Course: 12.3 水平和垂直单数

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水平和垂直单数

Horizontal and Vertical Asymptotes
::水平和垂直单数

We said that a horizontal asymptote is the value of $\begin{aligned} y \end{aligned}$ that the function approaches for large values of $\begin{aligned} | x | \end{aligned}$ . When we plug in large values of $\begin{aligned} x \end{aligned}$ in our function, higher powers of $\begin{aligned} x \end{aligned}$ get larger much quickly than lower powers of $\begin{aligned} x \end{aligned}$ . For example, consider:
::我们曾说过,水平的零点是 y 的值, y 的函数对 x 的大型值接近。当我们在函数中插入 x 的大值时, x 的较高功率比 x 的低功率大得多。例如, 考虑 :

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

::y= 2x2+x- 13x2- 4x+3

If we plug in a large value of $\begin{aligned} x \end{aligned}$ , say $\begin{aligned} x = 100 \end{aligned}$ , we get:
::如果我们插入大值的 x, 比如 x=100, 我们得到:

\begin{aligned} y = \frac{2 (100)^{2} + (100) - 1}{3 (100)^{2} - 4 (100) + 3} = \frac{20000 + 100 - 1}{30000 - 400 + 2} \end{aligned}

::y=2,1002+(100)-13(100)-2-4(100)+3=20000+100-130000-400+2

We can see that the beginning terms in the numerator and denominator are much bigger than the other terms in each expression . One way to find the horizontal asymptote of a rational function is to ignore all terms in the numerator and denominator except for the highest powers.
::我们可以看到,分子和分母中的起始词比每个表达式中的其他词要大得多。找到理性函数水平零点的一个方法就是忽略分子和分母中的所有词,但最高权力者除外。

In this example the horizontal asymptote is $\begin{aligned} y = \frac{2 x^{2}}{3 x^{2}} \end{aligned}$ , which simplifies to $\begin{aligned} y = \frac{2}{3} \end{aligned}$ .
::在此示例中, 水平单数为 y= 2x23x2, 简化为 y= 23 。

In the function above, the highest power of $\begin{aligned} x \end{aligned}$ was the same in the numerator as in the denominator. Now consider a function where the power in the numerator is less than the power in the denominator:
::在以上函数中, X 的最大功率在分子中与分母中相同。现在考虑一个函数,当分子中的力量小于分母中的力量时:

\begin{aligned} y = \frac{x}{x^{2} + 3} \end{aligned}

::y=x22+3 y=xx2+3

As before, we ignore all the terms except the highest power of $\begin{aligned} x \end{aligned}$ in the numerator and the denominator. That gives us $\begin{aligned} y = \frac{x}{x^{2}} \end{aligned}$ , which simplifies to $\begin{aligned} y = \frac{1}{x} \end{aligned}$ .
::和以前一样,我们忽略了所有术语,除了分子和分母中的 x 的最大功率之外。这使得我们有了y=x22, 它简化为y=1x。

For large values of $\begin{aligned} x \end{aligned}$ , the value of $\begin{aligned} y \end{aligned}$ gets closer and closer to zero. Therefore the horizontal asymptote is $\begin{aligned} y = 0 \end{aligned}$ .
::对于 x 的大数值, y 的值越来越接近零。因此, 水平的单数是 y=0 。

To summarize:
::总结如下:

Find vertical asymptotes by setting the denominator equal to zero and solving for $\begin{aligned} x \end{aligned}$ .
::通过设置等于零的分母并解析 x 的分母来查找垂直的单位数。

For , we must consider several cases:
- If the highest power of $\begin{aligned} x \end{aligned}$ in the numerator is less than the highest power of $\begin{aligned} x \end{aligned}$ in the denominator, then the horizontal asymptote is at $\begin{aligned} y = 0 \end{aligned}$ .
  ::如果分子中 x 的最大功率小于分母中 x 的最高功率,则水平静态为 y=0 。
- If the highest power of $\begin{aligned} x \end{aligned}$ in the numerator is the same as the highest power of $\begin{aligned} x \end{aligned}$ in the denominator, then the horizontal asymptote is at $\begin{aligned} y = \frac{coefficient of highest power of x}{coefficient of highest power of x} \end{aligned}$ .
  ::如果分子中x的最高功率与分母中x的最高功率相同,那么水平的静态与x最高功率的x系数最高功率的y=系数一致。
- If the highest power of $\begin{aligned} x \end{aligned}$ in the numerator is greater than the highest power of $\begin{aligned} x \end{aligned}$ in the denominator, then we don’t have a horizontal asymptote; we could have what is called an oblique (slant) asymptote, or no asymptote at all.
  ::如果分子中x的最大功率大于分母中x的最大功率,那么我们就不会有一个水平的无线点;我们可以有所谓的斜线(斜线)无线点,或者根本没有无线点。
::对于几种情况,我们必须考虑:如果分子中x的最高功率低于分母中x的最高功率,那么水平性静态在y=0上。如果分子中x的最高功率与分母中x的最高功率相同,那么水平性静态在x最高功率x最高功率的y=系数上。如果分子中x的最高功率大于分母中x的最高功率,那么我们没有水平性静态;我们可以有所谓的斜线(静态)静态,或者根本没有静态。

Finding Asymptotes
::查找无符号( A)

1. Find the vertical and horizontal asymptotes for $\begin{aligned} y = \frac{1}{x - 1} \end{aligned}$ .
::1. 查找y=1x-1的垂直和水平同源物。

Vertical asymptotes:
::垂直空位数 :

Set the denominator equal to zero. $\begin{aligned} x - 1 = 0 \Rightarrow x = 1 \end{aligned}$ is the vertical asymptote .
::将分母设为 0。 x- 1=0\\\\\ x=1 是垂直静态。

Horizontal asymptote:
::水平单点 :

Keep only the highest powers of $\begin{aligned} x \end{aligned}$ . $\begin{aligned} y = \frac{1}{x} \Rightarrow y = 0 \end{aligned}$ is the horizontal asymptote.
::只保留 x. y=1xy=0 的最高功率是水平单数。

2. Find the vertical and horizontal asymptotes for $\begin{aligned} y = \frac{3 x}{4 x + 2} \end{aligned}$ .
::2. 为 y= 3x4x+2 查找垂直和水平的静态和静态。

Vertical asymptotes:
::垂直空位数 :

Set the denominator equal to zero. $\begin{aligned} 4 x + 2 = 0 \Rightarrow x = - \frac{1}{2} \end{aligned}$ is the vertical asymptote.
::将分母设为 0。 4x+2=0\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\可以垂直的静态。

Horizontal asymptote:
::水平单点 :

Keep only the highest powers of $\begin{aligned} x \end{aligned}$ . $\begin{aligned} y = \frac{3 x}{4 x} \Rightarrow y = \frac{3}{4} \end{aligned}$ is the horizontal asymptote.
::只保留 x. y=3x4xy=34 的最高功率是水平静态。

3. Find the vertical and horizontal asymptotes for $\begin{aligned} y = \frac{x^{3}}{x^{2} - 3 x + 2} \end{aligned}$ .
::3. 查找y=x3x2-3x+2的垂直和水平同位数。

Vertical asymptotes:
::垂直空位数 :

Set the denominator equal to zero: $\begin{aligned} x^{2} - 3 x + 2 = 0 \end{aligned}$
::设置等于零的分母: x2 - 3x+2=0

Factor : $\begin{aligned} (x - 2) (x - 1) = 0 \end{aligned}$
::系数x-2)(x-1)=0

Solve: $\begin{aligned} x = 2 \end{aligned}$ and $\begin{aligned} x = 1 \end{aligned}$ are the vertical asymptotes.
::溶解: x=2 和 x=1 是垂直静态。

Horizontal asymptote. There is no horizontal asymptote because the power of the numerator is larger than the power of the denominator.
::水平渐变。没有水平渐变, 因为分子的功率大于分母的功率。

Notice the function in part d had more than one vertical asymptote. Here’s another function with two vertical asymptotes.
::注意 d 部分的函数有一个以上的垂直静态。这里有另一个函数,有两个垂直静态。

Graphing Functions
::图图函数

Graph the function $\begin{aligned} y = \frac{- x^{2}}{x^{2} - 4} \end{aligned}$ .
::函数 yx2x2 - 4 图形。

\begin{aligned} Set the denominator equal to zero: x^{2} - 4 = 0 \\ Factor: (x - 2) (x + 2) = 0 \\ Solve: x = 2, x = - 2 \end{aligned}

::设置分母等于 0: x2 - 4= 0 要素 : (x- 2) (x+2) = 0Solve: x=2, x% 2

We find that the function is undefined for $\begin{aligned} x = 2 \end{aligned}$ and $\begin{aligned} x = - 2 \end{aligned}$ , so we know that there are vertical asymptotes at these values of $\begin{aligned} x \end{aligned}$ .
::我们发现该函数未定义 x=2andx% 2 的值, 所以我们知道这些 x 的值是垂直的 asymptotes 。

We can also find the horizontal asymptote by the method we outlined above. It’s at $\begin{aligned} y = \frac{- x^{2}}{x^{2}} \end{aligned}$ , or $\begin{aligned} y = - 1 \end{aligned}$ .
::我们也可以通过上述方法找到横向的零点。在yx2x2, 或y1。

So, we start plotting the function by drawing the vertical and horizontal asymptotes on the graph.
::因此,我们开始通过在图形上绘制垂直和水平的零点来绘制函数。

Now, let’s make a table of values. Because our function has a lot of detail we must make sure that we pick enough values for our table to determine the behavior of the function accurately. We must make sure especially that we pick values close to the vertical asymptotes.
::现在,让我们来绘制一个数值表。因为我们的函数有很多细节,我们必须确保我们为表格选择足够的数值来准确确定函数的行为。我们必须特别确保我们选择接近垂直微粒的值。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = \frac{- x^{2}}{x^{2} - 4} \end{aligned}$
$\begin{aligned} - 5 \end{aligned}$	$\begin{aligned} y = \frac{- (- 5)^{2}}{(- 5)^{2} - 4} = \frac{- 25}{21} = - 1.19 \end{aligned}$
-4	$\begin{aligned} y = \frac{- (- 4)^{2}}{(- 4)^{2} - 4} = \frac{- 16}{12} = - 1.33 \end{aligned}$
-3	$\begin{aligned} y = \frac{- (- 3)^{2}}{(- 3)^{2} - 4} = \frac{- 9}{5} = - 1.8 \end{aligned}$
-2.5	$\begin{aligned} y = \frac{- (- 2.5)^{2}}{(- 2.5)^{2} - 4} = \frac{- 6.25}{2.25} = - 2.8 \end{aligned}$
-1.5	$\begin{aligned} y = \frac{- (- 1.5)^{2}}{(- 1.5)^{2} - 4} = \frac{- 2.25}{- 1.75} = 1.3 \end{aligned}$
-1	$\begin{aligned} y = \frac{- (- 1)^{2}}{(- 1)^{2} - 4} = \frac{- 1}{- 3} = 0.33 \end{aligned}$
0	$\begin{aligned} y = \frac{- 0^{2}}{(0)^{2} - 4} = \frac{0}{- 4} = 0 \end{aligned}$
1	$\begin{aligned} y = \frac{- 1^{2}}{(1)^{2} - 4} = \frac{- 1}{- 3} = 0.33 \end{aligned}$
1.5	$\begin{aligned} y = \frac{- {1.5}^{2}}{(1.5)^{2} - 4} = \frac{- 2.25}{- 1.75} = 1.3 \end{aligned}$
2.5	$\begin{aligned} y = \frac{- {2.5}^{2}}{(2.5)^{2} - 4} = \frac{- 6.25}{2.25} = - 2.8 \end{aligned}$
3	$\begin{aligned} y = \frac{- 3^{2}}{(3)^{2} - 4} = \frac{- 9}{5} = - 1.8 \end{aligned}$
4	$\begin{aligned} y = \frac{- 4^{2}}{(4)^{2} - 4} = \frac{- 16}{12} = - 1.33 \end{aligned}$
5	$\begin{aligned} y = \frac{- 5^{2}}{(5)^{2} - 4} = \frac{- 25}{21} = - 1.19 \end{aligned}$

Here is the resulting graph.
::以下是由此得出的图表。

Example
::示例示例示例示例

Example 1
::例1

Find the vertical and horizontal asymptotes for $\begin{aligned} y = \frac{x^{2} - 2}{2 x^{2} + 3} \end{aligned}$ .
::查找 y = x2- 22x2+3 的垂直和水平同位数。

Vertical asymptotes:
::垂直空位数 :

Set the denominator equal to zero: $\begin{aligned} 2 x^{2} + 3 = 0 \Rightarrow 2 x^{2} = - 3 \Rightarrow x^{2} = - \frac{3}{2} \end{aligned}$ . Since there are no solutions to this equation , there is no vertical asymptote.
::设置分母为零 : 2x2+3=0\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\没有此方程式的解决方案, 不存在垂直等同。

Horizontal asymptote:
::水平单点 :

Keep only the highest powers of $\begin{aligned} x \end{aligned}$ . $\begin{aligned} y = \frac{x^{2}}{2 x^{2}} \Rightarrow y = \frac{1}{2} \end{aligned}$ is the horizontal asymptote.
::只保留 x. y=x22x2y=12 的最高功率是水平的单数。

Review
::回顾

Find all the vertical and horizontal asymptotes of the following rational functions.
::查找以下理性函数的所有垂直和水平静态。

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

$\begin{aligned} y = \frac{5 x - 1}{2 x - 6} \end{aligned}$
::y=5x- 12x-6

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

$\begin{aligned} y = \frac{2}{x} - 5 \end{aligned}$
::y=2x-5

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

$\begin{aligned} y = \frac{4 x^{2}}{4 x^{2} + 1} \end{aligned}$
::y= 4x24x2+1

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

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

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

$\begin{aligned} y = \frac{2 x + 5}{x^{2} - 2 x - 8} \end{aligned}$
::y= 2x+5x2-2x-8

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

水平和垂直单数

Horizontal and Vertical Asymptotes ::水平和垂直单数

Finding Asymptotes ::查找无符号( A)

Graphing Functions ::图图函数

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Horizontal and Vertical Asymptotes
::水平和垂直单数

Finding Asymptotes
::查找无符号( A)

Graphing Functions
::图图函数

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)