Section outline

  • Learning Objectives
    ::学习目标

    • 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。
    • 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。
    • 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。
    • 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。
    • Identify even and odd functions from their graphs.
      ::从图表中识别偶数和奇数函数 。

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

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    In this graph of overlapping parent functions, which function overlaps the other half of g(x)=|x|?

    T he absolute value function  f ( x ) = | x 78 |   is a basic absolute value function  (you may recognize it from the prior lesson, ). The function f ( x ) = | x 78 |  is a transformation of the parent function  f ( x ) = | x | .  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。

    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。

     

     

    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 。

     


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

    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 f ( x ) + k   Shift up if   k > 0   Shift down if   k < 0   ( x , y + k )  
    Horizontal Shift f ( x h )   Shift right if   h > 0   Shift left if   h < 0   ( x h , y )  
    Vertical Reflection f ( x )       ( x , y )  
    Horizontal Reflection f ( x )       ( x , y )  
    Vertical Stretch/Shrink a f ( x )   Shrink if   0 < a < 1   Stretch if   a > 1   ( x , a y )  
    Horizontal Stretch/Shrink f ( b x )   Stretch if   0 < b < 1   Shrink if   b > 1   ( 1 b x , y )

    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。

    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。

     

     

     

     

     

     


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

    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  f ( x ) = x  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

    The commission * , in dollars, that a salesperson receives for x dollars in monthly sales can be modeled by the function  f ( x ) = x . 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 ) 。

    *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。

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

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

    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 。

    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。

    Answer:   g ( x ) = 0.05 ( x )
    ::答复:g(x)=0.05(x)

    This transformation can also be represented as  0.05 ( f ( x ) ) .
    ::这种转换也可以以0.05(f(x))表示。

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

    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。

    Answer:   h ( x ) = 0.05 ( x 2 , 000 )
    ::答复:h(x)=0.05(x-2 000)

    This can also be represented as  0.05 ( f ( x 2 , 000 ) ) .
    ::也可以以0.05(f(x)-2 000)表示。

     

     

     


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

    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 。

     

     

     


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

    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

     

     

     

     

     


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

     

     

     


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

    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。

     

     

    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。

     

     

     

       Summary
    ::摘要

    • 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。
    • 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。
    • 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。
    • 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。