1.12 笛卡尔平板上的功能

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  笛卡尔平板上的函数

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  Functions on a Cartesian Plane
  ::笛卡尔平板上的函数

  

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  We represent functions graphically by plotting points on a coordinate plane (also sometimes called the Cartesian plane ). The coordinate plane is a grid formed by a horizontal number line and a vertical number line that cross at a point called the origin . The origin has this name because it is the “starting” location; every other point on the grid is described in terms of how far it is from the origin.
  ::我们用图形方式在坐标平面(有时也称为笛卡尔平面)上绘制坐标点来表示函数。坐标平面是由水平数字线和垂直数字线组成的网格,该网格是在一个称为起源点的垂直数字线上交叉的。起源者有这个名称,因为它是“起始”位置;网格上的其他每个点都用它与起源的距离来描述。

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  The horizontal number line is called the $\begin{array}{r}x-\end{array}$ axis and the vertical line is called the $\begin{array}{r}y-\end{array}$ axis . We can represent each value of a function as a point on the plane by representing the $\begin{array}{r}x-\end{array}$ value as a distance along the $\begin{array}{r}x-\end{array}$ axis and the $\begin{array}{r}y-\end{array}$ value as a distance along the $\begin{array}{r}y-\end{array}$ axis. For example, if the $\begin{array}{r}y-\end{array}$ value of a function is 2 when the $\begin{array}{r}x-\end{array}$ value is 4, we can represent this pair of values with a point that is 4 units to the right of the origin (that is, 4 units along the $\begin{array}{r}x-\end{array}$ axis) and 2 units up (2 units in the $\begin{array}{r}y-\end{array}$ direction).
  ::水平数线称为 x- 轴,垂直线称为 y- 轴。我们可以通过在 y- 轴 上代表 x- 轴和 y- 轴 之间的距离来代表 函数在 y- 轴 上的每个值。例如, 当 x- 轴 值为 4 时, 一个函数的 y- 值为 2 , 我们可以在 y- 轴 右边代表 4 个 单位( 沿 x- 轴 4 个 单位) , 向上代表 2 个 单位 ( y- 方向为 2 个 单位) 。

  

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  We write the location of this point as (4, 2).
  ::我们将此点的位置写为 (4, 2) 。

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  Plotting Points on a Cartesian Plane
  ::笛卡尔平板上的绘图点

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  Plot the following coordinate points on the Cartesian plane.
  ::在笛卡尔飞机上绘制以下坐标点。

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  a) (5, 3)
  :a) (5,3)

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  b) (-2, 6)
  :b) (-2,6)

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  c) (3, -4)
  :c) (3,4)

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  d) (-5, -7)
  :d) (5-5-7)

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  Here are all the coordinate points on the same plot.
  ::这是同一阴谋上的所有坐标点。

  

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  Notice that we move to the right for a positive $\begin{array}{r}x-\end{array}$ value and to the left for a negative one, just as we would on a single number line. Similarly, we move up for a positive $\begin{array}{r}y-\end{array}$ value and down for a negative one.
  ::注意我们向右移动正 x - 值, 向左移动负值, 就像在单数字行一样 。 同样, 我们向上移动正 y - 值, 向下移动负值 。

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  The $\begin{array}{r}x-\end{array}$ and $\begin{array}{r}y-\end{array}$ axes divide the coordinate plane into four quadrants . The quadrants are numbered counter-clockwise starting from the upper right, so the plotted point for (a) is in the first quadrant , (b) is in the second quadrant, (c) is in the fourth quadrant, and (d) is in the third quadrant.
  ::x- 和 y- 轴将坐标平面分为四个四象方。四象体是从右上方开始的反时针编号,因此(a) 在第一个象体的绘图点,(b) 在第二个象体的绘图点,(c) 在第四个象体的绘图点,(d) 在第三个象体的绘图点。

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  Graph a Function From a Table
  ::图图a 从表格中的函数

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  If we know a rule or have a table of values that describes a function, we can draw a graph of the function. A table of values gives us coordinate points that we can plot on the Cartesian plane.
  ::如果我们知道一条规则,或者有一个描述函数的数值表,我们可以绘制函数的图。一个数值表给我们提供了坐标点,我们可以在笛卡尔平面上绘制这些坐标点。

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  Graphing a Function Given a Table of Values 
  ::图形化函数给定值表

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  1. Graph the function that has the following table of values.
  ::1. 绘制下列数值表的函数图。

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  $$\begin{array}{rl}& x-2-1012\\ & y68101214\end{array}$$

  
  ::x-2-10 1 2y6 8101214

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  The table gives us five sets of coordinate points: (-2, 6), (-1, 8), (0, 10), (1, 12), (2, 14).
  ::该表列出了五组协调点2、2、6、1、8、0、10)、1、12、2、14)。

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  To graph the function, we plot all the coordinate points. Since we are not told the domain of the function or given a real-world context, we can just assume that the domain is the set of all real numbers. To show that the function holds for all values in the domain, we connect the points with a smooth line (which, we understand, continues infinitely in both directions).
  ::要绘制函数图, 我们就可以绘制所有坐标点。 由于我们没有被告知函数的域或给定一个真实世界背景, 我们只能假设域是所有实际数字的集合。 要显示函数持有域内所有值, 我们就可以将点与平滑线连接( 我们知道, 平滑线会无穷无尽地延续到两个方向 ) 。

  

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  2. Graph the function that has the following table of values.
  ::2. 绘制下列数值表的函数。

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  $$\begin{array}{rl}& \text{Side of square}01234\\ & \text{Area of square}014916\end{array}$$

  
  ::平方01234面积0144916

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  The table gives us five sets of coordinate points: (0, 0), (1, 1), (2, 4), (3, 9), and (4, 16).
  ::该表列出了五组协调点:0,0,1,1,2,4,3,9,4,16。

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  To graph the function, we plot all the coordinate points. Since we are not told the domain of the function, we can assume that the domain is the set of all non-negative real numbers. To show that the function holds for all values in the domain, we connect the points with a smooth curve. The curve does not make sense for negative values of the independent variable , so it stops at $\begin{array}{r}x=0\end{array}$ , but it continues infinitely in the positive direction.
  ::绘制函数时, 我们绘制所有坐标点。 由于没有告知我们函数的域, 我们可以假设域是所有非负真实数字的一组。 要显示该函数持有域内的所有值, 我们用一个平滑的曲线连接点。 曲线对独立变量的负值没有意义, 因此它停留在 x=0 上, 但是它会无穷无尽地沿着正方向继续 。

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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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  Graph the function that has the following table of values.
  ::图形显示含有以下数值表格的函数。

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  $$\begin{array}{rl}& \text{Number of balloons}1213141516\\ & \text{Cost}4144475053\end{array}$$

  
  ::气球数量 1213141516

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  This function represents the total cost of the balloons delivered to your house. Each balloon is $3 and the store delivers if you buy a dozen balloons or more. The delivery charge is a $5 flat fee.
  ::此函数代表运到您家的气球的总成本。 每个气球为3美元, 商店购买十几个气球或更多气球, 交付费用为5美元。

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  The table gives us five sets of coordinate points: (12, 41), (13, 44), (14, 47), (15, 50), and (16, 53).
  ::该表列出了五组协调点12、41)、(13、44)、(14、47)、(15、50)和(16、53)。

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  To graph the function, we plot all the coordinate points. Since the $\begin{array}{r}x-\end{array}$ values represent the number of balloons for 12 balloons or more, the domain of this function is all integers greater than or equal to 12. In this problem, the points are not connected by a line or curve because it doesn’t make sense to have non-integer values of balloons.
  ::要绘制函数图, 我们绘制所有坐标点。 由于 x- 值代表气球的数量, 代表12个气球或12个以上, 此函数的域数全部大于或等于12个 。 在此问题上, 点没有被线或曲线连接, 因为没有气球的整数值是毫无意义的 。

  

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  In order to draw a graph of a function given the function rule , we must first make a table of values to give us a set of points to plot. Choosing good values for the table is a skill you will develop throughout this course. When you pick values, here are some of the things you should keep in mind.
  ::为了绘制一个函数的图形, 给函数规则, 我们首先必须绘制一个值表, 给我们一组点来绘制。 选择好的值是您在整个过程中将开发的技能。 当你选择值时, 这里就是您应该记住的一些事情 。

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  • Pick only values from the domain of the function.
    ::仅从函数的域中选择值。
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  • If the domain is the set of real numbers or a subset of , the graph will be a continuous curve.
    ::如果域是真实数字集或 的子集, 图形将是一个连续曲线 。
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  • If the domain is the set of integers of a subset of the integers, the graph will be a set of points not connected by a curve.
    ::如果域是整数子数的整数组,则该图将是一组没有被曲线连接的点。
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  • Picking integer values is best because it makes calculations easier, but sometimes we need to pick other values to capture all the details of the function.
    ::选取整数值是最好的,因为它使计算更容易,但有时我们需要选取其他数值来捕捉函数的所有细节。
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  • Often we start with one set of values. Then after drawing the graph, we realize that we need to pick different values and redraw the graph.
    ::通常我们先从一组数值开始。然后在绘制图表后,我们意识到我们需要选择不同的数值并重新绘制图表。

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  Review (Answers)
  ::回顾(答复)

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  For 1-5, plot the coordinate points on the Cartesian plane.
  ::1 -5,在笛卡尔飞机上绘制坐标点

  1. (5, -4)
  2. (2, 7)
  3. (-3, -5)
  4. (6, 3)
  5. (-4, 3)
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  6. Give the coordinates for each point in this Cartesian plane.
     ::给这架笛卡尔飞机上每个点的坐标

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  For 7-10, graph the function that has the following table of values.
  ::对于 7-10, 图表显示包含以下数值表格的函数。

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  7. $$\begin{array}{rl}& x-10-50510\\ & y-3-0.524.57\end{array}$$
     
     ::x-10 - 5 0 510y - 3 - 0.524.57
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  8. $$\begin{array}{rl}& \text{Side of cube (in.)}0123\\ & {\text{Volume (in}}^3)01827\end{array}$$
     
     ::立方体侧面 (in.) 012 3Volume (in3) 01827
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  9. $$\begin{array}{rl}& \text{Time (hours)}-2-1012\\ & \text{Distance from town center (miles)}502502550\end{array}$$
     
     ::时间(小时)-2-10 1 2 与市中心(英里)的距离50 25 02550
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  10. $$\begin{array}{r}x-2-1012\\ y-400100200300800\end{array}$$
       
      ::x-2 - 1 012y - 40000100-2003800

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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 。