节: 按司划分确定小数点 | 12.6 按司划分的确定年数

章节大纲

Determining Asymptotes by Division
::按司划分确定小数点

In the last section we saw how to find vertical and horizontal asymptotes . Remember, the horizontal asymptote shows the value of $\begin{aligned} y \end{aligned}$ that the function approaches for large values of $\begin{aligned} x \end{aligned}$ . Let’s review the method for finding horizontal asymptotes and see how it’s related to polynomial division .
::在最后一节中,我们看到了如何找到垂直和水平的静态。记住, 水平的静态显示了 y 的值, y 的函数是 x 的大值。让我们来审查寻找水平的静态的方法, 看看它与多元分法有何关系。

When it comes to finding asymptotes, there are basically four different types of rational functions .
::在寻找微粒时,基本上有四种不同的理性功能。

Finding Asymptotes
::查找无符号( A)

Case 1: The polynomial in the numerator has a lower degree than the polynomial in the denominator.
::案例1:分子中的多数值比分母中的多数值低。

Find the horizontal asymptote of $\begin{aligned} y = \frac{2}{x - 1} \end{aligned}$ .
::查找 y= 2x- 1 的水平单数。

We can’t reduce this fraction , and as $\begin{aligned} x \end{aligned}$ gets larger the denominator of the fraction gets much bigger than the numerator, so the whole fraction approaches zero.
::我们无法减少这个分数, 随着x的增大, 分数分母的分母会比分子大得多, 所以整个分数接近零。

The horizontal asymptote is $\begin{aligned} y = 0 \end{aligned}$ .
::水平点为 y=0 。

Case 2: The polynomial in the numerator has the same degree as the polynomial in the denominator.
::案例2:分子中的多数值与分母中的多数值具有相同的程度。

Find the horizontal asymptote of $\begin{aligned} y = \frac{3 x + 2}{x - 1} \end{aligned}$ .
::查找 y= 3x+2x- 1 的水平单数。

In this case we can divide the two :
::在这种情况下,我们可以将两者分开:

$\begin{aligned} \overset{3}{x - 1 \bar{) 3 x + 2}} \\ \underline{- 3 x + 3} \\ 5 \end{aligned}$

::x- 113x+2 3- 3x+3_ 5

So the expression can be written as $\begin{aligned} y = 3 + \frac{5}{x - 1} \end{aligned}$ .
::因此表达式可以以 y= 3+5x- 1 写入 y= 3+5x- 1 。

Because the denominator of the remainder is bigger than the numerator of the remainder, the remainder will approach zero for large values of $\begin{aligned} x \end{aligned}$ . Adding the 3 to that 0 means the whole expression will approach 3.
::由于剩余部分的分母大于剩余部分的分子数, X 的大值的剩余部分将接近零。加上 3 到 0 意味着整个表达式将接近 3 。

The horizontal asymptote is $\begin{aligned} y = 3 \end{aligned}$ .
::水平静态为 y= 3 。

Case 3: The polynomial in the numerator has a degree that is one more than the polynomial in the denominator.
::案例3:分子中的多数值具有比分母中的多数值多1的学位。

Find any asymptotes of $\begin{aligned} y = \frac{4 x^{2} + 3 x + 2}{x - 1} \end{aligned}$ .
::查找任何 Y= 4x2+3x+2x- 1 的 y= 4x2+3x+2x- 1 的 ymptotes 。

Solution:
::解决方案 :

We can do long division once again and rewrite the expression as $\begin{aligned} y = 4 x + 7 + \frac{9}{x - 1} \end{aligned}$ . The fraction here approaches zero for large values of $\begin{aligned} x \end{aligned}$ , so the whole expression approaches $\begin{aligned} 4 x + 7 \end{aligned}$ .
::我们可以再次做长的分隔, 重写表达式为 y=4x+7+9x- 1。这里的分数对 x 的大值接近零, 因此整个表达式接近 4x+7 。

When the rational function approaches a straight line for large values of $\begin{aligned} x \end{aligned}$ , we say that the rational function has an . In this case, then, the oblique asymptote is $\begin{aligned} y = 4 x + 7 \end{aligned}$ .
::当理性函数接近 x 的大值直线时, 我们就会说, 理性函数有一个。在这种情况下, 斜线是 y= 4x+7 。

Case 4: The polynomial in the numerator has a degree that is two or more than the degree in the denominator.
::案例4: 分子中的多数值具有两个或两个以上分母中的多数值。

Find any asymptotes of $\begin{aligned} y = \frac{x^{3}}{x - 1} \end{aligned}$ .
::查找任何 Y=x3x- 1 的 asymptotes 。

This is actually the simplest case of all: the polynomial has no horizontal or .
::这实际上是最简单的例子:多面性没有水平或。

Notice that a rational function will either have a horizontal asymptote, an oblique asymptote or neither kind. In other words, a function can’t have both; in fact, it can’t have more than one of either kind. On the other hand, a rational function can have any number of vertical asymptotes at the same time that it has horizontal or oblique asymptotes.
::注意理性函数要么具有水平性零点,要么具有倾斜性零点,要么没有同类。换句话说,一个函数不能同时具有两种功能;事实上,它不能同时具有两种类型中的一种以上。另一方面,一个理性函数在具有水平性或倾斜性零点的同时,也可以拥有任何数量的垂直静点。

Examples
::实例

Find the horizontal or oblique asymptotes of the following rational functions.
::查找以下理性函数的水平或倾斜性静态。

Example 1
::例1

$\begin{aligned} y = \frac{3 x^{2}}{x^{2} + 4} \end{aligned}$
::y= 3x2x2+4 y= 3x2x2+4

When we simplify the function, we get $\begin{aligned} y = 3 - \frac{12}{x^{2} + 4} \end{aligned}$ . There is a horizontal asymptote at $\begin{aligned} y = 3 \end{aligned}$ .
::当我们简化函数时, 我们得到 y= 3 - 12x2+ 4。在 y= 3 处有一个水平同位数。

Example 2
::例2

$\begin{aligned} y = \frac{x - 1}{3 x^{2} - 6} \end{aligned}$
::y=x - 13x2 - 6

We cannot divide the two polynomials. There is a horizontal asymptote at $\begin{aligned} y = 0 \end{aligned}$ .
::我们不能将两个多边形分隔开来。在 y=0 时, 存在着横向的空位。

Example 3
::例3

$\begin{aligned} y = \frac{x^{4} + 1}{x - 5} \end{aligned}$
::y=x4+1x-5

The power of the numerator is 3 more than the power of the denominator. There are no horizontal or oblique asymptotes.
::分子的功率比分母的功率高3倍。没有水平或斜度的单位数。

Example 4
::例4

$\begin{aligned} y = \frac{x^{3} - 3 x^{2} + 4 x - 1}{x^{2} - 2} \end{aligned}$
::y=x3- 3x2+4x- 1x2-2

When we simplify the function, we get $\begin{aligned} y = x - 3 + \frac{6 x - 7}{x^{2} - 2} \end{aligned}$ . There is an oblique asymptote at $\begin{aligned} y = x - 3 \end{aligned}$ .
::当我们简化函数时, 我们得到 y=x-3+6x-7x2-2 。在 y=x-3 上有一个斜线的单位。

Review
::回顾

Find all asymptotes of the following rational functions:
::查找以下理性函数的所有小数 :

$\begin{aligned} \frac{x^{2}}{x - 2} \end{aligned}$
::x2x-2

$\begin{aligned} \frac{1}{x + 4} \end{aligned}$
::1x+4 1x+4

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

$\begin{aligned} \frac{x - 4}{x^{2} - 9} \end{aligned}$
::x-4x2-9

$\begin{aligned} \frac{x^{2} + 2 x + 1}{4 x - 1} \end{aligned}$
::x2+2x+14x-1

$\begin{aligned} \frac{x^{3} + 1}{4 x - 1} \end{aligned}$
::x3+14x- 1

$\begin{aligned} \frac{x - x^{3}}{x^{2} - 6 x - 7} \end{aligned}$
::x- x3x2 - 6x-7

$\begin{aligned} \frac{x^{4} - 2 x}{8 x + 24} \end{aligned}$
::x4-2x8x+24

Graph the following rational functions. Indicate all asymptotes on the graph:
::绘制以下合理函数的图。在图形中显示所有小数点 :

$\begin{aligned} \frac{x^{2}}{x + 2} \end{aligned}$
::x2x+2

$\begin{aligned} \frac{x^{3} - 1}{x^{2} - 4} \end{aligned}$
::x3 - 1x2 - 4

$\begin{aligned} \frac{x^{2} + 1}{2 x - 4} \end{aligned}$
::x2+12x- 4

$\begin{aligned} \frac{x - x^{2}}{3 x + 2} \end{aligned}$
::x- xx23x+2

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

Determining Asymptotes by Division ::按司划分确定小数点

Finding Asymptotes ::查找无符号( A)

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Determining Asymptotes by Division
::按司划分确定小数点

Finding Asymptotes
::查找无符号( A)

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)