课程： 12.1 反逆变化模型

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反反变化模型

Inverse Variation Models
::反反变化模型

Many variables in real-world problems are related to each other by variations. A variation is an equation that relates a variable to one or more other variables by the operations of multiplication and division . There are three different kinds of variation: , inverse variation and joint variation .
::现实世界问题中的许多变数因变数而相互关联。变数是一个公式,通过乘法和分法的操作将变数与一个或多个其他变数联系起来。有三种不同的变数: 、反向变数和共同变数。

Distinguish Direct and Inverse Variation
::区分直接变化和反反变化

In direct variation relationships, the related variables will either increase together or decrease together at a steady rate . For instance, consider a person walking at three miles per hour. As time increases, the distance covered by the person walking also increases, at the rate of three miles each hour. The distance and time are related to each other by a direct variation:
::在直接变异关系中,相关变数会共同增加,或者以稳定的速度一起减少。例如,考虑一个人每小时行走3英里。随着时间的增加,行走者的距离也会增加,每小时行走3英里。距离和时间因直接变化而彼此相关:

\begin{aligned} d i s t a n c e = s p e e d \times t i m e \end{aligned}

$\begin{align*}distance = speed \times time\end{align*}$
::距离= 速度x时间

Since the speed is a constant 3 miles per hour, we can write: $\begin{align*}d = 3t\end{align*}$ .
::由于速度是恒定的每小时3英里, 我们可以写: d=3t 。

The general equation for a direct variation is $\begin{align*}y = kx\end{align*}$ , where $\begin{align*}k\end{align*}$ is called the constant of proportionality .
::直接变化的一般方程为y=kx, K称为相称性常数。

You can see from the equation that a direct variation is a linear equation with a $\begin{align*}y-\end{align*}$ intercept of zero. The graph of a direct variation relationship is a straight line passing through the origin whose is $\begin{align*}k\end{align*}$ , the constant of proportionality.
::您可以从方程式中看到,直接变异是一个直线方程式,有 y- interview 的 y- interview 零。直接变异关系图是一条直线, 直线通过源代码 k, 即相称性的常数。

A second type of variation is inverse variation . When two quantities are related to each other inversely, one quantity increases as the other one decreases, and vice versa.
::第二种变化是反向变化。当两个数量相互反向关联时,一个数量与另一个数量相对减少,反之亦然。

For instance, if we look at the formula $\begin{align*}distance = speed \times time\end{align*}$ again and solve for time, we obtain:
::例如,如果我们再看一次公式的距离=速度x时间并解决时间问题,我们就会得到:

\begin{aligned} t i m e = \frac{d i s t a n c e}{s p e e d} \end{aligned}

$\begin{align*}time = \frac{distance}{speed}\end{align*}$
::时间=远距速度

If we keep the distance constant, we see that as the speed of an object increases, then the time it takes to cover that distance decreases. Consider a car traveling a distance of 90 miles, then the formula relating time and speed is: $\begin{align*}t = \frac{90}{s}\end{align*}$ .
::如果我们保持距离不变, 我们就会看到随着物体速度的加快, 覆盖距离所需的时间就会减少。想象一下一辆行驶90英里的汽车, 那么与时间和速度相关的公式是: t=90s 。

The general equation for inverse variation is $\begin{align*}y=\frac{k}{x}\end{align*}$ , where $\begin{align*}k\end{align*}$ is the constant of proportionality .
::反差的一般方程式是y=kx, K是相称性的常数。

In this chapter, we’ll investigate how the graphs of these relationships behave.
::在本章中,我们将调查这些关系的图表如何运作。

Another type of variation is a joint variation . In this type of relationship, one variable may vary as a product of two or more variables.
::另一种变异是共同变异。在这种关系中,一种变异可能因两个或两个以上变异的产物而变异。

For example, the volume of a cylinder is given by:
::例如,圆柱体的体积由:

\begin{aligned} V = π R^{2} \cdot h \end{aligned}

$\begin{align*}V = \pi R^2 \cdot h\end{align*}$
::VR2h

In this example the volume varies directly as the product of the square of the radius of the base and the height of the cylinder. The constant of proportionality here is the number $\begin{align*}\pi\end{align*}$ .
::在此示例中,音量因基数半径方形和圆柱体高度的产物而直接变化。此处的相称性常数是 __。

In many application problems, the relationship between the variables is a combination of variations. For instance Newton’s Law of Gravitation states that the force of attraction between two spherical bodies varies jointly as the masses of the objects and inversely as the square of the distance between them:
::在许多应用问题中,变量之间的关系是各种变数的组合。例如,牛顿的《引力法》指出,两个球体之间的吸引力因对象质量而异,反之,因对象之间的距离而异:

\begin{aligned} F = G \frac{m_{1} m_{2}}{d^{2}} \end{aligned}

$\begin{align*}F = G \frac{m_1 m_2}{d^2}\end{align*}$
::F=Gm1m2d2

In this example the constant of proportionality is called the gravitational constant, and its value is given by $\begin{align*}G = 6.673 \times 10^{-11} \ N \cdot m^2 / kg^2\end{align*}$ .
::在这个例子中,相称性的常数称为引力常数,其值由G=6.673×10-11 Nm2/kg2给出。

Graph Inverse Variation Equations
::反向变化等量

We saw that the general equation for inverse variation is given by the formula $\begin{align*}y = \frac{k}{x}\end{align*}$ , where $\begin{align*}k\end{align*}$ is a constant of proportionality. We will now show how the graphs of such relationships behave. We start by making a table of values. In most applications, $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ are positive, so in our table we’ll choose only positive values of $\begin{align*}x\end{align*}$ .
::我们看到公式 y=kx 给出了反向变量的一般等式, k 是比例性常数。我们现在将显示这种关系的图表是如何表现的。我们从一个数值表开始。在大多数应用中, x 和 y 是正数, 所以在我们的表格中, 我们只能选择 x 的正数。

Graphing an Inverse Variation Relationship
::图图反反变化关系

Graph an inverse variation relationship with the proportionality constant $\begin{align*}k = 1\end{align*}$ .
::与相称性常数 k=1 的反变量关系图

$\begin{align}x\end{align}$	$\begin{align}y =\frac {1}{x}\end{align}$
0	$\begin{align}y=\frac {1}{0} = \text{undefined}\end{align}$
$\begin{align}\frac {1}{4}\end{align}$	$\begin{align}y =\frac {1}{\frac{1}{4}}=4\end{align}$
$\begin{align}\frac {1}{2}\end{align}$	$\begin{align}y=\frac {1}{\frac{1}{2}}=2\end{align}$
$\begin{align}\frac {3}{4}\end{align}$	$\begin{align}y =\frac {1}{\frac{3}{4}}=1.33\end{align}$
1	$\begin{align}y =\frac {1}{1}=1\end{align}$
$\begin{align}\frac {3}{2}\end{align}$	$\begin{align}y=\frac {1}{\frac{3}{2}}=0.67\end{align}$
2	$\begin{align}y =\frac {1}{2}=0.5\end{align}$
3	$\begin{align}y=\frac {1}{3}=0.33\end{align}$
4	$\begin{align}y =\frac {1}{4}=0.25\end{align}$
5	$\begin{align}y =\frac {1}{5}=0.2\end{align}$
10	$\begin{align}y=\frac {1}{10}=0.1\end{align}$

Here is a graph showing these points connected with a smooth curve.
::下面是一张图表,显示这些点与平滑曲线相连。

Both the table and the graph demonstrate the relationship between variables in an inverse variation. As one variable increases, the other variable decreases and vice versa.
::表格和图均以反向变量显示变量之间的关系。随着一个变量的增加,另一个变量减少,反之亦然。

Notice that when $\begin{align*}x = 0\end{align*}$ , the value of $\begin{align*}y\end{align*}$ is undefined . The graph shows that when the value of $\begin{align*}x\end{align*}$ is very small, the value of $\begin{align*}y\end{align*}$ is very big—so it approaches infinity as $\begin{align*}x\end{align*}$ gets closer and closer to zero.
::请注意, 当 x=0 时, y 的值是未定义的。图表显示, 当 x 的值非常小时, y 的值非常大, 所以它接近无穷, 因为 x 越来越接近零。

Similarly, as the value of $\begin{align*}x\end{align*}$ gets very large, the value of $\begin{align*}y\end{align*}$ gets smaller and smaller but never reaches zero. We will investigate this behavior in detail throughout this chapter.
::同样,随着 x 值变得非常大, y 值变得越来越小, 但永远不会达到零。我们将在整个本章中详细调查这种行为。

Writing Inverse Variation Equations
::书面反反变化等量

As we saw, an inverse variation fulfills the equation $\begin{align*}y = \frac{k}{x}\end{align*}$ . In general, we need to know the value of $\begin{align*}y\end{align*}$ at a particular value of $\begin{align*}x\end{align*}$ in order to find the proportionality constant. Once we know the proportionality constant, we can then find the value of $\begin{align*}y\end{align*}$ for any given value of $\begin{align*}x\end{align*}$ .
::正如我们所看到的那样,反差满足了y=kx的等式。一般来说, 我们需要知道 y 的值是 x 的某个特定值, 才能找到相称性常数。一旦我们知道相称性常数, 我们就可以找到 y 的值是 x 的任何给定值。

If $\begin{align*}y\end{align*}$ is inversely proportional to $\begin{align*}x\end{align*}$ , and if $\begin{align*}y = 10\end{align*}$ when $\begin{align*}x = 5\end{align*}$ , find $\begin{align*}y\end{align*}$ when $\begin{align*}x = 2\end{align*}$ .
::如果 y 与 x 成反比, 如果 y= 10 当 x= 5, 找到 y 当 x= 2 。

\begin{aligned} Since y is inversely proportional to x, then: & y = \frac{k}{x} \\ Plug in the values y = 10 and x = 5 : & 10 = \frac{k}{5} \\ Solve for k by multiplying both sides of the equation by 5 : & k = 50 \\ The inverse relationship is given by: & y = \frac{50}{x} \\ When x = 2 : & y = \frac{50}{2} or y = 25 \end{aligned}

$\begin{align*}\text{Since} \ y \ \text{is inversely proportional to} \ x, \text{then:} && y =\frac {k}{x} \\ \\ \text{Plug in the values} \ y = 10 \ \text{and} \ x = 5: && 10 =\frac {k}{5} \\ \\ \text{Solve for} \ k \ \text{by multiplying both sides of the equation by} \ 5: && k =50 \\ \\ \text{The inverse relationship is given by:} && y =\frac {50}{x} \\ \\ \text{When} \ x = 2: && y =\frac {50}{2} \ \text{or} \ y=25\end{align*}$
::y=kxPlug 在 y= 10 和 x= 5: 10=k5Solve 的 k 值中, 将方程两侧乘以 5: k= 50 以反比 y= 50 和 x= 10 和 x= 5: 10=k5Solve 乘以等式两侧乘以 5: k= 50 以 y= 50 x when x=2: y= 502 或y= 25 以 y= 50x when x= 2:y= 502 或y= 25 为反比

Compare Graphs of Inverse Variation Equations
::对比反反变化等量图

are the simplest example of rational functions . We saw that an inverse variation has the general equation: $\begin{align*}y= \frac{k}{x}\end{align*}$ . In most real-world problems, $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ take only positive values. Below, we will show graphs of three inverse variation functions.
::是一个最简单的理性函数示例。我们看到一个反差具有一般等式 : y=kx 。在大多数现实世界的问题中, x 和 y 只使用正值。下面我们将显示三个反差函数的图形。

Comparing Graphs
::比较图表

On the same coordinate grid , graph inverse variation relationships with the proportionality constants $\begin{align*}k = 1, k = 2,\end{align*}$ and $\begin{align*}k = \frac{1}{2}\end{align*}$ .
::在同一坐标网格上,图形反差关系与相称性常数k=1,k=2和k=12。

We’ll skip the table of values for this problem, and just show the graphs of the three functions on the same coordinate axes. Notice that for larger constants of proportionality, the curve decreases at a slower rate than for smaller constants of proportionality. This makes sense because the value of $\begin{align*}y\end{align*}$ is related directly to the proportionality constants, so we should expect larger values of $\begin{align*}y\end{align*}$ for larger values of $\begin{align*}k\end{align*}$ .
::我们跳过这一问题的数值表,而只是在同一坐标轴上显示三个函数的图表。请注意,对于较大的相称性常数,曲线的下降速度比较小的相称性常数的下降速度要慢。这有道理,因为y值与相称性常数直接相关,因此,我们应该期望,对于较大的 k 值, y值会更大。

Example
::示例示例示例示例

Example 1
::例1

If $\begin{align*}p\end{align*}$ is inversely proportional to the square of $\begin{align*}q\end{align*}$ , and $\begin{align*}p = 64\end{align*}$ when $\begin{align*}q = 3\end{align*}$ , find $\begin{align*}p\end{align*}$ when $\begin{align*}q = 5\end{align*}$ .
::如果 p 与 q 的平方成反比, 当 q = 3 时 p= 64, 当 q = 5 时 find p when q = 5 时。

\begin{aligned} Since p is inversely proportional to q^{2}, then: & p = \frac{k}{q^{2}} \\ Plug in the values p = 64 and q = 3 : & 64 = \frac{k}{3^{2}} or 64 = \frac{k}{9} \\ Solve for k by multiplying both sides of the equation by 9 : & k = 576 \\ The inverse relationship is given by: & p = \frac{576}{q^{2}} \\ When q = 5 : & p = \frac{576}{25} or y = 23.04 \end{aligned}

$\begin{align*}\text{Since} \ p \ \text{is inversely proportional to} \ q^2, \text{then:} && p =\frac {k}{q^2} \\ \\ \text{Plug in the values} \ p = 64 \ \text{and} \ q = 3: && 64 =\frac {k}{3^2} \ \text{or} \ 64=\frac {k}{9} \\ \\ \text{Solve for} \ k \ \text{by multiplying both sides of the equation by} \ 9: && k =576 \\ \\ \text{The inverse relationship is given by:} && p =\frac {576}{q^2} \\ \\ \text{When} \ q = 5: && p =\frac {576}{25} \ \text{or} \ y=23.04\end{align*}$
::由于 p 与 q2 反比, 那么: p=kq2Plug 在 p= 64 和 q= 3: 64=k32 或 64= k9Solve 的数值中, k 乘以方程的两边乘以 9: k= 576 。反比由 : p= 576q2 当 q= 5: p= 57625 或 y= 23. 04 给出。

Review
::回顾

For 1-4, graph the following inverse variation relationships.
::1-4, 如下图所示为反向变化关系。

$\begin{align*}y= \frac{3}{x}\end{align*}$
::y=3x y=3x

$\begin{align*}y= \frac{10}{x}\end{align*}$
::y=10x y=10x

$\begin{align*}y= \frac{1}{4x}\end{align*}$
::y=14x y=14x

$\begin{align*}y= \frac{5}{6x}\end{align*}$
::y=56x y=56x

If $\begin{align*}z\end{align*}$ is inversely proportional to $\begin{align*}w\end{align*}$ and $\begin{align*}z = 81\end{align*}$ when $\begin{align*}w = 9\end{align*}$ , find $\begin{align*}w\end{align*}$ when $\begin{align*}z = 24\end{align*}$ .
::如果 z 与 w 和 z= 81 成反比, w= 9, 发现 w 时 z= 24 。

If $\begin{align*}y\end{align*}$ is inversely proportional to $\begin{align*}x\end{align*}$ and $\begin{align*}y = 2\end{align*}$ when $\begin{align*}x = 8\end{align*}$ , find $\begin{align*}y\end{align*}$ when $\begin{align*}x = 12\end{align*}$ .
::如果 y 与 x 成反比, 当 x= 8 时, y 和 y= 2, 则在 x= 12 时会发现 y 。

If $\begin{align*}a\end{align*}$ is inversely proportional to the square root of $\begin{align*}b\end{align*}$ , and $\begin{align*}a = 32\end{align*}$ when $\begin{align*}b = 9\end{align*}$ , find $\begin{align*}b\end{align*}$ when $\begin{align*}a = 6\end{align*}$ .
::如果 a 与 b 的平方根成反比, b=9时为a=32, a=6时为b。

If $\begin{align*}w\end{align*}$ is inversely proportional to the square of $\begin{align*}u\end{align*}$ and $\begin{align*}w = 4\end{align*}$ when $\begin{align*}u = 2\end{align*}$ , find $\begin{align*}w\end{align*}$ when $\begin{align*}u = 8\end{align*}$ .
::如果 w 与 u 的平方成反比, 当 u = 2 时 w=4, 找到 u = 8 时 w = 8 。

If $\begin{align*}a\end{align*}$ is proportional to both $\begin{align*}b\end{align*}$ and $\begin{align*}c\end{align*}$ and $\begin{align*}a = 7\end{align*}$ when $\begin{align*}b = 2\end{align*}$ and $\begin{align*}c = 6\end{align*}$ , find $\begin{align*}a\end{align*}$ when $\begin{align*}b = 4\end{align*}$ and $\begin{align*}c = 3\end{align*}$ .
::a 如果b=2和c=6时,a与b和c和a=7成正比,b=2和c=6时,找到b=4和c=3时。

If $\begin{align*}x\end{align*}$ is proportional to $\begin{align*}y\end{align*}$ and inversely proportional to $\begin{align*}z\end{align*}$ , and $\begin{align*}x = 2\end{align*}$ when $\begin{align*}y = 10\end{align*}$ and $\begin{align*}z = 25\end{align*}$ , find $\begin{align*}x\end{align*}$ when $\begin{align*}y = 8\end{align*}$ and $\begin{align*}z = 35\end{align*}$ .
::如果 x 与 y 成比例, 与 z 成反比例, y= 10 和 z= 25 的 x=2, 找到 x y= 8 和 z= 35 的 x 。

If $\begin{align*}a\end{align*}$ varies directly with $\begin{align*}b\end{align*}$ and inversely with the square of $\begin{align*}c\end{align*}$ , and $\begin{align*}a = 10\end{align*}$ when $\begin{align*}b = 5\end{align*}$ and $\begin{align*}c = 2\end{align*}$ , find the value of $\begin{align*}a\end{align*}$ when $\begin{align*}b = 3\end{align*}$ and $\begin{align*}c = 6\end{align*}$ .
::如果 a 与 b 直接变化, 与 c 平方反差, 则a= 10, b=5 和 c=2, 发现 b=3 和 c=6 的值。

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。

$\begin{align}x\end{align}$	$\begin{align}y =\frac {1}{x}\end{align}$
0	$\begin{align}y=\frac {1}{0} = \text{undefined}\end{align}$
$\begin{align}\frac {1}{4}\end{align}$	$\begin{align}y =\frac {1}{\frac{1}{4}}=4\end{align}$
$\begin{align}\frac {1}{2}\end{align}$	$\begin{align}y=\frac {1}{\frac{1}{2}}=2\end{align}$
$\begin{align}\frac {3}{4}\end{align}$	$\begin{align}y =\frac {1}{\frac{3}{4}}=1.33\end{align}$
1	$\begin{align}y =\frac {1}{1}=1\end{align}$
$\begin{align}\frac {3}{2}\end{align}$	$\begin{align}y=\frac {1}{\frac{3}{2}}=0.67\end{align}$
2	$\begin{align}y =\frac {1}{2}=0.5\end{align}$
3	$\begin{align}y=\frac {1}{3}=0.33\end{align}$
4	$\begin{align}y =\frac {1}{4}=0.25\end{align}$
5	$\begin{align}y =\frac {1}{5}=0.2\end{align}$
10	$\begin{align}y=\frac {1}{10}=0.1\end{align}$

章节大纲

常规

反反变化模型

Inverse Variation Models ::反反变化模型

Distinguish Direct and Inverse Variation ::区分直接变化和反反变化

Graph Inverse Variation Equations ::反向变化等量

Graphing an Inverse Variation Relationship ::图图反反变化关系

Writing Inverse Variation Equations ::书面反反变化等量

Compare Graphs of Inverse Variation Equations ::对比反反变化等量图

Comparing Graphs ::比较图表

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)