Section: 简缩缩缩缩 | 2.11 中性静态

Section outline

When the degree of the numerator of a rational function exceeds the degree of the denominator by one then the function has oblique asymptotes . In order to find these asymptotes , you need to use polynomial long division and the non-remainder portion of the function becomes the . A natural question to ask is: what happens when the degree of the numerator exceeds that of the denominator by more than one?
::当一个理性函数的分子量超过分母的分母度时,该函数就会有一个斜形的微粒。要找到这些微粒,您需要使用多符号长的分割法,函数的非反射部分就会变成。一个自然的问题是要问:当分子量超过分母一个以上时会发生什么情况?

Oblique Asymptotes
::简缩缩缩缩

The following function is shown before and after polynomial long division is performed.
::以下函数在完成多面长分割之前和之后显示。

$\begin{aligned} f (x) = \frac{x^{4} + 3 x^{2} + 2 x + 14}{x^{3} - 3 x^{2}} = x + 3 + \frac{12 x^{2} + 2 x + 14}{x^{3} - 3 x^{2}} \end{aligned}$
::f( x) =x4+3x2+2x+14x3_3x2=x3+12x2+2x+14x3_3x2

Notice that the remainder portion will go to zero when $\begin{aligned} x \end{aligned}$ gets extremely large or extremely small because the power of the numerator is smaller than the power of the denominator. This means that while this function might go haywire with small absolute values of $\begin{aligned} x \end{aligned}$ , large absolute values of $\begin{aligned} x \end{aligned}$ are extremely close to the line $\begin{aligned} y = x + 3 \end{aligned}$ .
::请注意, 当 x 变得非常大或极小时, 其余部分会变为零, 因为分子的力量小于分母的力量。这意味着, 虽然这个函数可能会以小的绝对值 x 来擦去, 但是, x 的绝对值很大, 却非常接近 y=x+3 的线条 y=x+ 3 。

Oblique asymptotes are these slanted asymptotes that show exactly how a function increases or decreases without bound. Oblique asymptotes are also called slant asymptotes.
::微微微粒是这些倾斜的微粒,这些微粒确切显示函数如何在无约束的情况下增减。微微微粒也被称为斜斜的微粒。

Sometimes a function will have an asymptote that does not look like a line. Take a look at the following function:
::有时函数会有一个看起来不像线条的零星状态。查看以下函数 :

$\begin{aligned} f (x) = \frac{(x^{2} - 4) (x + 3)}{10 (x - 1)} \end{aligned}$
:xx) = (x2-4)(x+3) 10(x-1)

The degree of the numerator is 3 while the degree of the denominator is 1 so the slant asymptote will not be a line. However when the graph is observed, there is still a clear pattern as to how this function increases without bound as $\begin{aligned} x \end{aligned}$ approaches very large and very small numbers.
::分子的度为3, 分母的度为1, 这样倾斜的微粒就不会是一行。但是, 当观察到图形时, 仍然有一个清晰的规律, 说明随着 x 接近非常大和非常小的数字, 此函数是如何增加而不受约束的。

$\begin{aligned} f (x) = \frac{1}{10} (x^{2} + 4 x) - \frac{12}{10 (x - 1)} \end{aligned}$
:xx) = 110(x2+4x) - 1210(x-1)

As you can see, this looks like a parabola with a remainder. This rational function has a parabola backbone. A backbone is a function that a graph tends towards. This is not technically an oblique asymptote because it is not a line.
::正如您所看到的, 这看起来像一个配有剩余值的抛物线。这个理性函数有一个抛物线主干线。一个主干线是一个图表所偏向的函数。这在技术上并不是一个微缩的无线, 因为它不是直线。

Examples
::实例

Example 1
::例1

Earlier, you were asked what happens when the power of the numerator exceeds the power of the denominator by more than one. As seen above, when the power of numerator exceeds the power of the denominator by more than one, the function develops a backbone that be shaped like any polynomial . Oblique asymptotes are always lines.
::早些时候,有人问到当分子的力量超过分母的力量不止一个时会发生什么。如上所示,当分子的力量超过分母的力量超过一个以上的分母的力量时,函数会形成一个像任何多面体形状一样的脊柱。常态的微粒总是线条。

Example 2
::例2

Find the asymptotes and intercepts of the function:
::查找函数的空位和拦截 :

$\begin{aligned} f (x) = \frac{x^{3}}{x^{2} - 4} \end{aligned}$
:xx) =x3x2-4)

The function has vertical asymptotes at $\begin{aligned} x = \pm 2 \end{aligned}$ .
::该函数在 x @ @% 2 上具有垂直的空位。

After long division , the function becomes:
::长期分割后,该功能变成:

$\begin{aligned} f (x) = x + \frac{4}{x^{2} - 4} \end{aligned}$
:xx) =x+4x2-4

This makes the oblique asymptote at $\begin{aligned} y = x \end{aligned}$
::使 y=x 的倾斜渐移

Example 3
::例3

Identify the vertical and oblique asymptotes of the following rational function.
::识别以下理性函数的垂直和倾斜性静态。

$\begin{aligned} f (x) = \frac{x^{3} - x^{2} - x - 1}{(x - 3) (x + 4)} \end{aligned}$
:xx) =x3-x2-x1(x-3)(x+4)

After using polynomial long division and rewriting the function with a remainder and non-remainder portion it looks like this:
::在使用多面长的分隔线并用剩余部分和非重复部分重写函数后,它看起来是这样的:

$\begin{aligned} f (x) = x - 2 + \frac{13 x - 25}{x^{2} + x - 12} = x - 2 + \frac{13 x - 25}{(x - 3) (x + 4)} \end{aligned}$
:x) =x- 2+13x- 13x- 25x2+x-12=x-2+13x- 25(x-3)(x+4)

The oblique asymptote is $\begin{aligned} y = x - 2 \end{aligned}$ . The vertical asymptotes are at $\begin{aligned} x = 3 \end{aligned}$ and $\begin{aligned} x = - 4 \end{aligned}$ which are easier to observe in last form of the function because they clearly don’t cancel to become holes.
::斜体的渐微偏移是 y=x-2。垂直的渐微偏移在 x=3 和 x4 上, 比较容易以此函数的最后形式观察到, 因为它们显然不会取消, 成为空洞。

Example 4
::例4

Create a function with an oblique asymptote at $\begin{aligned} y = 3 x - 1 \end{aligned}$ , vertical asymptotes at $\begin{aligned} x = 2, - 4 \end{aligned}$ and includes a hole where $\begin{aligned} x \end{aligned}$ is 7.
::在 y= 3x-1, 创建 y= 3x- 4 垂直 asymptotes 以 y= 2, ~ 4 的斜线等量函数, 并包含 x 7 所在的洞。

While there are an infinite number of functions that match these criteria, one example is:
::虽然有无限数量的功能符合这些标准,但一个实例是:

$\begin{aligned} f (x) = 3 x - 1 + \frac{(x - 7)}{(x - 2) (x + 4) (x - 7)} \end{aligned}$
:xx)=3x-1+(x-7)(x-7)(x-7)(x-7)(x)

Example 5
::例5

Identify the backbone of the following function and explain why the function does not have an oblique asymptote.
::确定以下函数的主干线,并解释为何该函数没有倾斜的单点。

$\begin{aligned} f (x) = \frac{5 x^{5} + 27}{x^{3}} \end{aligned}$
:x)=5x5+27x3

While polynomial long division is possible, it is also possible to just divide each term by $\begin{aligned} x^{3} \end{aligned}$ .
::虽然可能存在多党制长期分歧,但也可以将每个术语除以x3。

$\begin{aligned} f (x) = \frac{5 x^{5} + 27}{x^{3}} = \frac{5 x^{5}}{x^{3}} + \frac{27}{x^{3}} = 5 x^{2} + \frac{27}{x^{3}} \end{aligned}$
::f(x)=5x5+27x3=5x5x3+27x3=5x5x3+27x3=5x2+27x3

The backbone of this function is the parabola $\begin{aligned} y = 5 x^{2} \end{aligned}$ . This is not an oblique asymptote because it is not a line.
::此函数的主干线是 parbola y= 5x2。这不是一个倾斜的单点, 因为它不是直线。

Summary

Oblique asymptotes occur when the degree of the numerator of a rational function is exactly one greater than the degree of the denominator.
::当一个理性函数的分子数量比分母数量高出一度时,就会出现尖锐的微粒。

Oblique asymptotes are slanted asymptotes that show how a function increases or decreases without bound.
::微微微微粒是倾斜的微粒,显示函数如何在无约束的情况下增减。

To find oblique asymptotes, use polynomial long division and the non-remainder portion of the function becomes the oblique asymptote.
::要找到斜线单线条, 使用多面长的分割和函数的非反向部分成为斜线单线条。

If the degree of the numerator exceeds the degree of the denominator by more than one, the function may have a backbone, which is a function that the graph tends towards.
::如果分子的程度超过分母的程度超过一个以上,则该函数可能有一个主干,这是图所倾向的函数。

A backbone is not technically an oblique asymptote if it is not a line.
::如果骨干不是一条线的话,从技术上讲,骨干并不是一个微弱的斜体。

Review
::回顾

1. What is an oblique asymptote?
::1. 什么是倾斜的消亡?

2. How can you tell by looking at the equation of a function if it will have an oblique asymptote or not?
::2. 如果某一函数的方程式有微弱的微弱状态,又如何通过查看该函数的方程式来判断该函数的方程式?

3. Can a function have both an oblique asymptote and a horizontal asymptote? Explain.
::3. 函数能否同时具有倾斜的零点和水平的零点?

For each of the following graphs, sketch the graph and then sketch in the oblique asymptote if it exists. If it doesn’t exist, explain why not.
::对于下图中的每一张图,请绘制图表的草图,然后在存在时以斜线线线绘制草图。如果它不存在,请解释为什么没有。

Find the equation of the oblique asymptote for each of the following rational functions. If there is not an oblique asymptote, explain why not and give an equation of the backbone of the function if one exists.
::查找以下每种理性函数的斜线单点等式。如果没有斜线单点, 请解释为什么没有, 如果函数有的话, 给出函数主干线的等式。

9. $\begin{aligned} f (x) = \frac{x^{3} - 7 x - 6}{x^{2} - 2 x - 15} \end{aligned}$
::9. f(x)=x3-7x-6x2-2-2x-15

10. $\begin{aligned} g (x) = \frac{x^{3} - 7 x - 6}{x^{4} - 3 x^{2} - 10} \end{aligned}$
::10. g(x)=x3-7x-6x4-3x2-10

11. $\begin{aligned} h (x) = \frac{x^{2} + 5 x + 6}{x^{2} + 2 x + 1} \end{aligned}$
::11. h(x) =x2+5x+6x2+2x+1

12. $\begin{aligned} k (x) = \frac{x^{4} + 9 x^{3} + 21 x^{2} - x - 30}{x^{2} + 2 x + 1} \end{aligned}$
::12. k(x) =x4+9x3+212x2-x-30x2+2x+1

13. Create a function with an oblique asymptotes at $\begin{aligned} y = 2 x - 1 \end{aligned}$ , a vertical asymptote at $\begin{aligned} x = 3 \end{aligned}$ and a hole where $\begin{aligned} x \end{aligned}$ is 7.
::13. 在y=2x-1、 x=3和 x为7 的洞处创建一个函数,在 y=2x-1、垂直等式中以斜线状的星点为 y=2x- 1 、垂直等式为 x=3 和洞处为 x 7 。

14. Create a function with an oblique asymptote at $\begin{aligned} y = x \end{aligned}$ , vertical asymptotes at $\begin{aligned} x = 1, - 3 \end{aligned}$ and no holes.
::14. 创建函数,在 y=x 、 x=1 =1 = 3 和无孔时,以斜线星点为 y=x 、垂直星点为垂直星点为 x=1 = 3 和无洞创建函数。

15. Does a parabola have an oblique asymptote? What about a cubic function?
::15. 抛物线是否具有微弱的微粒作用?

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

Oblique Asymptotes ::简缩缩缩缩

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)