1.14 基于图表的功能规则

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  基于图表的功能规则

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  Function Rules based on Graphs 
  ::基于图表的功能规则

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  In the last two Concepts, you learned how to graph a function from a table and from a function rule . Now, you will learn how to find coordinate points on a graph and to interpret the meaning. Recall that each point on the graph has an x-value and y-value. When given an x-value, you will be asked to find its y-value.
  ::在最后两个“概念”中,您学习了如何从表格和函数规则中绘制函数图。现在,您将学习如何在图表中找到坐标点并解释其含义。提醒注意,图形中的每个点都有 x 值和 Y 值。如果给定了 x 值,您将被要求查找其 y 值 。

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  Finding Coordinate Points on a Graph 
  ::图表上的查找坐标点

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  The students at a local high school took The Youth Risk Behavior Survey . The graph below shows the percentage of high school students who reported that they were current smokers. (A current smoker is anyone who has smoked one or more cigarettes in the past 30 days.) What percentage of high-school students were current smokers in the following years?
  ::当地一所高中的学生参加了 " 青年风险行为调查 " ,下图显示了中学生中报告自己是当前吸烟者的百分比。 (目前的吸烟者是过去30天中曾经吸烟一次或一次以上的人。 )在接下来的几年中,高中学生中目前吸烟者的比例是多少?

  1. 1991
  2. 2004

  

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  1. First, find the value 1991 on the x-axis . This appears to be the first red point on the left side of the graph. By looking at the y-axis , it looks like this point has a y-value of approximately 27. This means that in 1991, approximately 27% of high school students reported that they were current smokers.
  ::1. 首先,在X轴上找到1991年的值,这似乎是图左侧的第一个红色点,从Y轴上看,这个点的Y值大约为27,这意味着1991年大约27%的中学生报告说他们是当前的吸烟者。

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  2. Find the value 2004 on the x-axis, which appears to be between the two red points on the right. This has an approximate y-value of 22.5. This means that in 2004, approximately 22.5% of high school students reported that they were current smokers.
  ::2. 在X轴上找到2004年的值,这似乎是在右边两个红点之间,大约为y值22.5,这意味着2004年大约22.5%的中学生报告说他们是当前的吸烟者。

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  Write a Function Rule from a Graph
  ::从图表写入函数规则

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  Sometimes you’ll need to find the equation or rule of a function by looking at the graph of the function. Points that are on the graph can give you values of dependent and independent variables that are related to each other by the function rule. However, you must make sure that the rule works for all the points on the curve. In this course you will learn to recognize different kinds of functions and discover the rules for all of them. For now we’ll look at some simple examples and find patterns that will help us figure out how the dependent and independent variables are related.
  ::有时您需要通过查看函数的图形来找到函数的方程式或规则。 图形上的点可以提供函数规则所关联的依附性和独立变量的值。 但是, 您必须确保规则适用于曲线上的所有点。 在此过程中, 您将学会识别不同的函数, 并发现所有函数的规则。 现在, 我们将查看一些简单的例子, 并找到模式, 帮助我们弄清楚依附性和独立变量之间的关系 。

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  The graph to the right shows the distance that an ant covers over time. Find the function rule that shows how distance and time are related to each other.
  ::向右的图表显示蚂蚁随时间间隔所覆盖的距离。 查找显示距离和时间如何关联的函数规则 。

  

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  Let’s make a table of values of several coordinate points to see if we can spot how they are related to each other.
  ::让我们用几个坐标点的数值列表, 看看我们能否发现它们彼此之间的关系。

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  $$\begin{array}{rl}& \text{Time}0123456\\ & \text{Distance}01.534.567.59\end{array}$$

  
  ::时间 0 1 2 3 4 5 6 Disantance01.534.567.59

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  We can see that for every second the distance increases by 1.5 feet. We can write the function rule as
  ::我们可以看到,每秒,距离会增加1.5英尺。我们可以把功能规则写成

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  $$\begin{array}{r}\text{Distance}=1.5\times \text{time}\end{array}$$

  
  ::距离=1.5x时间

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  The equation of the function is $\begin{array}{r}f(x)=1.5x\end{array}$ .
  ::函数的方程式是 f(x)=1.5x。

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  Finding a Function Rule Described by a Graph 
  ::查找用图表描述的函数规则

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  Find the function rule that describes the function shown in the graph.
  ::查找描述图形中显示的函数的函数规则。

  

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  Again, we can make a table of values of several coordinate points to identify how they are related to each other.
  ::同样,我们可以列出几个协调点的数值表,以确定它们之间的关系。

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  $$\begin{array}{rl}& x-4-3-2-101234\\ & y84.52.50.524.58\end{array}$$

  
  ::x-4-3-2-101 2 3 4y 84.5 2 50.524.58

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  Notice that the values of $\begin{array}{r}y\end{array}$ are half of perfect squares: 8 is half of 16 (which is 4 squared), 4.5 is half of 9 (which is 3 squared), and so on. So the equation of the function is $\begin{array}{r}f(x)=\frac{1}{2}{x}^2\end{array}$ .
  ::注意 y 值为 完全方形的一半: 8 是 16 的 一半( 4 平方 ) , 4. 5 是 9 的 一半( 3 平方 ) , 等等 。 函数的方程式是 f( x) = 12x2 。

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  Example
  ::示例示例示例示例

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  Example 1
  ::例1

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  Find the function rule that shows the volume of a balloon at different times, based on the following graph:
  ::根据下图查找显示气球在不同时间的体积的函数规则:

  

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  (Notice that the graph shows negative time. The negative time can represent what happened on days before you started measuring the volume.)
  :注意图形显示负时间。 负时间可以代表您开始测量音量前几天发生的情况 。)

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  Once again, we make a table to spot the pattern :
  ::再一次,我们制作一张表格 来识别模式:

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  $$\begin{array}{rl}& \text{Time}-1012345\\ & \text{Volume}1052.51.20.60.30.15\end{array}$$

  
  ::时间-10 1 2 345Volume1052.51.20.60.3015

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  We can see that every day, the volume of the balloon is half what it was the previous day. On day 0, the volume is 5; on day 1, the volume is $\begin{array}{r}5\times \frac{1}{2}\end{array}$ ; on day 2, it is $\begin{array}{r}5\times \frac{1}{2}\times \frac{1}{2}\end{array}$ , and in general, on day $\begin{array}{r}x\end{array}$ it is $\begin{array}{r}5\times {\left(\frac{1}{2}\right)}^x\end{array}$ . The equation of the function is $\begin{array}{r}f(x)=5\times {\left(\frac{1}{2}\right)}^x\end{array}$ .
  ::我们可以看到,每天气球的体积是前一天的一半。在第0天,体积是5;在第1天,体积是5x12;在第2天,气球是5x12x12;一般而言,气球是5x(12)x。函数的方程式是 f(x)=5x(12)x。

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  Review 
  ::回顾

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  For 1-4, the graph below shows the average life-span of people based on the year in which they were born. This information comes from the National Vital Statistics Report from the Center for Disease Control.
  ::关于1-4,下图显示了根据出生年份得出的人口平均寿命,这一信息来自疾病控制中心的《国家生命统计报告》。

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  1. What is the average life-span of a person born in 1940?
     ::1940年出生的人的平均寿命是多少?
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  2. What is the average life-span of a person born in 1955?
     ::1955年出生的人的平均寿命是多少?
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  3. What is the average life-span of a person born in 1980?
     ::1980年出生的人的平均寿命是多少?
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  4. What is the average life-span of a person born in 1995?
     ::1995年出生的人的平均寿命是多少?

  

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  For 5-8, the graph below shows the median income of an individual based on his/her number of years of education. The top curve shows the median income for males and the bottom curve shows the median income for females. (Source: US Census, 2003.)
  ::关于5-8,下图根据个人受教育年限显示其中位收入,上曲线显示男性的中位收入,下曲线显示女性的中位收入。 (资料来源:美国人口普查,2003年)

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  5. What is the median income of a male that has 10 years of education?
     ::受过十年教育的男性的中位收入是多少?
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  6. What is the median income of a male that has 17 years of education
     ::受过17年教育的男性的中位收入是多少?
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  7. What is the median income of a female that has 10 years of education?
     ::受过十年教育的女性的中位收入是多少?
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  8. What is the median income of a female that has 17 years of education?
     ::受过17年教育的女性的中位收入是多少?

  

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  For 9-11, write the function rule for each graph.
  ::对于9-11, 写入每个图表的函数规则 。

  9. 
  10. 
  11. 

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。