2.9 数据定义的转换函数

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  由数据定义的转换函数

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  Introduction
  ::导言

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  Basic function families are useful guidelines to analyze mathematical models. However, some models can only be defined either graphically or with data points. Transformation techniques can still be applied to such models.  
  ::基本功能家庭是分析数学模型的有用指南,但有些模型只能用图形或数据点来定义。

  

   

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  Consider the following graph, which represents the distance that a delivery truck is  from a warehouse during one 8-hour trip. This will be used to define the function $\begin{align*}D\left(t\right)\end{align*}$ .
  ::考虑下图,该图表示运货卡车在一次8小时的旅程中从仓库到仓库的距离。这将用来定义函数D(t) 。

  

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  How does this function change if the driver completes the trip twice as fast , but only travels  $\begin{align*}\frac{1}{3}\end{align*}$ the distance away from the warehouse?  Using , the transformation is represented as  $\begin{align*}D(t) \to \frac{1}{3}D(2t) \end{align*}$ . From the previous section, we know this represents a horizontal shrink by a factor of $\begin{align*}2\end{align*}$ , and a vertical shrink by a factor of  $\begin{align*}3\end{align*}$ . The clearest way to represent the transformation is by moving points easily read from the 1st graph.
  ::如果驱动程序完成速度是行程的两倍,但只有距离仓库的13个距离才能到达。 使用 , 转换代表为 D( t)\\\\\\\\\\\\13D( 2t) 。 从上一节中, 我们知道这代表了水平缩进2倍, 垂直缩进3倍。 代表转换的最清楚的方式是移动点, 从第一个图中很容易读取 。

  Transformation Table

   

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  The transformed graph of the trip is:
  ::旅行的变形图是:

  

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  The changes required to create a new distance function are clearly satisfied. The trip took half the time because the driver completed the trip twice as fast, and the distance traveled away from the warehouse is  $\begin{align*}\frac{1}{3}\end{align*}$  the original distance.
  ::创建新距离功能所需的更改显然已经满足。 行程花了一半时间, 因为司机完成行程的速度是司机的两倍, 距离仓库的距离是13, 距离是最初的距离 。

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  Point Notation for Function Transformation
  ::功能转换的标记符号

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  A transformation can be written using function notation or point notation , which simply illustrates the  transformation of each key point. This method works well with tabular data  or a well-labeled graph. The process assigns a pairing between $\begin{align*}(x, y)\end{align*}$   and new coordinates based on the transformation. The form for  point notation is:
  ::转换可以使用函数符号或点符号来写,这些符号仅说明每个关键点的转换。该方法与表格数据或标签良好的图形运作良好。该过程在(x,y)和基于转换的新坐标之间指定了配对。点符号的窗体是:

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  $$\begin{align*}(x, y) \rightarrow (new \ x, new \ y)\end{align*}$$
  :x,y)(新x,newy)

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  Note that in function notation,  $\begin{align*}y=f(x)\end{align*}$ . The new  $\begin{align*}x\end{align*}$ - coordinate is the value you get when you solve the transformation  using the old  $\begin{align*}x\end{align*}$  input value. 
  ::请注意,在函数符号 y=f(x) 中。新的x坐标是使用旧的 x 输入值解决转换时得到的值。

    

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  Examples
  ::实例

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  Example 1
  ::例1

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  Write the following transformation using function notation and point notation. Then apply the transformation to the 3 points in the table. Transformation: Horizontal shift right 3 units, vertical shift up 4 units.
  ::使用函数符号和点符号写入以下转换。 然后将转换应用到表格中的 3 点。 转换: 水平向右移动 3 个单位, 垂直向上移动 4 个单位 。

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  $\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline 1 & 3 \\\hline 2 & 5 \\\hline 8 & \text{-}11 \\\hline \end{array}\end{align*}$
  ::Xy13258-11

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  Solution:
  ::解决方案 :

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  $\begin{align*}f(x) \rightarrow f(x-3)+4\end{align*}$
  :xx)f(x-3)+4

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  $\begin{align*}(x,y) \rightarrow (x+3,y+4)\end{align*}$
  :x,y)(x+3,y+4)

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  $\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline 1 & 3 \\\hline 2 & 5 \\\hline 8 & \text{-}11 \\\hline \end{array}\\\end{align*}$ $\begin{align*}\longrightarrow\end{align*}$    $\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline 4 & 7 \\\hline 5 & 9 \\\hline 11 & \text{-}7 \\\hline \end{array}\end{align*}$
  ::xy13258-11 xy475911-7

   

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  Example 2
  ::例2

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  Convert the following function notation into point notation:
  ::将下列函数符号转换为点符号:

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  $\begin{align*}f(x) \rightarrow \frac{1}{4} f\left(-(x+3)\right)-1\end{align*}$
  :xx)14f(-(x+3))-1

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  $\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline 0 & 0 \\\hline 1 & 4 \\\hline 2 & 8 \\\hline \end{array}\end{align*}$
  ::xy001428 xy001428

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  Solution:  
  ::解决方案 :

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  Point notation:  
  ::点符号 :

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    $\begin{align*}(x, y) \rightarrow \left ( - x-3,\frac{1}{4} y-1 \right )\end{align*}$
  :x,y)(-x-3,14y-1)

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  $\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline 0 & 0 \\\hline 1 & 4 \\\hline 2 & 8 \\\hline \end{array}\end{align*}$   $\begin{align*}\longrightarrow\end{align*}$   $\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline \text{-}3 & \text{-}1 \\\hline \text{-}4 & 0 \\\hline \text{-}5 & 1 \\\hline \end{array}\end{align*}$
  ::xy001428 xy-3-1-40-51

   

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  Example 3
  ::例3

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  Convert the following point notation to words and to function notation. Then apply the transformation to the tabular data:  
  ::将以下点符号转换为单词和函数符号。然后对表格数据应用转换:

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  $\begin{align*}(x,y) \rightarrow (2x,-y-1)\end{align*}$
  :x,y)(2x,-y-1)

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  $\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline 10 & 8 \\\hline 12 & 7 \\\hline 14 & 6 \\\hline \end{array}\end{align*}$
  ::xy108127146

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  Solution:
  ::解决方案 :

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  Transformation: Horizontal stretch by a factor of 2 , v ertical reflection across the  $\begin{align*}x\end{align*}$ - axis, and  vertical shift down 1 unit .
  ::转变:水平伸展乘以2, X轴垂直反射, 垂直向下移动 1 个单位 。

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  The point transformation in function notation is:
  ::函数符号中的点转换是:

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  $\begin{align*}f(x)\rightarrow - \left (f \left ( \frac{x}{2} \right )\right )-1\end{align*}$
  :xx)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  $\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline 10 & 8 \\\hline 12 & 7 \\\hline 14 & 6 \\\hline \end{array}\end{align*}$    $\begin{align*}\longrightarrow\end{align*}$   $\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline 20 & \text{-}9 \\\hline 24 & \text{-}8 \\\hline 28 & \text{-}7 \\\hline \end{array}\end{align*}$
  ::xy108127146 xy20-924-828-7

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  Example 4
  ::例4

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  D escribe  the following function notation and rewrite it using point notation. Apply the transformation to 3 points. 
  ::描述以下的函数符号, 并使用点符号重写它。 将转换应用到 3 个点 。

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  $\begin{align*}f(x) \rightarrow - 2f(x-1) + 4\end{align*}$
  :xx)%2f(x-1)+4

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  Solution:
  ::解决方案 :

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  $\begin{align*}f(x) \rightarrow - 2f(x-1) + 4\end{align*}$
  :xx)%2f(x-1)+4

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  Horizontal shift right 1 unit.  Vertical stretch by a factor of 2. Vertical reflection across the  $\begin{align*}x\end{align*}$ axis. Vertical shift 4 units. 
  ::水平向右移动 1 单位。垂直伸展以 2 系数为 2 垂直反射横跨 x 轴。 垂直倾角 4 单位。

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  $\begin{align*}(x,y)\rightarrow(x+1,- 2y+4)\end{align*}$
  :x,y)(x+1,--2y+4)

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  $\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline 0 & 5 \\\hline 1 & 6 \\\hline 2 & 7 \\\hline \end{array}\end{align*}$   $\begin{align*}\longrightarrow\end{align*}$   $\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline 1 & \text{-}6 \\\hline 2 & \text{-}8 \\\hline 3 & \text{-}10 \\\hline \end{array}\end{align*}$
  ::xy051627 xy1-62-83-10

   

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  Example 5
  ::例5

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  Convert the following point notation to function notation.  Describe the transformation:  
  ::将以下点符号转换为函数符号。描述转换:

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  $\begin{align*}(x,y)\rightarrow(3x+1,-y+7)\end{align*}$
  :x,y)(3x+1,-y+7)

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  Solution:
  ::解决方案 :

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  $\begin{align*}(x,y)\rightarrow(3x+1,-y+7)\end{align*}$
  :x,y)(3x+1,-y+7)

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  Horizontal stretch by a factor of 3 and then a horizontal shift right 1 unit. 
  ::水平伸展3乘以乘以3,然后水平向右移动1单位。

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  Vertical reflection over the  $\begin{align*}x\end{align*}$ axis and then a vertical shift 7 units up. 
  ::垂直反射在 x 轴上,然后垂直向上移动 7 个单位。

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  $\begin{align*}f(x)\rightarrow -f \left ( \frac{1}{3}x - \frac{1}{3} \right ) +7\end{align*}$
  :xx)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\7\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  Example 6
  ::例6

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  Convert the following function notation to point notation and describe the changes:
  ::将以下函数符号转换为点符号并描述变化:

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  $\begin{align*}f(x)\rightarrow -\frac{1}{2} f(x-1)+3\end{align*}$
  :xx)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\1+3

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  Solution:  
  ::解决方案 :

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  $\begin{align*}f(x)\rightarrow -\frac{1}{2} f(x-1)+3\end{align*}$
  :xx)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\1+3

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  Vertical reflection across the  $\begin{align*}x\end{align*}$ axis, vertical shrink by a factor of $\begin{align*}\frac{1}{2}\end{align*}$ , and shift up 3. Horizontal shift right 1 unit. 
  ::垂直反射横跨 x 轴, 垂直缩缩12 系数, 向上移动 3. 水平向右移动 1 单位 。

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  $\begin{align*}(x, y) \rightarrow \left ( x+1, -\frac{1}{2}y+3 \right )\end{align*}$
  :x,y)(x+1,-12y+3)

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  Summary
  ::摘要

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  • Transformation techniques can still be applied to such models.  
    ::变换技术仍可适用于此类模型。
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  • Point notation is used to translate key points based on a transformation of a function. It can be used either in place of or with functional notation.  
    ::指针用于根据函数转换来翻译关键点。它可以代替或用功能标记来使用。

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  Review
  ::回顾

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  Convert the following function notation into words and then point notation. Finally, apply the transformation to three example points:
  ::将以下函数符号转换为单词,然后点符号。最后,将转换应用到三个示例点:

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  $\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline 0 & 5 \\\hline 1 & 6 \\\hline 2 & 7 \\\hline \end{array}\end{align*}$
  ::xy051627

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  1. $\begin{align*}f(x) \rightarrow -\frac{1}{2} f(x+1)\end{align*}$
  ::1. f(x) 12f(x+1)

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  2. $\begin{align*}g(x)\rightarrow 2g(3x)+2\end{align*}$
  ::2. g(x)_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  3. $\begin{align*}h(x) \rightarrow - h(x-4)-3\end{align*}$
  ::3. h(x)h(x-4)-3

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  4. $\begin{align*}j(x) \rightarrow 3j(2x-4)+1\end{align*}$
  ::4. j(x)3j(2x-4)+1

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  5. $\begin{align*}k(x)\rightarrow -k(x-3)\end{align*}$
  ::5. k(x)\\\\ k(x- 3)

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  Convert the following functions in point notation to function notation: 
  ::将以下点符号函数转换为函数符号 :

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  6. $\begin{align*}(x,y) \rightarrow \left ( \frac{1}{2} x+3,y-4 \right )\end{align*}$
  ::6. (x,y)(12x+3,y-4)

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  7. $\begin{align*}(x,y) \rightarrow (2x+4,-y+1)\end{align*}$
  ::7.(x,y)(2x+4,-y+1)

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  8. $\begin{align*}(x,y) \rightarrow (4x,3y-5)\end{align*}$
  ::8. (x,y)(4x,3y-5)

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  9. $\begin{align*}(2x,y) \rightarrow (4x, - y+1)\end{align*}$
  ::9. (2x,y)(4x,-y+1)

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  10. $\begin{align*}(x+1,y-2) \rightarrow (3x+3, -y+3)\end{align*}$
  ::10. (x+1,y-2)(3x+3,-y+3)

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  Convert the following functions in function notation to point notation: 
  ::在函数符号中将下列函数转换为点符号 :

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  11. $\begin{align*}f(x) \rightarrow 3f(x-2)+1\end{align*}$
  ::11. f(x)3f(x-2)+1

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  12. $\begin{align*}g(x) \rightarrow -4g(x-1)+3\end{align*}$
  ::12. g(x)4g(x-1)+3

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  13. $\begin{align*}h(x) \rightarrow \frac{1}{2} h(2x+2)-5\end{align*}$
  ::13. h(x)12h(2x+2)-5

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  14. $\begin{align*}j(x) \rightarrow 5j \left ( \frac{1}{2} x-2 \right )-1\end{align*}$
  ::14. j(x)5j(12x-2)-1

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  15. $\begin{align*}k(x) \rightarrow \frac{1}{4} k(2x-4)\end{align*}$
  ::15. k( x) }14k( 2x- 4)

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  Review (Answers)
  ::回顾(答复)

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  Please see the Appendix.
  ::请参看附录。