7.17 线性方平方图

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  线性方形图

  

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  Dana is collecting information about caterpillars for science class. She’s comparing the lengths and widths of several caterpillars. Dana puts the data she has so far into a table. Dana is convinced there is a pattern . Can organize this information as a set of ordered pairs, graph it on a coordinate plane and write an equation that could model this?
  ::Dana正在收集科学类毛虫的信息。 她正在比较几只毛虫的长度和宽度。 Dana将她迄今掌握的数据放在一张表格中。 Dana相信有一个模式。 Dana可以将这些信息组织成一组定购配对,在坐标平面上绘制图表,并写出一个可以模拟这个模式的方程式?

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  $$\begin{array}{r}\begin{array}{|cc|}\hline x\text{ (width in cm)}& y\text{ (length in cm)}\\ 2& 2\\ 3& 4\\ 4& 6\\ 5& 8\\ 6& 10\\ \hline\end{array}\end{array}$$

   
  In this concept, you will learn to graph on the coordinate plane.
  ::在此概念中,您将学会在坐标平面上绘制图表。

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  Graphing Linear Functions
  ::直线函数

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  A linear function is a specific type of function . You may notice that the word “line” is part of the word “linear”. That fact can help you remember that when a linear function is graphed on a coordinate plane, its graph will be a straight line.
  ::线性函数是一种特定类型的函数。您可能会注意到,“线性”一词是“线性”一词的一部分。这一事实可以帮助您记住,当线性函数在坐标平面上绘制图时,其图将是一个直线线。

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  You can represent a function as a set of ordered pairs, through a table, and as an equation. You can also take the information in ordered pairs or in a table and represent a function as a graph.
  ::您可以通过表格和方程式代表一组有顺序的对子函数。您也可以以有顺序的对子或表格的形式接收信息,并代表一个图形函数。

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  Let’s look at an example.
  ::让我们举个例子。

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  The table of values below represents a function on a coordinate plane. On a coordinate plane, graph the linear function that is represented by the ordered pairs in the table below.
  ::下表所列数值表表示坐标平面上的函数。在坐标平面上,请绘制下表中定购对子所代表的线性函数图。

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  $$\begin{array}{r}\begin{array}{|cc|}\hline x& y\\ \text{-}4& 5\\ \text{-}2& 3\\ 0& 1\\ 2& \text{-}1\\ 4& \text{-}3\\ \hline\end{array}\end{array}$$

  
  ::Xy-45-23012-14-3

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  You can represent the information in this table as a set of ordered pairs  $\begin{array}{r}\{(-4,5),(-2,3),(0,1),(2,-1),(4,-3)\}\end{array}$ .
  ::您可以以一组有顺序的对{(- 4, 5, 5, (-2, 2, 3, 0, 1), (2, 1), (4, 3)} 来表示此表格中的信息 。

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  Plot those five points on the coordinate plane. Then, connect them as shown below.
  ::在坐标平面上标出这五个点,然后将它们连接起来,如下文所示。

  

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  Notice that the graph of this linear function is a straight line.
  ::注意此线性函数的图形为直线。

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  You can also graph a linear function if you are given an equation for that function. This will involve a few more steps. When you have an equation, you can use the equation to create a table. Then, plot several of the ordered pairs in the table and connect them with a line.
  ::如果给您给出了该函数的方程式,您也可以绘制线性函数。这将涉及几个步骤。如果您有一个方程式,您可以使用方程式来创建表格。然后,绘制表格中几个有顺序的对,并将它们与直线连接。

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  Here is another example.
  ::下面是另一个例子。

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  The equation $\begin{array}{r}y=2x-1\end{array}$  is a linear function. Graph that function on a coordinate plane.
  ::y=2x- 1 等式是一个线性函数。 图形在坐标平面上起作用 。

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  First, use the equation to create a table and find several ordered pairs for the function. It is a good idea to use some negative $\begin{array}{r}x\end{array}$ -values, some positive $\begin{array}{r}x\end{array}$ -values and 0. For example, you can create a table to find the values of $\begin{array}{r}y\end{array}$  when $\begin{array}{r}x\end{array}$  is equal to -2, -1, 0, 1, and 2.
  ::首先,使用方程式来为函数创建表格并找到多个有顺序的对。使用一些负 x 值、一些正 x 值和 0,这是一个好主意。例如,您可以创建一个表格,在 x 等于 - 2, - 1, 0, 1, 和 2 时查找 y 值。

    $\begin{array}{r}y=2x-1\end{array}$

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y\end{array}$
  $\begin{array}{r}-2\end{array}$	$\begin{array}{r}-5\end{array}$	$\begin{array}{r}2(-2)-1=-5\end{array}$
  $\begin{array}{r}-1\end{array}$	$\begin{array}{r}-3\end{array}$	$\begin{array}{r}2(-1)-1=-3\end{array}$
  	$\begin{array}{r}-1\end{array}$	$\begin{array}{r}2(0)-1=-1\end{array}$
  $\begin{array}{r}1\end{array}$	$\begin{array}{r}1\end{array}$	$\begin{array}{r}2(1)-1=1\end{array}$
  $\begin{array}{r}2\end{array}$	$\begin{array}{r}3\end{array}$	$\begin{array}{r}2(2)-1=3\end{array}$

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  The ordered pairs shown in the table are $\begin{array}{r}(-2,-5),(-1,-3),(0,-1),(1,1)\end{array}$  and $\begin{array}{r}(2,3)\end{array}$ .
  ::表中显示的有顺序配对是(-2,-5,(-1,-3,(0)-1,(1,1)和(2,3)。

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  Plot those five points on the coordinate plane. Then connect them as shown below.
  ::在坐标平面上标出这五个点,然后将其连接如下:

  

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about Dana’s project, which was comparing the lengths and widths of caterpillars.
  ::更早之前, Dana 的项目给您带来了问题, 比较毛虫的长度和宽度。

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  She’s put the data collected so far in a table (shown below). Can you plot these points and write the equation that models this information?
  ::她将迄今收集的数据放在一张表格中(如下表所示 ) 。 您能否绘制这些点和写出模拟这些信息的方程式?

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  $$\begin{array}{r}\begin{array}{|cc|}\hline x\text{ (width in cm)}& y\text{ (length in cm)}\\ 2& 2\\ 3& 4\\ 4& 6\\ 5& 8\\ 6& 10\\ \hline\end{array}\end{array}$$

   
  First, represent this information as a set of ordered pairs so that you can plot the points $\begin{array}{r}\{(2,2),(3,4),(4,6),(5,8),(6,10)\}\end{array}$ .
  ::首先,将这些信息作为一组有顺序的对子表示,以便您能够绘制点 {(2),2,(3),(4),(6),(5),(8),(6),(10)}。

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  Now, can you see a pattern in the table and then write the rule that describes it?
  ::现在,你能在表格中看到一个图案,然后写出描述它的规则吗?

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  Notice that as $\begin{array}{r}x\end{array}$  increases by 1, $\begin{array}{r}y\end{array}$  increases by 2. So, you know that $\begin{array}{r}2x\end{array}$  is involved in the equation. But $\begin{array}{r}y\end{array}$  is not quite $\begin{array}{r}2x\end{array}$ . It is $\begin{array}{r}2x-2\end{array}$ .
  ::注意当 x 增加 1 y 增加 2 。 所以, 您知道 2x 包含在方程中 。 但是 y 不是 相当 2x 。 它是 2x-2 。

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  So the equation that models this information is $\begin{array}{r}2x-2\end{array}$ .
  ::因此,这种信息模型的方程式是 2x-2 。

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  Next, plot the points on the coordinate plane and draw a line through them. The graph is shown below.
  ::接下来,绘制坐标平面上的点,然后通过它们绘制一条线条。图示如下。

  

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  Example 2
  ::例2

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  The table below represents inputs and outputs of a linear function. Can you represent this information as ordered pairs, figure out the equation for this function, and then graph the function?
  ::下表显示线性函数的输入和输出。您能否以定购对对表示此信息, 找出此函数的方程式, 然后绘制函数图 ?

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  $$\begin{array}{r}\begin{array}{|cc|}\hline x& y\\ 1& 5\\ 2& 10\\ 3& 15\\ 4& 20\\ \hline\end{array}\end{array}$$

  
  ::xy15210315420

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  You can extract information from the table and represent the same information as a set of ordered pairs. The $\begin{array}{r}x\end{array}$ -coordinate is the first value and the $\begin{array}{r}y\end{array}$ -coordinate is the second value.
  ::您可以从表格中提取信息,并代表与一组定购对对相同的信息。 x 坐标是第一个值, Y 坐标是第二个值。

  $$\begin{array}{r}\{(1,5),(2,10),(3,15),(4,20)\}\end{array}$$

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  Next, looking at the information in the table, you can see that when you multiply the $\begin{array}{r}x\end{array}$ -value by 5 you get the $\begin{array}{r}y\end{array}$ -value. The rule is multiply $\begin{array}{r}x\end{array}$  by 5 to get $\begin{array}{r}y\end{array}$ . You can write this as an equation.
  ::接下来,看看表格中的信息,你可以看到,当乘以 X 值乘以 5 时,你就会得到 Y 值。规则是乘以 x 乘以 5 以 y。你可以将此写成一个方程。

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  $$\begin{array}{r}y=5x\end{array}$$

  
  ::y=5x y=5x

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  You can graph plot the coordinates $\begin{array}{r}\{(1,5),(2,10),(3,15),(4,20)\}\end{array}$  and draw a line through them to see the graph.
  ::您可以绘制坐标 {(1, 5, (2, 10), (3, 15), (4, 20)) 并绘制一条线以查看图表 。

  

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  Answer the following questions about functions and coordinates.
  ::回答下列关于职能和坐标的问题。

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  Example 3
  ::例3

  

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  Is the function above increasing or decreasing?
  ::职能高于增加还是减少?

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  Notice that as $\begin{array}{r}x\end{array}$  increases $\begin{array}{r}y\end{array}$  increases. Notice that every time you increase $\begin{array}{r}x\end{array}$  by 1, $\begin{array}{r}y\end{array}$  will always increase. In this case, $\begin{array}{r}y\end{array}$  increases by two every time $\begin{array}{r}x\end{array}$  increases by 1.
  ::注意 x 增加 y 增加 y 增加 。 注意每次增加 x 1, y 总是增加 。 在这种情况下, y 每次增加 1 增加 2 y 增加 。

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  The answer is the function is increasing.
  ::答案是功能正在增加。

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  Example 4
  ::例4

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  In the point $\begin{array}{r}(-3,4)\end{array}$  is the $\begin{array}{r}x\end{array}$ -value positive or negative?
  ::在点(-3,4)中,X值是正值还是负值?

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  The $\begin{array}{r}x\end{array}$ -value is the first value in the coordinate. It is a negative number.
  ::x 值是坐标的第一个值。 这是一个负数 。

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  The answer is the $\begin{array}{r}x\end{array}$ -value is negative.
  ::答案是x值为负值。

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  Example 5
  ::例5

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  In $\begin{array}{r}(-6,-7)\end{array}$ , which value is $\begin{array}{r}y\end{array}$ -value?
  ::在(-6,-7)中,什么值是Y值?

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  The $\begin{array}{r}y\end{array}$ -value is the second value in a coordinate, and it is equal to -7.
  ::Y值是坐标中的第二个值,等于-7。

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  The answer is the $\begin{array}{r}y\end{array}$ -value is -7.
  ::答案是Y值是 -7

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  Review
  ::回顾

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  The information in the table represents points from a linear function. Plot the points in the table on a coordinate plane, and then draw a straight line through them to graph each function. Then identify the rule (equation) for the function.
  ::表格中的信息代表线性函数的点。在坐标平面上绘制表格中的点,然后通过它们绘制一条直线以图示每个函数。然后为函数指定规则(等距)。

  1.   

  Input	Output
  1	4
  2	5
  3	6
  4	7

  2.   

  Input	Output
  2	4
  3	6
  4	8
  5	10

  3.   

  Input	Output
  1	3
  2	6
  4	12
  5	15

  4.   

  Input	Output
  9	7
  7	5
  5	3
  3	1

  5.   

  Input	Output
  8	12
  9	13
  11	15
  20	24

  6.   

  Input	Output
  3	21
  4	28
  6	42
  8	56

  7.   

  Input	Output
  2	5
  3	7
  4	9
  5	11

  8.  

  Input	Output
  4	7
  5	9
  6	11
  8	15

  9.   

  Input	Output
  5	14
  6	17
  7	20
  8	23

  10.   

  Input	Output
  4	16
  5	20
  6	24
  8	32

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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 。

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  Resources
  ::资源