Section outline

  • Warm-up: Memes
    ::温热:

    lesson content

    Why do we need to learn about piecewise functions? Memes, of course!!
    ::为什么要学小字形功能?

    Who knew just a couple of functions and lines could make a meme? The key to this is the bounds of each of the lines.
    ::谁知道只有几条功能和线条 就能做一个meme吗?

    For example, the function labeled E looks like it's between 4.5 ≤ x ≤ 35.    
    ::例如,标签为 E 的函数看起来像是在4.5°x°35之间。

    Some lines may not be as easy to tell. Like line A , it has a so it might be harder to identify the bounds. Its bounds are − 9.5 ≤ x ≤ 11.
    ::有些线条可能不那么容易辨别。 像线A一样, 它可能更难辨别界限。 它的界限是- 9. 5x 11 。

    Combining two or more functions like this results in what is called a piecewise function . What are the rest of the bounds for the rest of the functions like B, C, D? 
    ::将两个或两个以上的函数合并在一起, 产生一个所谓的片段函数。 B、 C、 D 等函数的其余边框是什么 ?

     

    Activity 1: Model with Piecewise Functions
    ::活动1: 带有零散函数的模型

    Example 1-1
    ::例1-1

    Antonia is biking 200 kilometers on a mountain bike. Her  bike has an odometer that tells her  how far she's  gone, as well as a timer which tells her  how long  she's been riding. The first 3 hours of riding are over very difficult terrain, and she only covers  30 kilometers. What is her  average  speed over the 3 hours? Do you think that she travelled exactly this speed for 3 hours, or that her speed varied? Explain. Is her average speed a useful number? Why or why not?
    ::Antonia骑着200公里的山车骑着一辆山车。她的自行车有1兆米的仪表告诉她她走了多远,还有一位计时器告诉她她骑了多长时间。头3小时的骑车在非常困难的地形上,她只骑了30公里。她3小时的平均速度是多少?你认为她飞行了3小时,还是速度不同?解释一下。平均速度是有用的数字吗?为什么不是?

     

    Solution:   Since Antonia covered 30 kilometers in 3 hours, her average speed was 10 kilometers per hour. It's unlikely that she traveled exactly this speed for the 3 hours - her speed probably varied. This is because it is impossible to maintain a perfectly constant speed, especially given the likely variability in mountain terrain. Nevertheless, her average speed gives us a picture of her performance over this time period.  
    ::解答:自从Antonia在3小时中覆盖了30公里, 她的平均速度是每小时10公里。 她不可能在3小时中完全以这个速度行走—— 她的速度可能各不相同。 这是因为不可能保持一个完全不变的速度, 特别是考虑到山地可能的变化。 然而, 她的平均速度让我们看到她在这段时期的表现。

     

     

    Example 1-2
    ::例1-2

    Antonia will use her average speed over different sections of her ride to create a graph showing her performance. Here is a table showing her odometer readings at different times on the ride. How fast was she going for each section of the ride? 
    ::Antonia将使用她乘坐不同路段的平均速度来制作一个显示其性能的图表。 这是一张表格, 显示她在乘车的不同时间的超强计读数。 她骑车每个路段的速度有多快 ?

    t d 0 0 3 30 5 100 6 100 9 170 10.5 200

    ::00330510061009170100.5200吨

     

    Solution:  We can graph these points and connect them with straight segments, to create a simple model that accounts for some of the variability in her speed. 
    ::解答:我们可以绘制这些点的图表,并将它们与直线相连接, 从而建立一个简单的模型, 来计算她速度的某些变异性。

    The average speed over each interval is shown in the following table.
    ::下表显示每个间隔的平均速度。

    Interval Average Speed (mph) 0 ≤ x < 3 30 3 − 0 = 10 3 ≤ x < 5 100 − 30 5 − 3 = 35 5 ≤ x < 6 0 6 − 5 = 0 6 ≤ x < 9 70 9 − 6 = 23 1 3 9 ≤ x < 10.5 30 10.5 − 9 = 20

    ::双倍平均速度(mph)0x < 3303-00=103x < 5100-305-3=355x < 606-5=06x < 9709-6=23139x < 10.53010.5-9=20

     

     

    Work it Out
    ::工作出来

    A girl in a pink dress, riding a bike
    Valeria wants to ride 200 km in less than 12 hours
    1. Valeria has an ambitious goal of biking a 200-kilometer road route in less than 12 hours. Here is a graph showing the distance she has traveled as a piecewise function of time. Did she achieve her goal? How do you know? What was her average speed for each section of her ride? During some intervals, Valeria's speed was 0 kilometers per hour. What does this mean? What was her average speed for her ride overall?
      ::Valeria有一个雄心勃勃的目标,在不到12小时的时间内骑自行车走一条200公里长的公路。 这是一张图表, 显示她作为时间的片段函数所走过的路程。 她达到目标了吗? 你怎么知道? 她每段车程的平均速度是多少? 每隔一段时间, Valeria的速度是每小时0公里。 这是什么意思 ? 她骑车的平均速度是多少?
    A piecewise graph, increasing generally from left to right
    The graph of Valeria's bike ride
    1. Use Valeria's graph to determine her approximate distance at the 8-hour mark. Find the equation for section k, and use this equation to again determine a value for Valeria's distance at 8 hours. How confident are you that this number is accurate? Why or why not? Use this equation to approximate when she passed the 120-kilometer mark.
      ::使用 Valeria 的图来确定她大约在 8 小时的距离 。 查找 k 段的方程, 并使用此方程再次确定 8 小时的Valeria 距离值 。 您对这个数字是否准确有多有信心? 为什么或为什么没有? 使用此方程来估计她通过 120 公里标记时的距离 。

       Piecewise
    ::以小数计

    A piecewise linear function consists of line segments joined at their endpoints.
    ::片段线性函数由端点连接的线条段组成。

    E ach section may have intervals specified using interval notation. 
    ::每一节可能都有使用间距符号指定的间隔。

    To further describe the piecewise function, give the equation for the line containing each segment.
    ::要进一步描述片段函数,请给出包含每个段段的直线的方程。

     

     

    Activity 2 : Create and Interpret Piecewise Linear Functions
    ::活动2:创建和解释小数线性函数

    Work it Out
    ::工作出来

    1. The table below describes the piecewise function in the graph .
      ::下表说明图中按字形排列的函数。

    interval equation − 3 ≤ x < 1 y = − 2 x + 5 1 ≤ x < 3 y = 2 x + 1 3 ≤ x < 5 y = 7 5 ≤ x < 10 y = − x + 12


    ::eqquation - 3xx <1y_%2x+51_x <3y=2x+13_x <5y=75_x <10y_%x+12

    A piecewise graph, descending from left to right, ascending, then descending again
    A piecewise graph

    Calculate the slopes from the graph above to confirm that the slopes in the equations are correct. How many y-intercepts does this piecewise function have, and what are the coordinates ? Explain.
    ::从上图中计算斜度,以确认方程中的斜度是正确的。此平方形函数有多少 Y 界面,坐标是什么?解释 。

    1. Vashal takes a day off to drive in the countryside. Here is a graph representing his journey. Give a story that explains the  graph. Your explanation should include times, speeds, distances, and situations which justify the changes observed. 
      ::Vasshal在农村开车要休息一天。 这是一张代表他的旅程的图表。 给出一个解释图表的故事。 您的解释应该包括时间、 速度、 距离和情况, 说明所观察到的变化的合理性 。
    A piecewise graph, increasing generally from left to right, with multiple peaks.
    A graph representing Vashal's drive through the countryside.

    Interactive
    ::交互式互动

    Leti rides her bike from Eugene, OR into the Cascade Mountains. Her ride is shown in the interactive below. Use the slider for hours to see her distance traveled change. Leti is represented as a point on a horizontal segment representing her entire journey from the 0-kilometer mark to the 65-kilometer mark. The graph of her distance as a function of time is given as a piecewise function. How do the two representations of Leti's position over time relate to each other? How does each help you visualize Leti's speed over different intervals?
    ::Leti骑着她的自行车,从尤金, 或到卡斯卡德山。 她骑着自行车在下面的交互中显示。 使用滑动器数小时看她的距离变化 。 Leti 代表着她从 0 公里标记到 65 公里标记的整个旅程的横向段。 她的距离图作为时间的函数被用一个片断函数来表示 。 Leti 位置的两种表达方式如何互相关联? 每个人如何帮助你想象Leti 在不同时间段的速度 ?

     

    INTERACTIVE
    Piecewise Rate of Change
    minimize icon
    • In the interactive, move the red point to see how many kilometers Leti has traveled at different times.
      ::在互动中,移动红点以查看Leti在不同的时间旅行了多少公里。
    • The dashed lines are extensions of the segments in the piecewise function. 
      ::破折线是按片断函数各段的延伸。
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      Summary
    ::摘要

    • Piecewise linear functions may be used to model scenarios where a single linear function is inadequate to express changes in the rate of change.
      ::可使用小数线性函数来模拟单一线性函数不足以表示变化率变化的假设情况。
    • Piecewise functions may be expressed with tables, with graphs, and with equations that specify given intervals.
      ::小数函数可以用表格、图表和指定间隔的方程式表示。
    • Piecewise function equations may be written and solved to answer questions about the real world. 
      ::小数函数方程式可以被写入并解答 以解答关于真实世界的问题。