节: 比较线性、二次曲线和指数模型 | 3.14 线性、二次曲线和指数模型的比较-interactive

章节大纲

The Purpose of This Lesson
::本课程的目的

In this lesson, you will further explore recursive and explicit definitions of linear and to describe how these functions change over equal intervals. You'll compare the graphs, , and values returned by linear, exponential, and quadratic functions in the long run .
::在此教训中, 您将进一步探索线性重现和清晰的定义, 并描述这些函数如何在相等的间隔内变化。您将比较图表, 以及以线性、指数性和二次函数返回的数值, 从长远来看。

Introduction: Arithmetic, Geometric, and Quadratic
::导言: 电解学、几何学和二次曲线学

Work it Out
::工作出来

A scientist is researching three islands that are populated with turtles. Below are tables representing the population of the turtles over time. The first one is quadratic, and the remaining two are arithmetic or geometric. Find the explicit formulas for each. Find the recursive formula for the arithmetic and geometric sequences. Determine the number of turtles on each island in year 10.
::科学家正在研究三个海龟聚居的岛屿。下面是代表海龟随时间推移组成的表格。第一个是四边形, 其余两个是算术或几何。查找每个岛屿的明确公式。找到算术和几何序列的递归公式。确定10年每个岛屿的海龟数量。

\begin{aligned} \begin{array}{ll} n & a_{n} \\ 1 & 5 \\ 2 & 20 \\ 3 & 45 \\ 4 & 80 \\ 5 & 125 \end{array} \end{aligned}

\begin{aligned} \begin{array}{ll} n & a_{n} \\ 1 & 30 \\ 2 & 33 \\ 3 & 36 \\ 4 & 39 \\ 5 & 42 \end{array} \end{aligned}

\begin{aligned} \begin{array}{ll} n & a_{n} \\ 1 & 5 \\ 2 & 10 \\ 3 & 20 \\ 4 & 40 \\ 5 & 80 \end{array} \end{aligned}

Activity 1: Sequences
::活动1:顺序

Work it Out
::工作出来

Uncle Robert offers you a choice of 3 bank accounts that will accumulate more money over time. He presents you with tables that show you the expected amount of money in each account over the years. The tables only show the first few values, but they are representative of what you can expect. Determine the explicit formula for each, that is, the equation for the sequence that returns the amount of money for a given year. When possible, write the recursive formula for each sequence. Which account would you prefer and why? Support your answer with calculations.
::罗伯特叔叔向您提供3个银行账户的选择,这些账户将随着时间积累更多的资金。他向您提供表格, 显示您多年来每个账户的预期金额。表格只显示最初的几个数值, 但它们代表了您可以期望的数值。确定每个公式的明确公式, 即返回给定年份金额的序列的方程式。可能的话, 请为每个序列写入递归公式。您喜欢哪个账户, 以及为什么? 支持您的计算答案。

\begin{aligned} \begin{array}{cc} n & a_{n} \\ 1 & 100 \\ 2 & 1100 \\ 3 & 2100 \\ 4 & 3100 \end{array} \end{aligned}

\begin{aligned} \begin{array}{cc} n & a_{n} \\ 1 & 10 \\ 2 & 15 \\ 3 & 22.5 \\ 4 & 33.75 \end{array} \end{aligned}

\begin{aligned} \begin{array}{cc} n & a_{n} \\ 1 & 10 \\ 2 & 400 \\ 3 & 900 \\ 4 & 1600 \end{array} \end{aligned}

Use the recursive formulas from above to answer the following questions. How do arithmetic sequences grow? How do geometric sequences grow? In the long run, which type of growth will result in larger values--growth in an arithmetic sequence or growth in a geometric sequence ? Why?
::使用上面的递归公式回答下列问题。算术序列是如何增长的? 几何序列是如何增长的? 从长远看, 哪种增长会在算术序列或几何序列中产生更大的值增长? 为什么?

Arithmetic sequences are analogous to , and geometric sequences are analogous to exponential functions. For sequences a through d, write the continuous function to which it is analogous (two examples are provided for you). Write the recursive formulas for each sequence. Are any of these sequences decreasing? Why? Under what conditions do arithmetic and geometric sequences decrease? Which function will return greater values in the long run?
::亚学序列类似于 , 几何序列类似于指数函数。对于 a 至 d 序列, 请写上它相似的连续函数( 提供了两个示例 ) 。写入每个序列的循环公式。这些序列中是否有一个正在下降 ? 为什么? 在什么条件下算术和几何序列会下降? 从长远来看, 哪个函数会返回更大的值 ?

Sample 1:
::样本1:

Sample 2:
$\begin{aligned} \begin{array}{ll} a. & a_{n} = 5 (n - 1) - 14 \\ b. & a_{n} = 78 (1.5)^{n - 1} \\ c. & a_{n} = 32 (n - 1) + 20 \\ d. & a_{n} = 14 (0.9)^{n - 1} \end{array} \end{aligned}$
::样本2: a.an=5(n-1)-14b.an=78(1.5)n-1c.an=32(n-1)+20d.an=14(0.9)n-1

Arithmetic and Geometric Sequences
::亚学和几何序列

Arithmetic and geometric sequences are analogous to linear and exponential functions, shifted 1 unit to the right.
::测量和几何序列与直线和指数函数相似,将1个单位移到右边。

Arithmetic sequences grow according to a common difference , also known as the , which is the amount added to advance from one term to the next.
::亚理学序列根据共同差异增长, 也称为..., 即从一个任期向下一个任期递增的金额。

Geometric sequences grow according to a common ratio, also known as the multiplier, which is the amount by which a term is multiplied by in order to advance from one to the next.
::几何序列根据共同比率增长,也称为乘数,即一个术语乘以一个术语从一个向下一个术语前进的数值。

Arithmetic and geometric sequences can be increasing or decreasing. They increase if the common difference is positive or the common ratio is greater than 1.
::测量和几何序列可能会增加或减少,如果共同差值为正或共同比大于1,则会增加。

Geometric sequences will always surpass arithmetic sequences in the long run.
::从长远来看,几何序列将永远超过算术序列。

Activity 2: Proving How Linear and Exponential Functions Grow
::活动2:证明线性和指数性功能是如何成长的

Example 2-1
::例2-1

Consider the function $\begin{aligned} f (x) = 3 x + 7. \end{aligned}$ What kind of function is it? As $\begin{aligned} x \end{aligned}$ increases by $\begin{aligned} 1 \end{aligned}$ , how much does $\begin{aligned} y \end{aligned}$ increase by? Is this true for all values of $\begin{aligned} x ? \end{aligned}$ If $\begin{aligned} x \end{aligned}$ increases by 5, how much does $\begin{aligned} y \end{aligned}$ increase by? Is this true for all values of $\begin{aligned} x ? \end{aligned}$
::考虑函数 f( x) = 3x+ 7 。函数是什么? 作为 x 增加 1 、增加多少 y 增加多少? 这对 x 的所有值是否都是真实的? 如果 x 增加 5 、增加多少 y 增加多少? 这对 x 的所有值是否都是真实的 ?

Solution: This is a linear function . It's analogous to an arithmetic sequence with a common difference of 3. You can test a few values to confirm that as $\begin{aligned} x \end{aligned}$ increases by 1, $\begin{aligned} y \end{aligned}$ increases by a constant amount, namely 3:
::溶液 : 这是一个线性函数。它类似于一个算术序列, 共差为 3 。您可以测试几个值来确认 x 增加 1, y 增加为恒定数量, 即 3 :

From these experiments you observe that $\begin{aligned} f (x + 1) = f (x) + 3. \end{aligned}$
::在这些实验中,您观察到 f( x+1) = f( x) +3 。

If you perform the same experiment, but increasing $\begin{aligned} x \end{aligned}$ by 5, you observe that $\begin{aligned} f (x + 5) = f (x) + 15. \end{aligned}$
::如果您进行同样的实验,但将 x 增加 5, 您就会看到 f( x+5) = f( x)+15 。

Example 2-2
::例2-2

An increasing exponential function has the form $\begin{aligned} g (x) = a (b)^{x}, \end{aligned}$ with $\begin{aligned} b > 1. \end{aligned}$ Exponential functions do not increase additively, they increase by multiplication . Prove that an increase in $\begin{aligned} x \end{aligned}$ of 1 means the $\begin{aligned} y \end{aligned}$ -value increases by multiplying by $\begin{aligned} b . \end{aligned}$
::递增的指数函数有表g(x)=a(b)x, b>1. 指数函数不会增加,而是通过乘法增加。证明乘法增加 x 1 表示y值增加,乘法为b。

Solution:
::解决方案 :

Compare $\begin{aligned} g (x) \end{aligned}$ and $\begin{aligned} g (x + 1) : \end{aligned}$
::比较 g( x) 和 g( x+1) :

\begin{aligned} g (x) = a (b)^{x} \\ g (x + 1) = a (b)^{x + 1} = a (b^{x}) (b) \end{aligned}

::g(x)=a(b)xg(x+1)=a(b)x+1=a(b)x+1=a(b)x(b)

$\begin{aligned} g (x + 1) \end{aligned}$ is $\begin{aligned} g (x) \times b . \end{aligned}$ For every increase of 1 in $\begin{aligned} x, \end{aligned}$ then $\begin{aligned} g (x) \end{aligned}$ is multiplied by $\begin{aligned} b . \end{aligned}$
::g(x+1)是 g(x)xxb。每增加一英寸x,那么g(x)乘以 b。

Work it Out
::工作出来

The example above gives the structure for proving that linear functions of the form $\begin{aligned} f (x) = m x + b \end{aligned}$ ( where $\begin{aligned} m \end{aligned}$ is positive) grow additively by a constant amount. If $\begin{aligned} x \end{aligned}$ is increased from $\begin{aligned} x \end{aligned}$ to $\begin{aligned} x + 1, \end{aligned}$ the corresponding increase in $\begin{aligned} y \end{aligned}$ should be $\begin{aligned} m . \end{aligned}$ If $\begin{aligned} f (x) = m x + b, \end{aligned}$ what is $\begin{aligned} f (x + 1) ? \end{aligned}$ What is $\begin{aligned} f (x + 1) - f (x) ? \end{aligned}$ Explain how this shows that every increase in $\begin{aligned} x \end{aligned}$ of 1 results in an additive increase in $\begin{aligned} y \end{aligned}$ of $\begin{aligned} m \end{aligned}$ .
::以上示例为证明表f(x)=mx+b(Mm为正数)的线性函数以恒定数量递增增长提供了结构。如果x从x增长到x+1,那么y的相应增长应该是m。如果f(x)=mx+b,f(x)=mx+1是什么?f(x+1)是什么?f(x)+1)-f(x)是什么?解释这如何显示每增加1x中每增加1倍就会增加y。

If $\begin{aligned} f (x) = m x + b \end{aligned}$ (with positive slope $\begin{aligned} m \end{aligned}$ ) and $\begin{aligned} x \end{aligned}$ is increased by $\begin{aligned} d \end{aligned}$ , what is the corresponding increase in $\begin{aligned} y ? \end{aligned}$ Explain how this shows that every increase in $\begin{aligned} x \end{aligned}$ by a constant amount results in an additive increase in $\begin{aligned} y \end{aligned}$ by a constant amount.
::如果f(x)=mx+b(正斜坡m)和x增加d,那么y的相应增加是多少?解释这如何表明x每增加一个不变数量,就会使y增加一个不变数量。

Prove that for an increasing exponential function in the form $\begin{aligned} a (b)^{x}, \end{aligned}$ if $\begin{aligned} x \end{aligned}$ is increased by $\begin{aligned} d, \end{aligned}$ the values returned increase by multiplication by $\begin{aligned} b^{d} . \end{aligned}$
::证明对于以 a(b)x 形式显示的递增指数函数,如果 x 增加 d,则以 bd 乘法返回的数值会增加。

PLIX Interactive
::PLIX 交互式互动

Activity 3: Comparing Graphs of Linear, Exponential, and Quadratic Functions
::活动3:比较线性、指积和二次函数图

Example 3-1
::例3-1

Identify each as either linear, exponential, or quadratic. Explain your reasoning. Discuss the differences and similarities of each.
::解释你的推理。讨论每个不同和相似之处。

Solution: Linear functions are represented by a straight line, strictly increasing or decreasing by a constant rate . This matches the red graph, which is a straight line with a constant slope of 1/2.
::解决方案: 线性函数由直线表示, 以恒定速率严格递增或递减。这与红图相匹配, 红图为直线, 常数斜度为 1/2 。

Quadratic functions are represented by a parabolic curve, that increases and decreases, with a maximum or minimum value. This matches the blue graph, which is a u-shaped curve (decreases then increases) with a minimum at (0, 0).
::二次曲线的函数代表着一个有最大值或最小值增减的抛物线曲线。它与蓝色图相匹配,即以最小值(0,0)为单位的u形曲线(减少,然后增加)。

Exponential functions are represented by a curve that rapidly increases or decreases. This matches the green graph, which is rapidly increasing by a factor of 2.
::指数函数由快速增减的曲线表示,与正迅速增长2倍的绿图相匹配。

Use the questions below to learn about some similarities and differences between linear, exponential, and quadratic functions.
::利用下面的问题来了解线性、指数性和二次函数之间的一些相似和差异。

Activity 4: Comparing Rate of Change and Long Run Behavior
::活动4:比较变化率和长期运行行为

Work it Out
::工作出来

Given the following pairs of functions, explain which returns greater $\begin{aligned} y \end{aligned}$ -values as $\begin{aligned} x \end{aligned}$ progresses towards positive infinity. Explain your decision.
::根据以下的对函数, 请解释在 x 向正无限进进进时, 返回更大的 Y 值。请解释您的决定。

$\begin{aligned} \begin{array}{lll} a. & y_{1} = \frac{1}{3} x + 100 & y_{2} = 40 x - 1000 \\ b. & y_{1} = \frac{1}{3} x^{2} & y_{2} = 500 x + 1200 \\ c. & y_{1} = x & y_{2} = x + 100 \\ d. & y_{1} = x^{2} & y_{2} = x^{2} + 50 \\ e. & y_{1} = (x - 3)^{2} & y_{2} = x^{2} \\ f. & y_{1} = (x - 100)^{2} & y_{2} = 100 x \end{array} \end{aligned}$
::a.y1=13x+100y2=40x-1000b.y1=13x2y2=500x+1200c.y1=xy2=x+100d.y1=x2y2=x2y2=x2+50e.y1=(x-33)2y2=x2f.y1=(x-1002y2=100x)

Create a table of values for the following two functions, such that the $\begin{aligned} x \end{aligned}$ -values are consecutive integers . Find the slope between consecutive points in each table. For the quadratic, find the second differences. Interpret the increase in slope for the quadratic function , and explain the meaning of the second differences for the quadratic. Which function returns greater values as $\begin{aligned} x \end{aligned}$ approaches positive infinity?
::创建以下两个函数的值表, 使 x 值为连续的整数。在每个表格中查找连续点之间的斜度。对于二次曲线, 找到第二个差数。解释二次曲线函数的斜度增长, 并解释二次曲线差异的含义。哪个函数返回较大值, 如 x 接近正无穷度 ?

If $\begin{aligned} f (x) = a x^{2} + b x + c, \end{aligned}$ find a simplified expression for the difference between $\begin{aligned} y \end{aligned}$ -values returned for consecutive integer $\begin{aligned} x \end{aligned}$ -values. Specifically, find and simplify $\begin{aligned} f (x + 1) - f (x) . \end{aligned}$ Confirm your formula is correct by checking your work with quadratic used in the previous problem. How does this expression show that the slope of an increasing quadratic function will always increase as $\begin{aligned} x \end{aligned}$ increases?
::如果 f( x) = ax2+bx+c, 请为连续整数 x 值返回的 y 值之间的差数找到一个简化表达式。具体来说, 查找并简化 f( x+1) - f( x) 。通过检查您在前一个问题中使用的二次曲线的工作来确认您的公式是正确的。这个表达式如何显示, 递增的二次函数的斜度会随着 x 的增加而增加 ?

If the difference between the $\begin{aligned} y \end{aligned}$ -values in the last problem is $\begin{aligned} d (x) = a x + a, \end{aligned}$ Find the difference between the differences. That is, find $\begin{aligned} d (x + 1) - d (x) . \end{aligned}$ What do you observe? How does this confirm the relationship between the leading coefficient of a quadratic and its rate of acceleration? Does the acceleration change or stay constant as $\begin{aligned} x \end{aligned}$ increases? Explain.
::如果最后一个问题中的 Y 值之间的差别是 d( x) = ax+a, 找出差异之间的差别。也就是说, 找到 d( x+1) - d( x) 。您观察什么 ? 这如何确认二次曲线主要系数与其加速率之间的关系? 加速度是随着 x 的增加而变化还是保持不变? 解释。

Create a table of values for the following functions, such that the $\begin{aligned} x \end{aligned}$ -values are consecutive integers. Find the differences, second differences, and the third differences for each. What do you observe? Contrast this with your observations regarding quadratics in the last two problems. Based on these observations, which type of increasing function will return greater values as $\begin{aligned} x \end{aligned}$ approaches positive infinity? Explain.
::为以下函数创建一个数值表, 使 x 值为连续整数。查找每个函数的差异、第二次差异和第三个差异。您观察了什么? 与您对后两个问题中的二次曲线的观察对比。基于这些观察, 哪种类型的递增函数会随着 x 接近正无穷度而返回更大的值 ? 解释。

$\begin{aligned} \begin{array}{ll} a. & y_{1} = 2^{x} \\ b. & y_{2} = 3^{x} \\ c. & y_{3} = 512 (1.5)^{x} \end{array} \end{aligned}$
::a.y1=2xb.y2=3xc.y3=512(1.5)x

Given a linear function, a quadratic function, and an exponential function, all of which are increasing, which will return the largest values in the long run? Why? Which will return the second largest values in the long run? Why?
::如果给定一个线性函数、二次函数和指数函数,所有这些都在增加,它们将返回长期最大的值;为什么?从长远看,它们将返回第二大值?为什么?为什么?从长远看,它们将返回第二大值?为什么?

Interactive
::交互式互动

Use the interactive to experiment with the parameters for exponential and quadratic functions. Is it possible to create a quadratic with a positive leading coefficient that returns values exceeding those of an increasing exponential function? Why or why not?
::使用互动来实验指数函数和二次函数的参数。是否有可能创建一个具有正前导系数的二次曲线, 返回超过指数函数增加值的数值? 为什么或为什么没有?

INTERACTIVE

Comparing Quadratic and Exponential Functions

Move the blue point to change the stretch factor of the parabola.
::移动蓝色点以改变抛物线的伸展系数。

Notice that the exponential curve overtakes the parabola in the long run.
::注意指数曲线从长远来看超过抛物线。

Summary
::摘要

Linear functions have a constant slope.
::线性函数有一个固定的斜坡。

Quadratic functions have a constant acceleration.
::二次曲线函数具有恒定加速度。

Increasing exponential functions have an increasing acceleration.
::成倍增长的函数加速度越来越快。

If a linear function, a quadratic function, and an exponential function are all increasing, as $\begin{aligned} x \end{aligned}$ approaches positive infinity:
::如果线性函数、二次函数和指数函数都在增加,因为x接近正无穷度:

The exponential function will return greater values than the quadratic function.
::指数函数返回的数值大于二次函数。

The quadratic function will return greater values than the linear function.
::二次函数返回的数值大于线性函数。

章节大纲

Introduction: Arithmetic, Geometric, and Quadratic ::导 言: 电解学、几何学和二次曲线学

Work it Out ::工作出来

Activity 1: Sequences ::活动1:顺序

Work it Out ::工作出来

Arithmetic and Geometric Sequences ::亚学和几何序列

Activity 2: Proving How Linear and Exponential Functions Grow ::活动2:证明线性和指数性功能是如何成长的

Example 2-1 ::例2-1

Example 2-2 ::例2-2

Work it Out ::工作出来

PLIX Interactive ::PLIX 交互式互动

Activity 3: Comparing Graphs of Linear, Exponential, and Quadratic Functions ::活动3:比较线性、指积和二次函数图

Example 3-1 ::例3-1

Activity 4: Comparing Rate of Change and Long Run Behavior ::活动4:比较变化率和长期运行行为

Interactive ::交互式互动

Summary ::摘要

Introduction: Arithmetic, Geometric, and Quadratic
::导言: 电解学、几何学和二次曲线学

Work it Out
::工作出来

Activity 1: Sequences
::活动1:顺序

Work it Out
::工作出来

Arithmetic and Geometric Sequences
::亚学和几何序列

Activity 2: Proving How Linear and Exponential Functions Grow
::活动2:证明线性和指数性功能是如何成长的

Example 2-1
::例2-1

Example 2-2
::例2-2

Work it Out
::工作出来

PLIX Interactive
::PLIX 交互式互动

Activity 3: Comparing Graphs of Linear, Exponential, and Quadratic Functions
::活动3:比较线性、指积和二次函数图

Example 3-1
::例3-1

Activity 4: Comparing Rate of Change and Long Run Behavior
::活动4:比较变化率和长期运行行为

Interactive
::交互式互动

Summary
::摘要