节: 指数增长和衰减 | 12.17 指数增长和衰减

章节大纲

A condominium complex charges $185 per month for the homeowners’ association fee. The rates can rise every year because of inflation but they promise not to raise the rates more than 10% each year. Keep in mind, though, that if they raise the rate by 10% the first year, the second year is now more expensive. If they raise the maximum again, they are increasing the original $185 plus the first year’s adjustment by 10%. Graph the situation for 10 years.
::公寓综合体每月收费185美元,以支付房主协会费。利率可以因通货膨胀而逐年上涨,但保证利率不会每年提高10 % 。但是,记住,如果第一年将利率提高10 % , 第二年现在更昂贵。如果他们再次提高最高利率,则将原来的185美元加第一年的调整增加10 % 。分析10年的情况。

How much could the homeowners’ fee be in ten years? Use the function $\begin{aligned} f = 185 \times {1.1}^{t} \end{aligned}$ where $\begin{aligned} f \end{aligned}$ is the fee after $\begin{aligned} t \end{aligned}$ years.
::10年内房主的收费能有多少?使用这个功能f=185x1.1t,f是年之后的收费。

In this concept, you will learn to distinguish between and .
::在这一概念中,你会学会区分和区分。

Exponential Growth and Decay
::指数增长和衰减

Sometimes you will need to identify whether a function is an exponential function . If your function can be written in the form $\begin{aligned} y = a b^{x} \end{aligned}$ , where $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ are constants, $\begin{aligned} a \neq 0, b > 0 \end{aligned}$ , and $\begin{aligned} b \neq 1 \end{aligned}$ , then it must be exponential. In quadratic equations, your functions were always to the 2 ^nd power. In , the exponent is a variable . Their graphs will have a characteristic curve either upward or downward.
::有时您需要确定函数是否是一个指数函数。如果您的函数可以以 y=abx 的形式写成, 其中 a和 b 是常数, a0, b>0, 和 b1, 那么它必须是指数。在二次方程式中, 您的函数总是在第二倍。在中, 引号是一个变量。它们的图表将有一个向上或向下的特性曲线。

Therefore the function $\begin{aligned} c = 4 \times 10^{a} \end{aligned}$ is an exponential function, but $\begin{aligned} y = - 6 \times 0^{x} \end{aligned}$ is not because $\begin{aligned} b \leq 1 \end{aligned}$ .
::因此,函数 c=4x10a 是一个指数函数,但 y=6x0x 不是因为 b1 。

In some cases with exponential functions, as the $\begin{aligned} x \end{aligned}$ value increased, the $\begin{aligned} y \end{aligned}$ value increased, too. This was a direct relationship known as exponential growth . As the $\begin{aligned} x \end{aligned}$ value increases, the $\begin{aligned} y \end{aligned}$ value grows at a very fast rate!
::在某些具有指数函数的情况下, 随着 x 值的增加, y 值也增加了。这是直接的关系, 被称为指数增长。随着 x 值的增加, y 值以非常快的速度增长 !

In other cases, as the $\begin{aligned} x \end{aligned}$ value increased, the $\begin{aligned} y \end{aligned}$ value decreased. This relationship is an inverse relationship known as exponential decay . The graphs of these functions are opposites, reflected on the $\begin{aligned} y \end{aligned}$ -axis.
::在其他情况下,随着 x 值的增加, Y 值下降。此关系是一种反向关系, 被称为指数衰减。这些函数的图形是相反的, 反映在 Y 轴上。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the rising condominium fees. You need to determine how much the homeowners’ fee will be in ten years using the function $\begin{aligned} f = 185 \times {1.1}^{t} \end{aligned}$ where $\begin{aligned} f \end{aligned}$ is the fee after $\begin{aligned} t \end{aligned}$ years.
::早些时候,您曾遇到有关公寓费上涨的问题。您需要使用 F=185x1.1t 的功能来决定十年内房主费的多少, F 是 T 年之后的f 收费。

First, make a table of values for the function $\begin{aligned} f = 185 \times {1.1}^{x} \end{aligned}$ .
::首先,绘制函数 f=185x1.1x 的值表。

$\begin{aligned} t \end{aligned}$	0	1	2	3	4	5	6	7	8
$\begin{aligned} f \end{aligned}$	185	203.50	223.85	246.24	270.86	297.94	327.74	360.51	396.56

Next, graph the function.
::下一位, 绘制函数图。

Example 2
::例2

Does the following function represent an exponential function? Graph the function.
::以下函数是否代表指数函数?图形显示函数。

\begin{aligned} y = 3 \times 1^{x} \end{aligned}

::y=3x1x y=3x1x

First, answer the question.
::首先,回答问题。

No, this function does not represent an exponential function because the $\begin{aligned} b \end{aligned}$ value is 1.
::否,此函数不代表指数函数,因为 b 值为 1。

Next, graph the function.
::下一位, 绘制函数图。

Example 3
::例3

Graph the function $\begin{aligned} y = {(\frac{1}{2})}^{x} \end{aligned}$ and tell whether it will represent exponential growth or decay.
::绘制函数y=( 12) x 的图, 并显示它代表指数增长还是衰变。

First, graph the function.
::首先,绘制函数图。

Next, answer the question.
::下一个,回答问题。

From the graph, the function $\begin{aligned} y = {(\frac{1}{2})}^{x} \end{aligned}$ represents exponential decay.
::从图形中,函数 y=( 12)x 表示指数衰减。

Example 4
::例4

Graph the function $\begin{aligned} y = 4^{x} \end{aligned}$ and tell whether it represents exponential growth or decay.
::绘制 y= 4x 函数图,并显示该函数代表指数增长还是衰变。

First, graph the function.

::首先,绘制函数图。

Next, answer the question.
::下一个,回答问题。

From the graph, the function $\begin{aligned} y = 4^{x} \end{aligned}$ represents exponential growth.
::从图中,函数y=4x代表指数增长。

Example 5
::例5

Graph the function $\begin{aligned} y = 5^{x} \end{aligned}$ and tell whether it represents exponential growth or decay.
::绘制 y= 5x 函数图,并显示该函数代表指数增长还是衰变。

First, graph the function.

::首先,绘制函数图。

Next, answer the question.
::下一个,回答问题。

From the graph, the function $\begin{aligned} y = 5^{x} \end{aligned}$ represents exponential growth.
::从图中,函数 y=5x 表示指数增长。

Review
::回顾

Graph each function. Then say whether it represents economic growth or decay.
::每个函数图。然后说明它是代表经济增长还是代表衰退。

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

$\begin{aligned} y = {(\frac{1}{2})}^{x} \end{aligned}$
::y=( 12) x

$\begin{aligned} y = {(\frac{1}{3})}^{x} \end{aligned}$
::y=( 13) x

$\begin{aligned} y = 7^{x} \end{aligned}$
::y=7x y=7x

$\begin{aligned} y = 5^{x} \end{aligned}$
::y=5x y=5x

$\begin{aligned} y = 2^{x} \end{aligned}$
::y=2x y=2x

$\begin{aligned} y = {(\frac{1}{4})}^{x} \end{aligned}$
::y=( 14) x

$\begin{aligned} y = {(\frac{3}{4})}^{x} \end{aligned}$
::y=( 34)x

$\begin{aligned} y = 6^{x} \end{aligned}$
::y=6x y=6x

$\begin{aligned} y = 11^{x} \end{aligned}$
::y=11x y=11x

$\begin{aligned} y = 9^{x} \end{aligned}$
::y=9x y=9x

$\begin{aligned} y = {(\frac{1}{8})}^{x} \end{aligned}$
::y=( 18) x

$\begin{aligned} y = 12^{x} \end{aligned}$
::y=12x y=12x

$\begin{aligned} y = {(\frac{2}{5})}^{x} \end{aligned}$
::y=( 25) x

$\begin{aligned} y = 13^{x} \end{aligned}$
::y=13x y=13x

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。

章节大纲

Exponential Growth and Decay ::指数增长和衰减

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)

Exponential Growth and Decay
::指数增长和衰减

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)