2.2 功能转换-interactive

章节大纲

• 选择节常规

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  常规

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  • 选择活动新闻通告

    

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• 选择节函数转换

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  函数转换

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  Learning Objectives
  ::学习目标

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  • Identify the effect on the graph of a function after a transformation .
    ::识别转换后函数图图中的影响。
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  • Identify the parent function given the graph or equation of a transformed function.
    ::给已转换函数的图形或方程式指定父函数。
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  • Identify how a function has been translated given the graph or equation.
    ::确定根据图形或方程式,函数是如何被翻译的。
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  • Construct a function to model a linear relationship between two quantities from a description.
    ::构造一个函数以模拟描述中两个数量之间的线性关系。
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  • Identify even and odd functions from their graphs.
    ::从图表中识别偶数和奇数函数 。

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  Introduction: Transforming the World Around Us
  ::导言:改变我们周围的世界

  

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  T he absolute value function  $\begin{align*}f(x) = |x - 78|\end{align*}$   is a basic absolute value function  (you may recognize it from the prior lesson, ). The function $\begin{align*}f(x)=|x-78|\end{align*}$  is a transformation of the parent function  $\begin{align*}f(x) = |x|.\end{align*}$  A parent function is the simplest function that preserves a specific shape. All transformations of a parent function are known as the function family . Function transformations are commonly used to manipulate a parent function to model a real-world object, context, or relationship.
  ::绝对值函数 f(x)\\\\\\\\\\\\\\\\\\\\\\\\基本绝对值函数( 您可以从上一个课程中识别它) 。 函数 f( x)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\F\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\可以转换父函数父函数。 父函数是保存特定形状的最简单函数。 父函数的所有转换都被称为函数家族。 函数转换通常用于操作父函数来模拟真实世界对象、 对象、 或关系。

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  Use the interactive below to explore the transformations which take place when modeling an object in the real world.
  ::使用下面的互动来探索在真实世界中模拟物体时发生的变异。

   

   

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  Discussion : How does each variable affect the model? Use the point at the center of the v-shape, called the vertex , as a reference point for your explanation.
  ::讨论: 每个变量如何影响模型? 使用 v 形状中心点, 称为顶点, 作为您解释的参考点 。

   

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  Activity 1: Reviewing Transformations
  ::活动1:审查转型

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  The following table provides a summary of the transformations explored above and derived in Algebra 1.
  ::下表汇总了上文所探讨和从代数1中得出的变换。

  Transformation	Notation	Condition 1	Condition 2	Points
  Vertical Shift	$\begin{align*}f(x)+k\end{align*}$	Shift up if   $\begin{align*}k \gt 0\end{align*}$	Shift down if   $\begin{align*}k \lt 0\end{align*}$	$\begin{align*}(x, y+k)\end{align*}$
  Horizontal Shift	$\begin{align*}f(x-h)\end{align*}$	Shift right if   $\begin{align*}h \gt 0\end{align*}$	Shift left if   $\begin{align*}h \lt 0\end{align*}$	$\begin{align*}(x-h, y)\end{align*}$
  Vertical Reflection	$\begin{align*}-f(x)\end{align*}$			$\begin{align*}(x, -y)\end{align*}$
  Horizontal Reflection	$\begin{align*}f(-x)\end{align*}$			$\begin{align*}(-x, y)\end{align*}$
  Vertical Stretch/Shrink	$\begin{align*}af(x)\end{align*}$	Shrink if   $\begin{align*}0 \lt a\lt 1\end{align*}$	Stretch if   $\begin{align*}a \gt 1\end{align*}$	$\begin{align*}(x, ay)\end{align*}$
  Horizontal Stretch/Shrink	$\begin{align*}f(bx)\end{align*}$	Stretch if   $\begin{align*}0 \lt b \lt 1\end{align*}$	Shrink if   $\begin{align*}b \gt 1\end{align*}$	$\begin{align*}(\tfrac{1}{b}x, y)\end{align*}$

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  Understanding these rules can help quickly graph and write function models for real-world situations.
  ::理解这些规则可以帮助快速地为现实世界局势绘制和写出功能模型。

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  Use the interactive below to practice deriving and transforming a parent function.
  ::使用以下互动方式来实践产生和改变父函数的做法。

   

   

   

   

   

   

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  Activity 2 : Modeling With Transformations
  ::活动2:随着变革而建模

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  While the examples above are focused on using transformations to manipulate the graph of a parent function, transformations can be used to fit a function to a context. In this case, the linear parent function  $\begin{align*}f(x) = x\end{align*}$  will be used.
  ::上述示例侧重于使用变换来操纵父函数的图形,但变换可以用来适应上下文的函数。在这种情况下,将使用线性母函数f(x)=x。

  

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  Example
  ::示例示例示例示例

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  The commission * , in dollars, that a salesperson receives for x dollars in monthly sales can be modeled by the function  $\begin{align*}f(x) = x.\end{align*}$ This very basic (parent) version of the function states that for every $1 the salesperson sells, they make $1 in commission (a commission that high would make a very happy salesperson).
  ::佣金* 以美元计, 销售人员每月得到的x美元销售额可以通过函数 f(x)=x来模拟。 这个功能的基本( 母) 版本表示, 每售出一美元, 销售人员就拿出一美元佣金( 如此高的佣金会让销售人员非常快乐 ) 。

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  *A commission is money that is paid to an employee often as the result of selling a certain amount of goods or services. This amount is usually given as a percentage of the revenue earned from the goods sold.
  ::* 佣金是指通常因出售一定数量的货物或服务而支付给雇员的金钱,通常按从所售货物所得收入的百分比给予这一数额。

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  Consider this (more realistic) commission-based scenario:
  ::考虑这一(更现实的)基于委员会的设想:

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  a )  A salesperson makes 5%, or 5 cents per dollar, on his or her total monthly sales in commission. Transform the function $\begin{align*}f(x) = x\end{align*}$  to model this scenario.
  :a) 销售人员按其每月佣金销售总额计算5%,即每美元5美分。

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  The input of the function is the number of sales, and the output is the commission. The 5% only applies to the input, the sales made .  
  ::函数的输入是销售数量, 产出是佣金。 5%仅适用于投入, 即销售量 。

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  Notice that t he salesperson is earning his or her commission at a slower rate from the parent function. This change can be seen on a graph as a decrease in the steepness of the function.
  ::注意销售者从父函数中以较慢的速度挣取佣金。 在图表中可以将这一变化视为该函数的陡峭性降低。

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  Answer:   $\begin{align*}g(x) = 0.05(x)\end{align*}$
  ::答复:g(x)=0.05(x)

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  This transformation can also be represented as  $\begin{align*}0.05(f(x)).\end{align*}$
  ::这种转换也可以以0.05(f(x))表示。

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  b )  A salesperson makes 5% commission on total monthly sales over $2,000. Transform the function  $\begin{align*}f(x) = x\end{align*}$  to model this scenario.
  :b) 销售人员每月总销售额超过2 000美元的佣金占5%。

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  The 5% will have the same effect on the parent function as in part A. However, by only applying it to sales over $2,000, a change is being made to the input. The commission  is  only being applied to the sales after removing $2,000. By decreasing the input, this will have the visual effect of moving the function to the right on a graph.
  ::5%对母体功能的影响与A部分相同。 但是,仅对超过2 000美元的销售适用5%。 但是,对输入进行修改。 只有在删除2 000美元之后,才对销售适用委员会。 通过减少输入,这将产生将函数在图中移到右侧的视觉效果。

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  Answer:   $\begin{align*}h(x) = 0.05(x - 2,000)\end{align*}$
  ::答复:h(x)=0.05(x-2 000)

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  This can also be represented as  $\begin{align*}0.05(f(x - 2,000)).\end{align*}$
  ::也可以以0.05(f(x)-2 000)表示。

   

   

   

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  Activity 3 : Putting It All Together
  ::活动3:将 " 团结在一起 "

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  Use the interactive below to practice applying multiple transformations.
  ::使用下面的交互操作操作, 应用多个变换 。

   

   

   

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  Activity 4 : Even and Odd Functions
  ::活动4:偶数和奇数函数

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  There are many ways to describe a function. One such example of this is continuous or discrete ,  seen in Functions. Another way to describe a function is as being even or odd. An even function is a function that is symmetric about the y-axis .
  ::函数的描述有很多种方式。其中一个例子是连续的或离散的,在函数中可以看到。另一种描述函数的方式是偶数或奇数。一个偶数函数是对 Y 轴的对称函数。

  

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  An odd function is a function that is symmetric about the origin.
  ::奇数函数是对源代码的对称函数。

   

  

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  A function is neither even nor odd if neither of the conditions above is satisfied.
  ::3⁄4 ̄ ̧漯B

   

   

   

   

   

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  Wrap-Up : Review Questions
  ::总结:审查问题

   

   

   

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  Extension :   Interactive Practice
  ::扩展:交互式做法

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  Use the interactive below to review vertical and horizontal shifts.
  ::使用下面的交互数据来审查纵向和横向变化。

   

   

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  Use the interactive below to review stretches and reflections . 
  ::使用下面的交互效果来审查伸展和反省。

   

   

   

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     Summary
  ::摘要

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  • A parent function is the simplest function that preserves a specific shape.
    ::父函数是保存特定形状的最简单的函数。
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  • All transformations of that function are known as the function family . 
    ::此函数的所有变换都称为函数族。
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  • A change to the output will result in a vertical transformation. A change to the input will result in a horizontal transformation.
    ::产出的改变将导致垂直转换。输入的改变将导致横向转换。
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  • An even function is a function that is symmetric about the y-axis. An odd function is a function that is symmetric about the origin.
    ::even 函数是对 Y 轴的对称函数。奇数函数是对原函数的对称函数。