1}{4}\\ -1 & \frac12\\ 0 & 1\\ 1 & 2\\ 2 & 4\\ 3 & 8\\ 4 & 16\\ 5 & 32\\ \end{array}\end{align*}$
::指数函数的表格在下面显示。是否在此显示您已经观察到的相同模式为真实? 请解释。您观察的其他模式是什么 ? 解释。绘制这个指数函数。显示该图形是否曾经穿过 x 轴? 为什么会出现这种情况? xy - 5132 - 4112 - 4116 - 318 - 214 - 11201122484 165

Complete the suggested table below and graph the rational function $\begin{align*}y=\frac{5}{x}\end{align*}$ . Do the same patterns you already observed in rational functions continue to hold true here? What additional patterns do you observe? Explain.

$\begin{aligned} \begin{array}{rc} x & y \\ - 10 \\ 10 \\ - 5 \\ 5 \\ - 1 \\ 1 \\ - \frac{1}{2} \\ \frac{1}{2} \\ - \frac{1}{5} \\ \frac{1}{5} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{r|c} x & y\\ \hline -10 & \\ 10 & \\ -5 & \\ 5 & \\ -1 & \\ 1 & \\ -\frac12 & \\ \frac12 & \\ -\frac15 & \\ \frac15 & \\ \end{array}\end{align*}$
::完成下面建议的表格, 并绘制合理函数 y=5x 的图形。您在合理函数中观察到的相同模式是否在此继续维持为真实 ? 您观察到的其他模式是什么 ? 请解释。 xy- 1010- 55- 55- 11- 1212- 15

Which of the above graphs (in questions 1, 2, and 3) features a vertical line across which the reflected graph maps to itself? This means the graph has vertical . Why does it have vertical symmetry? What is the equation for the line across which the graph is symmetrical ?
::上面哪个图表(在问题1、2和3中)带有垂直线,反射图图本身的分布图是垂直的?这意味着图形是垂直的。为什么它有垂直对称?图形对称线的公式是什么?

Which of the above graphs, if rotated 180 degrees, maps to itself? This means the graph has rotational symmetry . Why does it have rotational symmetry? What are the coordinates of the point around which the graph has rotational symmetry?
::上面哪个图形, 如果旋转 180 度, 则向它自己绘制地图? 这意味着图形具有旋转对称性。为什么它具有旋转对称性? 图形旋转对称的点的坐标是多少?

(extension) Compare and contrast the $\begin{align*}y\end{align*}$ -values for each function above as you substitute larger and larger positive $\begin{align*}x\end{align*}$ -values. Explain.
::比较和对比以上每个函数的 Y 值,因为您正在替换较大和更大的正 x 值。请解释。

Activity 2: Compare Parameters and Function Behavior
::活动2:比较参数和函数行为

Interactive
::交互式互动

Use the interactive below to compare and contrast how changing the parameters of each function type changes the behavior of the function. Discuss and record your observations. Click or touch the arrows to change the function type.
::使用下面的交互效果来比较和比较每个函数类型参数的变化如何改变函数类型的行为。讨论并记录您的观测结果。单击箭头或触摸箭头以改变函数类型。

Work it Out
::工作出来

Make a table for each of the following functions, building the table from an $\begin{align*}x\end{align*}$ -value of 0 to an $\begin{align*}x\end{align*}$ -value of 5. Find the change in $\begin{align*}y\end{align*}$ -values as you go down the table for each function. Compare and contrast these results for each function.

$\begin{aligned} \begin{array}{l} y = 3 x + 5 \\ y = 3^{x} \\ y = 3 x^{2} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{l} y=3x+5\\ y=3^x\\ y=3x^2\\ \end{array}\end{align*}$
::为以下每个函数绘制表格, 将表格从 X 值 0 构建为 x 值 5 。查找您在下表为每个函数查找 Y 值的变化。比较并对比每个函数的这些结果。 y= 3x+5y=3xy=3xy=3x2

(extension) Create a linear function and a quadratic function. Consider the $\begin{align*}y\end{align*}$ -values returned as $\begin{align*}x\end{align*}$ approaches positive infinity. Is it possible to make a linear function and a quadratic function such that the linear function is never exceeded by the quadratic as $\begin{align*}x\end{align*}$ approaches infinity? Why or why not?
:扩展) 创建线性函数和二次函数。将返回的 Y 值视为x 接近正无穷。是否有可能做出线性函数和二次函数, 使线性函数在x 接近无穷时从未被二次函数超过? 为什么或为什么没有?

Summary
::摘要

Linear Functions
::线线函数

Slope-intercept form is $\begin{align*}y=mx+b\end{align*}$ , where $\begin{align*}m\end{align*}$ is the slope and $\begin{align*}b\end{align*}$ is the $\begin{align*}y\end{align*}$ -intercept (the value for $\begin{align*}y\end{align*}$ when $\begin{align*}x\end{align*}$ is 0).
::缩略图截取形式是 y=mx+b, 其中 m 是斜度, b 是 y 截取( x 是 0 的 y 值) 。

M odel scenarios featuring a constant rate of change. The constant slope of a linear function means the function increases or decreases at a constant rate.
::以恒定变化率为特征的模型假想。线性函数的常态斜度是指函数以恒定速率增减。

Graph as a straight line due to the constant rate of change.
::由于不变的变动率,图表为直线。

Simple Exponential Functions
::简单指数函数

Have the structure $\begin{align*}y=a(b)^x\end{align*}$ , where $\begin{align*}a\end{align*}$ represents the initial value and $\begin{align*}b\end{align*}$ represents the multiplier.
::结构 y=a(b)x,其中,a 表示初始值,b 表示乘数。

M odel scenarios where a quantity is doubling, halving, or otherwise increasing or decreasing by a constant multiplier, therefore v alues can increase very rapidly.
::因此,一个数量翻一番、减半或以其他方式以不变乘数增加或减少的模型假设,其数值可迅速增加。

D o not have a constant slope, but have constant ratio.
::没有固定的斜坡, 但有恒定比例。

Simple ones of the structure $\begin{align*}y=a(b)^x\end{align*}$ do not cross the $\begin{align*}x\end{align*}$ -axis.
::结构的 y=a(b)x 的简单 y=a(b)x 不越过 x 轴。

Simple Quadratic Functions
::简单二次曲线函数

Have the structure $\begin{align*}y=ax^2\end{align*}$ .
::拥有结构y=ax2。

Can be used to model anything changing according to a constant rate of acceleration. The coefficient $\begin{align*}a\end{align*}$ is half the rate of acceleration.
::系数a为加速率的一半。

G raphs have vertical symmetry (more complex quadratic graphs may have horizontal symmetry instead).
::图表具有垂直对称性(较复杂的二次方形图可能具有水平对称性)。

D o not have a constant slope, but have constant second differences because acceleration is constant.
::没有固定的斜坡, 但有恒定的第二差, 因为加速度是恒定的。

Simple Rational Functions
::简单理性函数

Have the structure $\begin{align*}y=\frac{a}{x}\end{align*}$ .
::拥有结构y=ax。

M odel the relationship between density and volume, and other inverse relationships (often seen in physics).
::模拟密度和体积之间的关系,以及其他反向关系(通常见于物理学)。

As $\begin{align*}x\end{align*}$ gets large, the $\begin{align*}y\end{align*}$ values become very small. Small $\begin{align*}x\end{align*}$ -values return large $\begin{align*}y\end{align*}$ -values.
::随着 x 变大, Y 值变小。小 x 值返回大 y 值。

Graphs of rational functions have rotational symmetry.
::理性函数的图形具有旋转对称性。

PLIX Interactive
::PLIX 交互式互动

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