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  • 以多面函数建模

    Lesson Objectives
    ::经验教训目标

    • Identify the effect on the graph of a cubic function after a transformation.
      ::显示转换后立方体函数对图形的影响。
    • Fit a nonlinear function to a scatter plot.
      ::非线性函数适合散射图 。
    • Write a polynomial within a real-world or mathematical context.  
      ::在现实世界或数学背景下写一个多边名词。

    Introduction: Roller Coasters
    ::一. 导言:越野车

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    T he coefficients of a polynomial function impact its values and its shape when graphed. The rise and fall of polynomial values make them ideal for modeling functions that rise and fall. An exciting example of this is a roller coaster. As you rush through the drops and turns of a roller coaster, you are feeling the effects of changing gravity. A human standing on the earth feels 1g of gravity (normal gravity), but on a roller coaster, this can increase greatly. On the Tower of Terror at Gold Reef City in South Africa, you can feel as much as 6.3 g’s! Some countries place a safety limit on the amount of g-force that you can feel on a roller coaster at around 4 g’s. Use the interactive below to design a roller coaster using a polynomial function model. Manipulate the coefficients to change the shape of the roller coaster.
    ::多角度函数的系数在图形化时会影响其值和形状。多数值的上升和下降使它们成为模型功能的理想。多数值的上升和衰落使他们成为模型功能的理想。一个令人振奋的例子就是滚动的海岸。当你冲过滚动的海岸时,你会感受到重力变化的影响。一个人在地球上的立方体感觉到1克重力(正常重力),但是在滚动的海岸上,这种情况会大大增加。在南非金礁城的恐怖塔上,你可以感觉到6.3克!有些国家对四克左右的滚动海岸上可以感觉到的G力量设置了安全限度。在下面用交互作用设计一个使用多角度函数模型的滚动海岸。操纵系数来改变滚动海岸的形状。

     

    Activity 1: Thinking Outside the Box
    ::活动1:在方框外思考

    A rectangular piece of cardboard is folded into a box. The folding is made possible by cutting squares of equal area out of the four corners of the cardboard. How would you represent each side of the box as a function of the square side length cut from the corner? 
    ::纸板的长方形块被折叠成一个盒子。折叠可以通过从纸板的四角切开平方块来做到。您如何将盒子的两侧作为从角落切开的平方长度的函数来表示 ?

    Below is a diagram of what the flattened cardboard box looks like. The long side of the box, once folded, can be expressed as  ( 12 − 2 x ) .  What about the short side of the box, or the box height? The expressions for the three sides can be used to create a formula for the volume which the box can contain  given that   l e n g t h ⋅ w i d t h ⋅ h e i g h t = v o l u m e .
    ::下面是平坦的纸板框的外观图。框的长边一旦折叠,可以表示为(12-2x) 。 框的短边或框的高度如何? 3边的表达式可以用来创建一个公式, 该框可以包含这个公式, 因为它的长度为“ width”= 体积 。

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    Models of Cubic Polynomial Functions
    ::立体多边函数模型

    A cubic model uses a cubic function (of the form  a x 4 + b x 3 + c x + d ) to model real-world situations. They can be used to model three-dimensional objects to allow you to identify a missing dimension or explore the result of changes to one or more dimensions .

    What would be an equation for the volume of the box as a function of x ?   What is the maximum possible box volume if the cardboard measures 12" by 8"? What would the domain and range of this function be given the context?
    ::框的体积是 x 函数的方程式是什么? 如果纸板的量度为 12 乘 8 ? 这个函数的域和范围会给上下文什么?

     

    Activity 2: Modeling Using Different Forms
    ::活动2:利用不同形式建模

    One of the easiest ways to write a function model is by using transformations . T he parent cubic function  f ( x ) = x 3 ,  can be written using transformation notation as  f ( x ) = a ( b ( x − h ) ) 3 + k .  Use the interactive below to practice transforming the function  f ( x ) = x 3 .
    ::写函数模型最容易的方法之一是使用变换。父立方函数 f( x) =x3 可以用 f( x) = a( b( x- h)) 3+k 的变换符号写成。 使用下面的交互方式来练习转换函数 f( x) =x3 。

    Discussion Question : In every transformation of the parent function with an odd degree , there is only one x- intercept . Why? How would it be possible to write a transformation with more than one?
    ::讨论问题:在以奇异程度对父函数进行每次转换时,只有一次 X 拦截。 为什么? 如何写出不止一次的转换?

     

    A polynomial model can also be written in factored form as a ( x − p ) ( x − q )  ... where a  is the a - coefficient and p , q  ,… are the factors of the polynomial. This form is ideal for situations where you need to write a model based on the solutions. Use the interactive below to practice writing a cubic function in factored form.
    ::多式模型也可以以(x-p)(x-q)(a) 参数形式写成,作为(x-p)(x-q)(a) 系数和p,q(p,q) 系数是多式模型的因子。对于需要根据解决方案写出模型的情况,该表形式是理想的。使用下面的交互式模式练习以系数形式写立方函数。

    Discussion Questions :
    ::讨论问题:

    1. How could you shift the model up or down?
      ::你怎么能把模型上上下移动?
    2. What is a situation where factored form would be more useful than transformation form and vice versa?
      ::在何种情况下,要素形式比变换形式更为有用,反之亦然?

     

    Activity 3: Modeling Data
    ::活动3:建模数据

    Thus far, you have modeled to real-world scatter plot data. Use the strategies that you have learned throughout this chapter to write a model for the data below.
    ::到目前为止,您已经模拟了真实世界的分布图数据。使用您在整个本章中学到的战略来为以下的数据写一个模型。

    Discussion Question : What are the domain and range of the function in context? Will they continue to model the data appropriately beyond the window of the interactive?
    ::讨论问题:在上下文中该职能的范畴和范围是什么?它们是否将继续在互动窗口之外适当模拟数据?

      

    Activity 4: Writing  Polynomial Functions
    ::活动4:写入多面函数

    The interactive below displays a pyramid built with cubes. Use the interactive to add layers to the pyramid. What patterns do you see?
    ::下面的交互显示显示一个用立方体建造的金字塔。 使用交互来添加金字塔的层层。 您看到什么模式 ?

     

    Discussion Question : Which of the roots of the model are viable solutions? Why?
    ::讨论问题:模式的哪些根源是可行的解决办法?为什么?

     

    Extension: Rates of Rates of Change!
    ::扩展:变化率率!

    The a- term in a quadratic is  1 2   the second difference , the a -term in a cubic is  1 6   the third difference. Make a conjecture about what the a -term of a quartic would be and test your conjecture on a quartic function.
    ::二次曲线中的一个期数是 12 的第二个差数, 立方体中的一个期数是 16 的第三个差数是 16 的第三个差数。对一个二次曲线的期数进行猜测, 并对一个二次曲线函数中的推测进行测试 。

     

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

       Summary
    ::摘要

    • f ( x ) = a ( b ( x − h ) ) 3 + k  is a transformed version of the function f ( x ) = x 3 .  
      ::f(x) =a(b(x-h)) 3+k 是函数 f(x) =x3 的转换版本 。
    • Cubic functions, a x 3 + b x + c ,  can be used to model three-dimensional objects to allow you to identify a missing dimension.
      ::Cubic 函数, ax3+bx+c,可用于模拟三维天体,以便您识别缺失的维度。