2. 12 理性函数图的签名测试

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  合理函数图的签名测试

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  The asymptotes of a rational function provide a very rigid structure in which the function must live.  Once the asymptotes are known you must use the sign testing procedure to see if the function becomes increasingly positive or increasingly negative near the asymptotes.  A driving question then becomes how close does near need to be in order for the sign test to work? 
  ::理性函数的微粒提供了一个非常僵硬的结构, 函数必须在其中运行。 一旦知道这些微粒, 您就必须使用符号测试程序来检查该函数是否在小粒子附近变得日益正或越来越负。 然后一个驱动问题就变成了要进行标志测试需要多久才能接近?

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  Sign Test for Rational Functions
  ::合理函数的签名测试

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  Consider mentally substituting the number 2.99999 into the following rational expression . 
  ::考虑在精神上用以下合理表达方式取代2.99999。

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  $$\begin{array}{r}f(x)=\frac{(x-1)(x+3)(x-5)(x+10)}{(x+2)(x-4)(x-3)}\end{array}$$

  
  :xx) = (x- 1) (x+3) (x-5) (x-5) (x+10) (x+2) (x-4) (x-3)

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  Without doing any of the arithmetic, simply note the sign of each term :
  ::在不做任何算术的情况下,只需指出每个术语的符号:

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  $$\begin{array}{r}f(x)=\frac{(+)\cdot (+)\cdot (-)\cdot (+)}{(+)\cdot (-)\cdot (-)}\end{array}$$

  
  :x) = (+) = (+) = (+) = (+) = (-) = (+) = (+) = (+) = (+) = (+) + (+) + (+) + (+) + (- (-)

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  The only term where the value is close to zero is $\begin{array}{r}(x-3)\end{array}$  but careful subtraction still indicates a negative sign.  The product of all of these signs is negative.  This is strong evidence that this function approaches negative infinity as $\begin{array}{r}x\end{array}$  approaches 3 from the left. 
  ::值接近零的唯一术语是 (x-3) , 但仔细减法仍然表示一个负符号。 所有这些符号的产物都是负符号。 这有力地证明这个函数接近负无限, 如左侧x 3 接近x 3 一样 。

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  Next consider mentally substituting 3.00001 and going through the same process. 
  ::接下来考虑在精神上取代3.0001, 并经历同样的过程。

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  $$\begin{array}{r}f(x)=\frac{(+)\cdot (+)\cdot (-)\cdot (+)}{(+)\cdot (-)\cdot (+)}\end{array}$$

  
  :x) = (+) = (+) = (+) +) = (-) = (+) = (+) = (+) = (+) = (+) + (+) + (+) + (-) + (-) + (+)

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  The product of all of these signs is positive which means that from the right this function approaches positive infinity instead. This technique is called the sign test. The sign test is a procedure for determining only whether a function is above or below the axis at a particular value.
  ::所有这些符号的产物都是正的, 这意味着从右侧这个函数向正的无限方向走。 这个技术被称为符号测试。 符号测试是一种程序, 仅用于确定一个函数是否高于或低于轴, 其值是特定值 。

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  The sign test helps you sketch and graph a function. Look at the following function:
  ::符号测试有助于您绘制图像和绘制函数图。请查看以下函数:

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  $$\begin{array}{r}f(x)=\frac{1}{(x+2{)}^2\cdot (x-1)}\end{array}$$

  
  ::f(xx) = 1(x+2) 2(x- 1)

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  Your first step for sketching this is to identify the vertical asymptotes. The vertical asymptotes occur at $\begin{array}{r}x=-2\end{array}$  and $\begin{array}{r}x=1\end{array}$ .  Then, you use the asymptotes to perform a sign test. The points to use the sign testing procedure with are -2.001, -1.9999, 0.9999, 1.00001.  The number of decimals does not matter so long as the number is sufficiently close to the asymptote.  Note that any real number squared is positive. 
  ::您绘制草图的第一步是识别垂直小数。 垂直小数出现在 x2 和 x=1 。 然后, 您使用小数来进行符号测试。 使用符号测试程序的点数是 - 2. 001, - 1. 999, 0. 999, 0. 00001。 只要小数与小数足够接近, 小数数数并不重要。 请注意, 任何实际数字正方形 。

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  $$\begin{array}{rl}f(-2.001)& =\frac{(+)}{(+)\cdot (-)}=-\\ f(-1.9999)& =\frac{(+)}{(+)\cdot (-)}=-\\ f(0.9999)& =\frac{(+)}{(+)\cdot (-)}=-\\ f(1.0001)& =\frac{(+)}{(+)\cdot (+)}=+\end{array}$$

  
  ::f( - 2. 001) = (+)(+) +(-) (-) (-) 1.999) = (+)(+) +(-) (-) f( 0. 9999) = (+) +(+) +(-) (-) f(1.000) = (+)(+) +(+) +(+) (+) +(-) (-) (-)

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  Later when you sketch everything you will use your knowledge of zeroes and intercepts .  For now, focus on just the portions of the graph near the asymptotes.   Note that the graph below is NOT complete.
  ::稍后, 当您绘制所有您将使用对零和拦截知识的图像时。 目前, 请只关注小行星附近的图形部分。 注意下面的图形不完整 。

  

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked how close the numbers need to be to perform the sign test. In order to truly answer the question about how close the numbers need to be, calculus should be used.  For the purposes of PreCalculus, the testing number should be closer to the vertical asymptote than any other number in the problem.  If the vertical asymptote occurs at 3 and 3.01 is in the problem elsewhere, do not choose 3.1 as a sign test number. 
  ::早些时候, 有人询问了要进行符号测试需要多少数字才能进行标记测试。 为了真正回答关于数字需要多近的问题, 应该使用微积分。 为了预显计算的目的, 测试数字应该比问题中任何其他数字更接近垂直的静态。 如果在3点和3. 01点发生垂直的静态, 在别处的问题是问题, 不要选择3.1作为符号测试数字 。

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  Example 2
  ::例2

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  Identify the vertical asymptotes and use the sign testing procedure to roughly sketch the nature of the function near the vertical asymptotes. 
  ::识别垂直静脉喷发,并使用标志测试程序大致勾画垂直静脉喷发附近函数的性质。

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  $$\begin{array}{r}f(x)=\frac{(x+1)(x-4{)}^2(x-1)(x+3{)}^3}{100(x-1{)}^2(x+2)}\end{array}$$

  
  ::f(x) = (x+1) (x) = (x) = (x+1) = (x) = (x) = (x) = (x) = 4) 2(x-1 (x) = (x) +1)(x) +3) 3100 (x- 1) 2(x) 2(x+2)

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  Note that $\begin{array}{r}x=-2\end{array}$  is clearly an asymptote.  It may be initially unclear whether $\begin{array}{r}x=1\end{array}$  is an asymptote or a hole .  Just like holes have priority over zeroes, asymptotes have priority over holes.  The four values to use the sign testing procedure are -2.001, -1.9999, 0.9999, 1.00001.
  ::注意 x2 明显是一个小洞。 最初可能不清楚 x=1 是小洞还是洞。 就像洞比零优先一样, 小洞比洞优先。 使用符号测试程序的四种值是 - 2. 001, - 1. 999, 0. 999, 0. 999, 1. 0001 。

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  $$\begin{array}{rl}f(-2.001)& =\frac{(-)\cdot (+)\cdot (-)\cdot (+)}{(+)\cdot (-)}=-\\ f(-1.9999)& =\frac{(-)\cdot (+)\cdot (-)\cdot (+)}{(+)\cdot (+)}=+\\ f(0.9999)& =\frac{(+)\cdot (+)\cdot (-)\cdot (+)}{(+)\cdot (+)}=-\\ f(1.0001)& =\frac{(+)\cdot (+)\cdot (+)\cdot (+)}{(+)\cdot (+)}=+\end{array}$$

  
  ::f(-)_(-)_(+)_(+)_(+)_(+)_(+)_(+)_(-)_(f)(-)_(-)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)(+)_(+)_(+)_(+)_(+)(+)_(+)(+)_(+)_(+)_(+)(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_(+)_)_(+(+)_(+)_(+)_(+)_(+)_)_)_(+++++++(+++++++++++++++++++++++++_)_)_)_

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  A sketch of the behavior of this function near the asymptotes is:
  ::此函数在小行星附近的行为草图如下:

  

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  Example 3
  ::例3

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  Create a function with two vertical asymptotes at 3 and -2 such that the function approaches positive infinity from both directions at both vertical asymptotes. 
  ::在 3 和 - 2 时创建两个垂直小数的函数, 使该函数在两个垂直小数和 - 2 时都从两个方向接近正无穷度 。

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  Earlier, there was a function that approached negative infinity from both sides of the asymptote.  This occurred because the term was squared in the denominator. An even powered term will always produce a positive term. 
  ::早些时候,有一个功能接近了小行星两侧的负无限性。 之所以出现这个功能,是因为该词在分母中正方形。 一个甚至有权势的词总会产生一个正面的词。

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  $$\begin{array}{r}f(x)=\frac{1}{(x-3{)}^2(x+2{)}^2}\end{array}$$

  
  :xx)=1(x-3)2(x+2)2

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  Example 4
  ::例4

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  Create a function with three vertical asymptotes such that the function approaches negative infinity for large and small values of  $\begin{array}{r}x\end{array}$ and has an .
  ::创建函数, 3个垂直小数点, 使函数对 x 的大小值接近负无穷, 并有一个 。

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  There are an infinite number of possible solutions.  The key is to create a function that may work and then use the sign testing procedure to check.  Here is one possibility. 
  ::有很多可能的解决方案。 关键是创建一个可能有效的函数, 然后使用符号测试程序来检查。 这里有一个可能性 。

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  $$\begin{array}{r}f(x)=\frac{-x^7}{10(x-1{)}^2(x-2{)}^2(x-4{)}^2}\end{array}$$

  
  :xx)x710(x-1)2(x-2)(x-4)2

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  Example 5
  ::例5

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  Identify the vertical asymptotes and use the sign testing procedure to roughly sketch the nature of the function near the vertical asymptotes.
  ::识别垂直静脉喷发,并使用标志测试程序大致勾画垂直静脉喷发附近函数的性质。

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  $$\begin{array}{r}f(x)=\frac{(x-2{)}^3(x-1{)}^2(x+1)(x+3)}{x^3\left(x+\frac{1}{2}\right)(x-1)(x-2{)}^2}\end{array}$$

  
  ::f(x) = (x- 2) 3(x- 1) 2(x+1)(x+3) x3(x+12)(x-1)(x-2) 2

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  The vertical asymptotes occur at $\begin{array}{r}x=0,-\frac{1}{2}\end{array}$ .  Therefore the  $\begin{array}{r}x\end{array}$ values to sign test are -.001, 0.001, 3.999, 4.0001.
  ::垂直微粒在 x=0 = -12 时发生。 因此,要签名测试的x值是 - 001, 0.001, 3.999, 4.0001 。

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  $$\begin{array}{rl}f(-0.001)& =+\\ f(0.001)& =-\\ f(3.999)& =-\\ f(4.0001)& =+\end{array}$$

  
  ::f(- 0.0001)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ -\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

  Summary

  播放段落 The sign test is a procedure for determining whether a function is above or below the axis at a particular value. ::符号测试是一种程序,用以确定一个函数是否在轴上或下,是否有特定值。 播放段落 To perform the sign test, mentally substitute a number very close to the asymptote and note the sign of each term in the rational expression. ::为了进行标志测试, 精神上可以替换非常接近于无位数的数字, 并注意理性表达中每个词的符号 。 播放段落 The sign test is useful for sketching and graphing a function, as it helps to determine the behavior of the function near the asymptotes. ::符号测试有助于绘制和绘制函数图示,因为它有助于确定该函数在小行星附近的行为。

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  Review
  ::回顾

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  Consider the function below for questions 1-4.
  ::为问题1-4考虑以下职能。

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  $$\begin{array}{r}f(x)=\frac{(x-2{)}^4(x+1)(x+3)}{x^3(x+3)(x-4)}\end{array}$$

  
  ::f(x) = (x- 2) 4(x+1)(x+3) x3(x+3)(x- 4)

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  1.  Identify the vertical asymptotes.
  ::1. 确定垂直零点数。

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  2.  Will this function have an oblique asymptote?  A horizontal asymptote?  If so, where?
  ::2. 此函数是否具有倾斜的无线状态? 水平的无线状态? 如果有,在哪里?

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  3.  What values will you need to use the sign test with in order to help you make a sketch of the graph?
  ::3. 您需要使用什么值来使用符号测试来帮助您绘制图表草图?

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  4.  Use the sign test and sketch the graph near the vertical asymptotes.
  ::4. 使用标志测试,在垂直微粒附近绘制图示。

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  Consider the function below for questions 5-8.
  ::审议以下关于问题5-8的职能。

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  $$\begin{array}{r}g(x)=\frac{3(x-2{)}^2(x-1{)}^2(x+1)(x+3)}{15x^2(x+5)(x+1)(x-3{)}^2}\end{array}$$

  
  ::g(x) = 3(x- 2) 2(x- 1) 2(x+1)(x+3) 15x2(x+5)(x+1)(x-3) 2

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  5.  Identify the vertical asymptotes.
  ::5. 确定垂直零点数。

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  6.  Will this function have an oblique asymptote?  A horizontal asymptote?  If so, where?
  ::6. 此函数是否具有倾斜的无线状态? 水平的无线状态? 如果有,在哪里?

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  7.  What values will you need to use the sign test with in order to help you make a sketch of the graph?
  ::7. 要帮助您绘制图表草图,您需要使用什么值来进行符号测试?

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  8.  Use the sign test and sketch the graph near the vertical asymptote(s).
  ::8. 使用标志测试,在垂直静态附近绘制图形草图。

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  Consider the function below for questions 9-12.
  ::审议下面的问题9-12的功能。

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  $$\begin{array}{r}h(x)=\frac{9x^4-102x^3+349x^2-340x+100}{x^3-9x^2+24x-16}\end{array}$$

  
  ::h(x) = 9x4 - 102x3+ 349x2-340x+ 100x3 - 9x2+24x-16

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  9.  Identify the vertical asymptotes.
  ::9. 确定垂直零点数。

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  10.  Will this function have an oblique asymptote?  A horizontal asymptote?  If so, where?
  ::10. 此函数是否具有倾斜的无症状? 水平的无症状? 如果是,在哪里?

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  11.  What values will you need to use the sign test with in order to help you make a sketch of the graph?
  ::11. 您需要使用什么值来使用符号测试来帮助您绘制图表草图?

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  12.  Use the sign test and sketch the graph near the vertical asymptotes.
  ::12. 使用标志测试,在垂直小行星附近绘制图示。

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  Consider the function below for questions 13-16.
  ::审议下面关于问题13-16的职能。

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  $$\begin{array}{r}k(x)=\frac{2x^3-5x^2-11x-4}{3x^3+11x^2+5x-3}\end{array}$$

  
  :x) = 2x3 - 5x2 - 11x - 43x3+11x2+5x3

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  13.  Identify the vertical asymptotes.
  ::13. 确定垂直静态。

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  14.  Will this function have an oblique asymptote?  A horizontal asymptote?  If so, where?
  ::14. 此函数是否具有倾斜的零星状态? 水平的零星状态? 如果有,在哪里?

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  15.  What values will you need to use the sign test with in order to help you make a sketch of the graph?
  ::15. 您需要用什么值来使用符号测试来帮助您绘制图表草图?

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  16.  Use the sign test and sketch the graph near the vertical asymptotes.
  ::16. 使用标志测试,在垂直微粒附近绘制图示。

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  Review (Answers)
  ::回顾(答复)

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  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 。