Course: 12.2 理性功能图 | The Way To Learn

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逻辑函数图

Graphs of Rational Functions
::逻辑函数图

Graphs of rational functions are very distinctive, because they get closer and closer to certain values but never reach those values. This behavior is called asymptotic behavior, and we will see that rational functions can have horizontal asymptotes , vertical asymptotes or oblique (or slant) asymptotes.
::理性函数的图形非常独特, 因为它们越来越接近某些值, 但从未达到这些值。这种行为被称为“ 无症状行为 ” , 而我们会发现理性函数可能具有水平性无症状、垂直无症状或倾斜性( 倾斜性) 。

Now we’ll extend the domain and range of rational equations to include negative values of $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ . First we’ll plot a few rational functions by using a table of values, and then we’ll talk about the distinguishing characteristics of rational functions that can help us make better graphs.
::现在,我们将扩大理性方程式的域域和范围, 以包括x和y的负值。首先,我们将使用一个数值表来绘制一些合理的函数, 然后我们再讨论理性函数的显著特征, 以便帮助我们制作更好的图表。

As we graph rational functions, we need to always pay attention to values of $\begin{aligned} x \end{aligned}$ that will cause us to divide by 0. Remember that dividing by 0 doesn’t give us an actual number as a result.
::当我们绘制理性函数时,我们需要始终关注x的值,这将导致我们除以0。记住,除以0不会给我们一个实际数字。

Graphing Functions
::图图函数

1. Graph the function $\begin{aligned} y = \frac{1}{x} \end{aligned}$ .
::1. 绘制函数y=1x。

Before we make a table of values, we should notice that the function is not defined for $\begin{aligned} x = 0 \end{aligned}$ . This means that the graph of the function won’t have a value at that point. Since the value of $\begin{aligned} x = 0 \end{aligned}$ is special, we should make sure to pick enough values close to $\begin{aligned} x = 0 \end{aligned}$ in order to get a good idea how the graph behaves.
::在绘制数值表之前,我们应该注意,函数不是为 x=0 定义的。这意味着函数的图形在那个点上不会有值。由于 x=0 值是特殊的, 我们应该确定选择接近 x=0 的足够值, 以便很好地了解图表的运行方式。

Let’s make two tables: one for $\begin{aligned} x - \end{aligned}$ values smaller than zero and one for $\begin{aligned} x - \end{aligned}$ values larger than zero.
::让我们做两个表格:一个为零以下的x-值,一个为零以下的x-值。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = \frac{1}{x} \end{aligned}$	$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = \frac{1}{x} \end{aligned}$
$\begin{aligned} - 5 \end{aligned}$	$\begin{aligned} y = \frac{1}{- 5} = - 0.2 \end{aligned}$	0.1	$\begin{aligned} y = \frac{1}{0.1} = 10 \end{aligned}$
-4	$\begin{aligned} y = \frac{1}{- 4} = - 0.25 \end{aligned}$	0.2	$\begin{aligned} y = \frac{1}{0.2} = 5 \end{aligned}$
-3	$\begin{aligned} y = \frac{1}{- 3} = - 0.33 \end{aligned}$	0.3	$\begin{aligned} y = \frac{1}{0.3} = 3.3 \end{aligned}$
-2	$\begin{aligned} y = \frac{1}{- 2} = - 0.5 \end{aligned}$	0.4	$\begin{aligned} y = \frac{1}{0.4} = 2.5 \end{aligned}$
-1	$\begin{aligned} y = \frac{1}{- 1} = - 1 \end{aligned}$	0.5	$\begin{aligned} y = \frac{1}{0.5} = 2 \end{aligned}$
-0.5	$\begin{aligned} y = \frac{1}{- 0.5} = - 2 \end{aligned}$	1	$\begin{aligned} y = \frac{1}{1} = 1 \end{aligned}$
-0.4	$\begin{aligned} y = \frac{1}{- 0.4} = - 2.5 \end{aligned}$	2	$\begin{aligned} y = \frac{1}{2} = 0.5 \end{aligned}$
-0.3	$\begin{aligned} y = \frac{1}{- 0.3} = - 3.3 \end{aligned}$	3	$\begin{aligned} y = \frac{1}{3} = 0.33 \end{aligned}$
-0.2	$\begin{aligned} y = \frac{1}{- 0.2} = - 5 \end{aligned}$	4	$\begin{aligned} y = \frac{1}{4} = 0.25 \end{aligned}$
-0.1	$\begin{aligned} y = \frac{1}{- 0.1} = - 10 \end{aligned}$	5	$\begin{aligned} y = \frac{1}{5} = 0.2 \end{aligned}$

We can see that as we pick positive values of $\begin{aligned} x \end{aligned}$ closer and closer to zero, $\begin{aligned} y \end{aligned}$ gets larger, and as we pick negative values of $\begin{aligned} x \end{aligned}$ closer and closer to zero, $\begin{aligned} y \end{aligned}$ gets smaller (or more and more negative).
::我们可以看到,当我们选择x的正值接近零时,y会变大,当我们选择x的负值接近零时,y会变小(或越来越消极)。

Notice on the graph that for values of $\begin{aligned} x \end{aligned}$ near 0, the points on the graph get closer and closer to the vertical line $\begin{aligned} x = 0 \end{aligned}$ . The line $\begin{aligned} x = 0 \end{aligned}$ is called a vertical asymptote of the function $\begin{aligned} y = \frac{1}{x} \end{aligned}$ .
::图形上的注意,对于接近 0 的 x 值, 图形上的点会越来越接近垂直线x= 0。直线 x=0 被称为函数 y= 1x 的垂直同位数。

We also notice that as the absolute values of $\begin{aligned} x \end{aligned}$ get larger in the positive direction or in the negative direction, the value of $\begin{aligned} y \end{aligned}$ gets closer and closer to $\begin{aligned} y = 0 \end{aligned}$ but will never gain that value. Since $\begin{aligned} y = \frac{1}{x} \end{aligned}$ , we can see that there are no values of $\begin{aligned} x \end{aligned}$ that will give us the value $\begin{aligned} y = 0 \end{aligned}$ . The horizontal line $\begin{aligned} y = 0 \end{aligned}$ is called a horizontal asymptote of the function $\begin{aligned} y = \frac{1}{x} \end{aligned}$
::我们也注意到,随着x的绝对值在正方向或负方向上变得更大,y的绝对值越来越接近y=0,但永远不会获得该值。由于 y=1x,我们可以看到,没有x的绝对值能给我们的数值y=0。水平线 y=0被称为函数y=1x的水平单点。

Asymptotes are usually denoted as dashed lines on a graph. They are not part of the function; instead, they show values that the function approaches, but never gets to. A horizontal asymptote shows the value of $\begin{aligned} y \end{aligned}$ that the function approaches (but never reaches) as the absolute value of $\begin{aligned} x \end{aligned}$ gets larger and larger. A vertical asymptote shows that the absolute value of $\begin{aligned} y \end{aligned}$ gets larger and larger as $\begin{aligned} x \end{aligned}$ gets closer to a certain value which it can never actually reach.
::Asymptotes 通常被记为图形中虚线行。它们不是函数的一部分; 相反, 它们显示函数接近但永远无法到达的值。水平的零点显示函数接近的 y 值, 因为x 的绝对值越来越大( 但从未达到) 。垂直的零点显示, y 的绝对值随着x 接近一个它实际上无法真正达到的某个值, y 的绝对值越大越大。

Now we’ll show the graph of a rational function that has a vertical asymptote at a non-zero value of $\begin{aligned} x \end{aligned}$ .
::现在我们将显示一个合理函数的图, 该函数的垂直空位值为x的非零值。

2. Graph the function $\begin{aligned} y = \frac{1}{(x - 2)^{2}} \end{aligned}$ .
::2. 绘制函数y=1(x-2)图。

We can see that the function is not defined for $\begin{aligned} x = 2 \end{aligned}$ , because that would make the denominator of the fraction equal zero. This tells us that there should be a vertical asymptote at $\begin{aligned} x = 2 \end{aligned}$ , so we can start graphing the function by drawing the vertical asymptote.
::我们可以看到该函数没有为 x=2 定义定义, 因为这将使分数分母为零。这告诉我们在 x=2 时应该有一个垂直的静态, 这样我们就可以通过绘制垂直静态来开始图形化函数。

Now let’s make a table of values.
::现在让我们绘制一个数值表。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = \frac{1}{(x - 2)^{2}} \end{aligned}$
0	$\begin{aligned} y = \frac{1}{(0 - 2)^{2}} = \frac{1}{4} \end{aligned}$
1	$\begin{aligned} y = \frac{1}{(1 - 2)^{2}} = 1 \end{aligned}$
1.5	$\begin{aligned} y = \frac{1}{(1.5 - 2)^{2}} = 4 \end{aligned}$
2	$\begin{aligned} undefined \end{aligned}$
2.5	$\begin{aligned} y = \frac{1}{(2.5 - 2)^{2}} = 4 \end{aligned}$
3	$\begin{aligned} y = \frac{1}{(3 - 2)^{2}} = 1 \end{aligned}$
4	$\begin{aligned} y = \frac{1}{(4 - 2)^{2}} = \frac{1}{4} \end{aligned}$

Here’s the resulting graph:
::以下是由此得出的图表:

Notice that we didn’t pick as many values for our table this time, because by now we have a pretty good idea what happens near the vertical asymptote.
::注意我们这次没有为桌子选择那么多的值, 因为现在我们非常清楚在垂直无序状态附近会发生什么。

We also know that for large values of $\begin{aligned} | x | \end{aligned}$ , the value of $\begin{aligned} y \end{aligned}$ could approach a constant value. In this case that value is $\begin{aligned} y = 0 \end{aligned}$ : this is the horizontal asymptote.
::我们还知道,对于 x 的大值来说, y 的值可以接近一个恒定值。在这种情况下, 值是 y=0 : 这是水平的单数。

A rational function doesn’t have to have a vertical or horizontal asymptote. The next example shows a rational function with no vertical asymptotes.
::理性函数不必有垂直或水平的空点。下一个示例显示的是没有垂直空点的理性函数。

3. Graph the function $\begin{aligned} y = \frac{x^{2}}{x^{2} + 1} \end{aligned}$ .
::3. 绘制函数 y=x2x2+1 。

We can see that this function will have no vertical asymptotes because the denominator of the fraction will never be zero. Let’s make a table of values to see if the value of $\begin{aligned} y \end{aligned}$ approaches a particular value for large values of $\begin{aligned} x \end{aligned}$ , both positive and negative.
::我们可以看到,这个函数将没有垂直的零位数,因为分数的分母永远不会是零。让我们绘制一个数值表,看看y值是否接近X(正值和负值)的大值的特定值。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = \frac{x^{2}}{x^{2} + 1} \end{aligned}$
$\begin{aligned} - 3 \end{aligned}$	$\begin{aligned} y = \frac{(- 3)^{2}}{(- 3)^{2} + 1} = \frac{9}{10} = 0.9 \end{aligned}$
-2	$\begin{aligned} y = \frac{(- 2)^{2}}{(- 2)^{2} + 1} = \frac{4}{5} = 0.8 \end{aligned}$
-1	$\begin{aligned} y = \frac{(- 1)^{2}}{(- 1)^{2} + 1} = \frac{1}{2} = 0.5 \end{aligned}$
0	$\begin{aligned} y = \frac{(0)^{2}}{(0)^{2} + 1} = \frac{0}{1} = 0 \end{aligned}$
1	$\begin{aligned} y = \frac{(1)^{2}}{(1)^{2} + 1} = \frac{1}{2} = 0.5 \end{aligned}$
2	$\begin{aligned} y = \frac{(2)^{2}}{(2)^{2} + 1} = \frac{4}{5} = 0.8 \end{aligned}$
3	$\begin{aligned} y = \frac{(3)^{2}}{(3)^{2} + 1} = \frac{9}{10} = 0.9 \end{aligned}$

Below is the graph of this function.
::下面是此函数的图形。

The function has no vertical asymptote. However, we can see that as the values of $\begin{aligned} | x | \end{aligned}$ get larger, the value of $\begin{aligned} y \end{aligned}$ gets closer and closer to 1, so the function has a horizontal asymptote at $\begin{aligned} y = 1 \end{aligned}$ .
::函数没有垂直的零星状态。然而, 我们可以看到随着 {x}} 的值越来越大, y 的值越来越接近 1, 所以函数在 y= 1 上有一个水平的零星状态。

Example
::示例示例示例示例

Example 1
::例1

Graph the function $\begin{aligned} y = \frac{3}{x - 1} \end{aligned}$ .
::函数 y= 3x- 1 图形。

Start by making a table of values:
::以建立一个数值表开始 :

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = \frac{3}{x - 1} \end{aligned}$
$\begin{aligned} - 2 \end{aligned}$	$\begin{aligned} y = \frac{3}{- 2 - 1} = \frac{3}{- 3} = - 1 \end{aligned}$
-1	$\begin{aligned} y = \frac{3}{- 1 - 1} = \frac{3}{- 2} = - 1.5 \end{aligned}$
0	$\begin{aligned} y = \frac{3}{0 - 1} = \frac{3}{- 1} = - 3 \end{aligned}$
1	undefined
2	$\begin{aligned} y = \frac{3}{2 - 1} = \frac{3}{1} = 3 \end{aligned}$
3	$\begin{aligned} y = \frac{3}{3 - 1} = \frac{3}{2} = 1.5 \end{aligned}$
4	$\begin{aligned} y = \frac{3}{4 - 1} = \frac{3}{3} = 1 \end{aligned}$

Next, graph the points. Recall that the function $\begin{aligned} y = \frac{1}{x} \end{aligned}$ has two curves, that are on either side of the vertical asymptote, which is where the function is undefined. The same is true for this function.
::下一页请绘制各个点的图表。回顾函数 y=1x 有两条曲线, 这两条曲线位于垂直单点的两侧, 即函数未定义的位置。此函数同样如此。

Review
::回顾

Graph the following rational functions. Draw dashed vertical and horizontal lines on the graph to denote asymptotes.
::绘制以下合理函数的图形。在图形上绘制垂直和水平虚线以表示无符号。

$\begin{aligned} y = \frac{2}{x - 3} \end{aligned}$
::y=2x-3

$\begin{aligned} y = \frac{3}{x^{2}} \end{aligned}$
::y=3x2 y=3x2

$\begin{aligned} y = \frac{x}{x - 1} \end{aligned}$
::y=xx-1

$\begin{aligned} y = \frac{2 x}{x + 1} \end{aligned}$
::y=2xx+1 y=2xx+1

$\begin{aligned} y = \frac{- 1}{x^{2} + 2} \end{aligned}$
::y1x2+2 y1x2+2

$\begin{aligned} y = \frac{x}{x^{2} + 9} \end{aligned}$
::y=x22+9

$\begin{aligned} y = \frac{x^{2}}{x^{2} + 1} \end{aligned}$
::y=x2x2+1 y=x2x2+1

$\begin{aligned} y = \frac{1}{x^{2} - 1} \end{aligned}$
::y= 1x2 - 1

$\begin{aligned} y = \frac{2 x}{x^{2} - 9} \end{aligned}$
::y=2x22-9

$\begin{aligned} y = \frac{x^{2}}{x^{2} - 16} \end{aligned}$
::y=x2x2 - 16

$\begin{aligned} y = \frac{3}{x^{2} - 4 x + 4} \end{aligned}$
::y= 3x2 - 4x+4

$\begin{aligned} y = \frac{x}{x^{2} - x - 6} \end{aligned}$
::y=x22-x-6 y=x22-x-6

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。

Section outline

General

逻辑函数图

Graphs of Rational Functions ::逻辑函数图

Graphing Functions ::图图函数

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾