Course: 8.1 指数增长职能

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指数增长功能

A population of 10 mice grows at a rate of 300% every month. How many mice are in the population after six months?
::10只小鼠每月以300%的速度生长。 6个月后,人口中有多少小鼠?

Exponential Growth Functions
::指数增长功能

An exponential function has the variable in the exponent of the expression . All have the form: $\begin{align*}f(x)=a \cdot b^{x-h}+k\end{align*}$ , where $\begin{align*}h\end{align*}$ and $\begin{align*}k\end{align*}$ move the function in the $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ directions respectively, much like the other functions we have seen in this text. $\begin{align*}b\end{align*}$ is the base and $\begin{align*}a\end{align*}$ changes how quickly or slowly the function grows. Let’s take a look at the parent graph , $\begin{align*}y=2^x\end{align*}$ .
::指数函数在表达式的引言中含有变量。所有函数都有窗体 : f(x) = abx-h+k, 其中 h 和 k 分别移动 x 和 y 方向中的函数, 与我们在此文本中看到的其他函数非常相似。 b 是函数增长的基数和变化速度或速度。让我们查看父图形, y= 2x 。

Let's graph $\begin{align*}y=2^x\end{align*}$ and find the $\begin{align*}y\end{align*}$ - intercept .
::让我们绘制 y=2x 并找到 Y 接口。

Let’s start by making a table. Include some positive and negative values for $\begin{align*}x\end{align*}$ and zero.
::让我们先做一个表格。包括一些x和0的正值和负值。

$\begin{align}x\end{align}$	$\begin{align}2^x\end{align}$	$\begin{align}y\end{align}$
3	$\begin{align}2^3\end{align}$	8
2	$\begin{align}2^2\end{align}$	4
1	$\begin{align}2^1\end{align}$	2
0	$\begin{align}2^0\end{align}$	1
-1	$\begin{align}2^{-1}\end{align}$	$\begin{align}\frac{1}{2}\end{align}$
-2	$\begin{align}2^{-2}\end{align}$	$\begin{align}\frac{1}{4}\end{align}$
-3	$\begin{align}2^{-3}\end{align}$	$\begin{align}\frac{1}{8}\end{align}$

This is the typical shape of an exponential growth function . The function grows “exponentially fast”. Meaning, in this case, the function grows in powers of 2. For an exponential function to be a growth function, $\begin{align*}a > 0\end{align*}$ and $\begin{align*}b > 1\end{align*}$ and $\begin{align*}h\end{align*}$ and $\begin{align*}k\end{align*}$ are both zero $\begin{align*}(y=ab^x)\end{align*}$ . From the table, we see that the $\begin{align*}y\end{align*}$ -intercept is (0, 1).
::这是指数增长函数的典型形状。函数会“ 迅速增长 ” 。意思是, 如果指数函数是增长函数, 则该函数的功率会增长为 2 。 a> 0 和 b> 1 和 h 和 k 是零 (y=abx) 。从表格中, 我们可以看到 y 的干扰是 0, 1 。

Notice that the function gets very, very close to the $\begin{align*}x\end{align*}$ -axis, but never touches or passes through it. Even if we chose $\begin{align*}x=-50, \ y\end{align*}$ would be $\begin{align*}2^{-50}=\frac{1}{2^{50}}\end{align*}$ , which is still not zero, but very close. In fact, the function will never reach zero, even though it will get smaller and smaller. Therefore , this function approaches the line $\begin{align*}y=0\end{align*}$ , but will never touch or pass through it. This type of boundary line is called an asymptote . In the case with all exponential functions, there will be a horizontal asymptote . If $\begin{align*}k=0\end{align*}$ , then the asymptote will be $\begin{align*}y=0\end{align*}$ .
::注意该函数非常非常接近 x 轴, 但从不触碰或通过它。即使我们选择了 x50, y 会是 2- 50=1250, 但它仍然不是零, 但非常接近。事实上, 该函数永远不会达到零, 即使它会越来越小。因此, 此函数会接近 y=0 线, 但永远不会触碰或通过它。这种类型的边界线被称为“ 静态 ” 。在使用所有指数函数的情况下, 将会有一个水平的静态。如果 k=0, 那么, 单位将是一个 y=0 。

Now, let's graph $\begin{align*}y=3^{x-2}+1\end{align*}$ and find the $\begin{align*}y\end{align*}$ -intercept, asymptote, domain and range .
::现在,让我们来绘制 y=3x-2+1 并找到 y 界面、空点、域名和范围。

This is not considered a growth function because $\begin{align*}h\end{align*}$ and $\begin{align*}k\end{align*}$ are not zero. To graph something like this (without a calculator), start by graphing $\begin{align*}y=3^x\end{align*}$ and then shift it $\begin{align*}h\end{align*}$ units in the $\begin{align*}x\end{align*}$ -direction and $\begin{align*}k\end{align*}$ units in the $\begin{align*}y\end{align*}$ -direction.
::这不是一个增长函数, 因为 h和 k 不是零。要绘制这样的图表( 没有计算器 ) , 请先绘制 y= 3x 的图形, 然后在 y 方向的 x 方向和 k 单位中移动 h 单位。

Notice that the point (0, 1) from $\begin{align*}y=3^x\end{align*}$ gets shifted to the right 2 units and up one unit and is (2, 2) in the translated function, $\begin{align*}y=3^{x-2}+1\end{align*}$ . Therefore, the asymptote is $\begin{align*}y=1\end{align*}$ . To find the $\begin{align*}y\end{align*}$ -intercept, plug in $\begin{align*}x=0\end{align*}$ .
::请注意 y= 3x 的点 (0, 1) 被移到右 2 个单位和上方一个单位, 在翻译函数 y= 3x-2+1 中为 2, 2 个单位。因此, y= 1 的表示是y 。要找到 y 接口, 插入 x=0 。

\begin{aligned} y = 3^{0 - 2} + 1 = 3^{- 2} + 1 = 1 \frac{1}{9} = 1. \bar{1} \end{aligned}

$\begin{align*}y=3^{0-2}+1=3^{-2}+1=1 \frac{1}{9} = 1. \overline{1}\end{align*}$
::y=30-2+1=3-2+1=119=1.1

The domain of all exponential functions is all real numbers. The range will be everything greater than the asymptote. In this problem, the range is $\begin{align*}y > 1\end{align*}$ .
::所有指数函数的域都是真实数字。范围将比无位数大。在此问题上, 范围是 y> 1 。

Finally, let's graph the function $\begin{align*}y= -\frac{1}{2} \cdot 4^x\end{align*}$ and determine if it is an exponential growth function.
::最后,让我们用图表显示 y124x 的函数, 并确定它是否是一个指数增长函数。

In this problem , we will outline how to use the graphing calculator to graph an exponential function. First, clear out anything in Y=. Next, input the function into Y1= -(1/2)4^X and press GRAPH . Adjust your window accordingly.
::在此问题上, 我们将概要描述如何使用图形计算器来绘制指数函数。首先, 清除 Y= 中的任何内容。下一步, 将函数输入到 Y1 = - (1/2)4x 中, 并按 GRAPH 键。调整您的窗口。

This is not an exponential growth function, because it does not grow in a positive direction. By looking at the definition of a growth function, $\begin{align*}a>0\end{align*}$ , and it is not here.
::这不是一个指数增长函数, 因为它不是朝着正方向增长。通过查看增长函数的定义, a>0, 它不在这里。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the number of mice that are in the population after six months.
::早些时候,你被要求去寻找六个月后在人口中的老鼠数量。

This is an example of exponential growth, so we can use the exponential form $\begin{align*}f(x)=a \cdot b^{x-h}+k\end{align*}$ . In this case, a = 10, the starting population; b = 300% or 3, the rate of growth; x-h = 6 the number of months, and k = 0.
::这是指数增长的一个例子, 所以我们可以使用指数表 f( x) = abx-h+k。在这种情况下, a = 10, 起始人口; b = 300% 或 3, 增长率; x-h = 6 个月数; k = 0 。

\begin{aligned} P = 10 \cdot 3^{6} \\ = 10 \cdot 729 = 7290 \end{aligned}

$\begin{align*}P = 10 \cdot 3^6\\ = 10 \cdot 729 = 7290\end{align*}$
::P=1036=10729=7290

Therefore, the mouse population after six months is 7,290.
::因此,六个月后的鼠标人数为7 290人。

For Examples 2-4, graph the exponential functions. Determine if they are growth functions. Then, find the $\begin{align*}y\end{align*}$ -intercept, asymptote, domain and range. Use an appropriate window.
::对于示例 2-4, 图形显示指数函数。确定它们是否为增长函数。然后, 找到 y 界面、零点、域和范围。使用一个合适的窗口。

Example 2
::例2

$\begin{align*}y=3^{x-4}-2\end{align*}$
::y=3x-4-2

This is not a growth function because $\begin{align*}h\end{align*}$ and $\begin{align*}k\end{align*}$ are not zero. The $\begin{align*}y\end{align*}$ -intercept is $\begin{align*}y=3^{0-4}-2=\frac{1}{81}-2=-1\frac{80}{81}\end{align*}$ , the asymptote is at $\begin{align*}y=-2\end{align*}$ , the domain is all real numbers and the range is $\begin{align*}y>-2\end{align*}$ .
::这不是一个增长函数, 因为 h和 k 不是零。 y 界面是 y= 30 - 4 - 2 - 2= 181 - 2\\\\\\\\\\ 18081, 单位是 y @ 2, 域是全部真实数字, 范围是 y# 2 。

Example 3
::例3

$\begin{align*}f(x)=(-2)^{x+5}\end{align*}$
:xx)=(-2)x+5

This is not a growth function because $\begin{align*}h\end{align*}$ is not zero. The $\begin{align*}y\end{align*}$ -intercept is $\begin{align*}y=(-2)^{0+5}=(-2)^5=-32\end{align*}$ , the asymptote is at $\begin{align*}y=0\end{align*}$ , the domain is all real numbers and the range is $\begin{align*}y>0\end{align*}$ .
::这不是一个增长函数, 因为 h 不是零。 y 界面是 y= (-2) 0+5 = (-2) 5 = (-2) 5 @ 32 , 单位是 y= 0 , 域是全部真实数字, 范围是 y> 0 。

Example 4
::例4

$\begin{align*}f(x)=5^x\end{align*}$
:fx)=5x

This is a growth function.
::这是一个增长功能。

The $\begin{align*}y\end{align*}$ -intercept is $\begin{align*}y=5^\circ =1\end{align*}$ , the asymptote is at $\begin{align*}y=0\end{align*}$ , the domain is all real numbers and the range is $\begin{align*}y>0\end{align*}$ .
::Y 界面是 y= 5\\\ 1, y= 0. 范围是 y= 0, 域是全部真实数字, 范围是 y> 0 。

Example 5
::例5

Abigail is in a singles tennis tournament. She finds out that there are eight rounds until the final match. If the tournament is single elimination, how many games will be played? How many competitors are in the tournament?
::艾比盖尔在单人网球锦标赛中。她发现在决赛前有八轮比赛。如果比赛是一次性淘汰, 将玩多少场比赛? 比赛中有多少竞争对手?

If there are eight rounds to single’s games, there are will be $\begin{align*}2^8=256\end{align*}$ competitors. In the first round, there will be 128 matches, then 64 matches, followed by 32 matches, then 16 matches, 8, 4, 2, and finally the championship game. Adding all these all together, there will be $\begin{align*}128+64+32+16+8+4+2+1\end{align*}$ or 255 total matches.
::如果单场比赛分为八轮,则将出现28=256竞争对手。在第一轮比赛中,将出现128场比赛,然后是64场比赛,接下来是32场比赛,然后是16场比赛,最后是8、4、2场比赛,最后是冠军赛。加上所有这些,将出现128+64+32+16+8+4+2+1,或者255场总比赛。

Review
::回顾

Graph the following exponential functions. Find the $\begin{align*}y\end{align*}$ -intercept, the equation of the asymptote and the domain and range for each function.
::图形显示以下的指数函数。查找 y 界面、空点的方程以及每个函数的域和范围。

$\begin{align*}y=4^x\end{align*}$
::y=4x y=4x

$\begin{align*}y=(-1)(5)^x\end{align*}$
::y= (- 1) (5)x

$\begin{align*}y=3^x-2\end{align*}$
::y=3x-2

$\begin{align*}y=2^x+1\end{align*}$
::y=2x+1 y=2x+1

$\begin{align*}y=6^{x+3}\end{align*}$
::y=6x+3 y=6x+3

$\begin{align*}y= -\frac{1}{4}(2)^x+3\end{align*}$
::y14(2)x+3 y14(2)x+3

$\begin{align*}y=7^{x+3}-5\end{align*}$
::y= 7x+3- 5

$\begin{align*}y=-(3)^{x-4}+2\end{align*}$
::y(3)x-4+2

$\begin{align*}y=3(2)^{x+1}-5\end{align*}$
::y=3(2)x+1-5

What is the y -intercept of $\begin{align*}y=a^x\end{align*}$ ? Why is that?
::y=ax的 Y 界面是什么?为什么呢?

What is the range of the function $\begin{align*}y=a^{x-h}+k\end{align*}$ ?
::函数 y=ax-h+k 的范围是什么 ?

March Madness is a single-game elimination tournament of 64 college basketball teams. How many games will be played until there is a champion? Include the championship game.
::March Manness是一场由64个大学篮球队组成的单场淘汰赛。在冠军出现之前,将玩多少场比赛? 包括冠军赛。

In 2012, the tournament added 4 teams to make it a field of 68 and there are 4 "play-in" games at the beginning of the tournament. How many games are played now?
::2012年,锦标赛增加了4个球队,使之成为68个球队的球场,比赛开始时有4场“玩耍”比赛。现在有多少场比赛?

An investment grows according the function $\begin{align*}A=P(1.05)^t\end{align*}$ where $\begin{align*}P\end{align*}$ represents the initial investment, $\begin{align*}A\end{align*}$ represents the value of the investment and $\begin{align*}t\end{align*}$ represents the number of years of investment. If $10,000 was the initial investment, how much would the value of the investment be after 10 years, to the nearest dollar?
::P代表初始投资,A代表投资的价值, t代表投资年数。如果10 000美元是初始投资,投资价值在10年后将达到多少美元?

How much would the value of the investment be after 20 years, to the nearest dollar?
::20年后投资价值到最近的美元是多少?

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

$\begin{align}x\end{align}$	$\begin{align}2^x\end{align}$	$\begin{align}y\end{align}$
3	$\begin{align}2^3\end{align}$	8
2	$\begin{align}2^2\end{align}$	4
1	$\begin{align}2^1\end{align}$	2
0	$\begin{align}2^0\end{align}$	1
-1	$\begin{align}2^{-1}\end{align}$	$\begin{align}\frac{1}{2}\end{align}$
-2	$\begin{align}2^{-2}\end{align}$	$\begin{align}\frac{1}{4}\end{align}$
-3	$\begin{align}2^{-3}\end{align}$	$\begin{align}\frac{1}{8}\end{align}$

Section outline

General

指数增长功能

Exponential Growth Functions ::指数增长功能

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)