节: 协调计划中的要点 | 4.1 协调计划中的要点 | The Way To Learn

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Points in the Coordinate Plane
::协调计划中的要点

Lydia lives 2 blocks north and one block east of school; Travis lives three blocks south and two blocks west of school. What’s the shortest line connecting their houses?
::Lydia住在学校以北两个街区和以东一个街区;Travis住在学校以南三个街区和以西两个街区。

The Coordinate Plane
::协调计划

The coordinate plane can be thought of as two number lines that meet at right angles. The horizontal line is called the $\begin{aligned} x \end{aligned}$ -axis and the vertical line is the $\begin{aligned} y \end{aligned}$ -axis. Together the lines are called the axes , and the point at which they cross is called the origin . The axes split the coordinate plane into four quadrants , which are numbered sequentially (I, II, III, IV) moving counter-clockwise from the upper right.
::坐标平面可被视为在右角度相交的两条数字线。水平线称为 X 轴, 垂直线称为 Y 轴。横线加起来称为轴, 其交叉点称为原点。轴将坐标平面分割成四个四重方, 从右上方按顺序编号( 一、二、三、四) , 从右上方逆时针移动。

Identify Coordinates of Points
::确定各点的坐标

When given a point on a coordinate plane, it’s easy to determine its coordinates . The coordinates of a point are two numbers , and written together they are called an ordered pair . The numbers describe how far along the $\begin{aligned} x \end{aligned}$ -axis and $\begin{aligned} y \end{aligned}$ -axis the point is. The ordered pair is written in " data-term="Parentheses" role="term" tabindex="0"> parentheses , with the $\begin{aligned} x \end{aligned}$ coordinate (also called the abscissa ) first and the $\begin{aligned} y \end{aligned}$ coordinate ( also known as the ordinate ) second.
::当在坐标平面上给定一个点时,很容易确定坐标。一个点的坐标是两个数字,并一起写成,它们被称为一个定对。数字说明了点在x轴和y轴之间的距离。定对以括号写,X坐标(也称为abscissa)第一,Y坐标(也称为坐标)第二。

$\begin{aligned} (1, 7) & An ordered pair with an x -value of one and a y -value of seven \\ (0, 5) & An ordered pair with an x -value of zero and a y -value of five \\ (- 2.5, 4) & An ordered pair with an x -value of -2.5 and a y -value of four \\ (- 107.2, - 0.005) & An ordered pair with an x -value of -107.2 and a y -value of - 0.005 \end{aligned}$

:1,7)一对订购,X值为1,Y值为7(0,0,5)一对订购,X值为零,Y值为5(2.5,4)一对订购,X值为-2.5,Y值为4(107.2,-0.005)一对订购,X值为-107.2,Y值为-0.005)一对

Identifying coordinates is just like reading points on a number line, except that now the points do not actually lie on the number line!
::识别坐标与数字行的读点一样,但现在,数字线上的点实际上并不在数字线上!

Finding the Coordinates of a Point
::查找点的坐标

1.

Find the coordinates of the point labeled $\begin{aligned} P \end{aligned}$ in the diagram above
::在上面的图表中找到点 P 标签点的坐标

Imagine you are standing at the origin (the point where the $\begin{aligned} x \end{aligned}$ -axis meets the $\begin{aligned} y \end{aligned}$ -axis ). In order to move to a position where $\begin{aligned} P \end{aligned}$ was directly above you, you would move 3 units to the right (we say this is in the positive $\begin{aligned} x \end{aligned}$ direction).
::想象一下您站在原点( X 轴与 Y 轴相匹配的点 ) 。为了移动到 P 直接在你上方的位置, 您会移动到右边的 3 个单位( 我们表示这是正向方向 ) 。

The $\begin{aligned} x \end{aligned}$ coordinate of $\begin{aligned} P \end{aligned}$ is +3.
::P的x协调为+3。

Now if you were standing at the 3 marker on the $\begin{aligned} x \end{aligned}$ -axis, point $\begin{aligned} P \end{aligned}$ would be 7 units above you (above the axis means it is in the positive $\begin{aligned} y \end{aligned}$ direction).
::现在,如果你站在X轴上的3个标记处,P点在你上面7个单位(在轴上方意味着它正朝着积极方向前进)。

The $\begin{aligned} y \end{aligned}$ coordinate of $\begin{aligned} P \end{aligned}$ is +7.
::P的 Y 坐标是+ 7 。

The coordinates of point $\begin{aligned} P \end{aligned}$ are (3, 7).
::P点的坐标是(3,7)

2.

Find the coordinates of the points labeled $\begin{aligned} Q \end{aligned}$ and $\begin{aligned} R \end{aligned}$ in the diagram.
::查找图表中标有Q和R的点的坐标。

In order to get to $\begin{aligned} Q, \end{aligned}$ move three units to the right (the positive $\begin{aligned} x \end{aligned}$ direction) then two units down (the negative $\begin{aligned} y \end{aligned}$ direction). The $\begin{aligned} x \end{aligned}$ coordinate of $\begin{aligned} Q \end{aligned}$ is +3, the $\begin{aligned} y \end{aligned}$ coordinate of $\begin{aligned} Q \end{aligned}$ is -2, so the ordered pair is (3, -2).
::为了到达Q,请将三个单位移到右边(正x方向),然后将两个单位下移(负方向)。Q的x坐标是+3,Q的Y坐标是-2,所以订购的对是(3,2-2)。

The coordinates of $\begin{aligned} R \end{aligned}$ are found in a similar way. The $\begin{aligned} x \end{aligned}$ coordinate is +5 (five units in the positive $\begin{aligned} x \end{aligned}$ direction) and the $\begin{aligned} y \end{aligned}$ coordinate is again -2, so the ordered pair is (5, -2)
::R 的坐标也以类似的方式找到。 x 坐标是+5 +5 (正向 x 方向的五个单位), Y 坐标是 - 2, 所以定购对( 5, 2) 。

The coordinates of $\begin{aligned} Q \end{aligned}$ are (3, -2). The coordinates of $\begin{aligned} R \end{aligned}$ are (5, -2).
::Q座标是(3,2),R座标是(5,2)

Finding the Coordinates of Vertices
::寻找顶点的坐标

Triangle $\begin{aligned} A B C \end{aligned}$ is shown in the diagram to the right. Find the coordinates of the vertices $\begin{aligned} A, B \end{aligned}$ and $\begin{aligned} C \end{aligned}$ .
::三角 ABC 显示在向右的图表中。查找 A、 B 和 C 顶点的坐标。

Point $\begin{aligned} A \end{aligned}$ :
::A点:

$\begin{aligned} x -coordinate = - 2 \end{aligned}$
::x坐标坐标=-2

$\begin{aligned} y -coordinate = + 5 \end{aligned}$
::y- 坐标% 5

Point $\begin{aligned} B \end{aligned}$ :
::B点:

$\begin{aligned} x -coordinate = + 3 \end{aligned}$
::x 坐标% 3

$\begin{aligned} y -coordinate = - 3 \end{aligned}$
::Y 坐标=-3

Point $\begin{aligned} C \end{aligned}$ :
::C点:

$\begin{aligned} x -coordinate = - 4 \end{aligned}$
::x坐标坐标=-4

$\begin{aligned} y -coordinate = - 1 \end{aligned}$
::Y- 坐标=-1

$\begin{aligned} A (- 2, 5) \end{aligned}$
:A-2,5)

$\begin{aligned} B (3, - 3) \end{aligned}$
::B(3,3,3-3)

$\begin{aligned} C (- 4, - 1) \end{aligned}$
::C (4-4-1)

Plotting Points in a Coordinate Plane
::坐标平面中的绘图点

Plotting points is simple, once you understand how to read coordinates and read the scale on a graph. As a note on scale, in the next two examples pay close attention to the labels on the axes.
::绘图点很简单, 一旦您了解如何在图表中读取坐标和比例尺。作为比例说明, 在接下来的两个示例中, 密切关注轴上的标签。

Plot the following points on the coordinate plane.
::在坐标飞机上标出以下各点。

$\begin{aligned} A (2, 7) B (- 4, 6) D (- 3, - 3) E (0, 2) F (7, - 5) \end{aligned}$
::A(2,7)B(4,6)D(3,3)E(0,2)F(7,5)

Point $\begin{aligned} A (2, 7) \end{aligned}$ is 2 units right, 7 units up. It is in Quadrant I.
::A点( 2, 7) 是 2 个单位右侧, 向上 7 个单位。在 Quadrant I 中。

Point $\begin{aligned} B (- 4, 6) \end{aligned}$ is 4 units left, 6 units up. It is in Quadrant II.
::B点( 4, 6) 是剩下4个单元, 向上 6个单元, 在 Quadrant II 中。

Point $\begin{aligned} D (- 3, - 3) \end{aligned}$ is 3 units left, 3 units down. It is in Quadrant III.
::D( 3 - 3) 点是左3个单位, 向下3个单位, 在 Quadrant III 中。

Point $\begin{aligned} E (0, 2) \end{aligned}$ is 2 units up from the origin. It is right on the $\begin{aligned} y \end{aligned}$ -axis, between Quadrants I and II.
::点E( 0. 2) 是从源起向上移动的 2 个单位。它位于 y 轴上, 介于夸德兰特 II 和 II 之间。

Point $\begin{aligned} F (7, - 5) \end{aligned}$ is 7 units right, 5 units down. It is in Quadrant IV.
::F(7,5)点是右7个单位,向下5个单位,在Quadrant IV。

Examples
::实例

Example 1
::例1

Plot the following points on the coordinate plane.
::在坐标飞机上标出以下各点。

$\begin{aligned} A (2.5, 0.5) B (π, 1.2) C (2, 1.75) D (0.1, 1.2) E (0, 0) \end{aligned}$
::A(2.5,0.5)B(,1.2)C(2,1.75D(0.1,1.2)E(0,0)

Here we see the importance of choosing the right scale and range for the graph. In Example 4, our points were scattered throughout the four quadrants. In this case, all the coordinates are positive, so we don’t need to show the negative values of $\begin{aligned} x \end{aligned}$ or $\begin{aligned} y \end{aligned}$ . Also, there are no $\begin{aligned} x \end{aligned}$ -values bigger than about 3.14, and 1.75 is the largest value of $\begin{aligned} y \end{aligned}$ . We can therefore show just the part of the coordinate plane where $\begin{aligned} 0 \leq x \leq 3.5 \end{aligned}$ and $\begin{aligned} 0 \leq y \leq 2 \end{aligned}$ .
::这里我们可以看到为图形选择正确比例和范围的重要性。在例4中, 我们的点分布在四个四位数中。在此情况下, 所有坐标都是正的, 所以我们不需要显示 x 或 y 的负值。另外, 没有大于 3. 14 的 x 值, 且 1. 75 是y 的最大值。因此, 我们只能显示坐标平面中 0Qx 3.5 和 0Q = 2 的部分。

Here are some other important things to notice about this graph:
::以下是有关这个图的其他重要信息:

The tick marks on the axes don’t correspond to unit increments (i.e. the numbers do not go up by one each time). This is so that we can plot the points more precisely.
::轴上的刻痕与单位递增不符(即数字不会每次上升一个 ) 。这样我们可以更精确地绘制点数。

The scale on the $\begin{aligned} x \end{aligned}$ -axis is different than the scale on the $\begin{aligned} y \end{aligned}$ -axis, so distances that look the same on both axes are actually greater in the $\begin{aligned} x \end{aligned}$ direction. Stretching or shrinking the scale in one direction can be useful when the points we want to plot are farther apart in one direction than the other.
::X 轴的比值与 y 轴的比值不同, 所以两个轴的距离看起来都一样, 实际上在 x 方向上要大得多。当我们想要绘制的点在一个方向上更相距远时, 向一个方向拉伸或缩缩缩的比值可能是有用的。

Review
::回顾

Identify the coordinates of each point, $\begin{aligned} A - F \end{aligned}$ , on the graph below.

::在下图中标明每个点的坐标A-F。

Draw a line on the above graph connecting point $\begin{aligned} B \end{aligned}$ with the origin. Where does that line intersect the line connecting points $\begin{aligned} C \end{aligned}$ and $\begin{aligned} D \end{aligned}$ ?
::绘制上图中连接点B和来源的线条。该线条在哪里交叉连接点C和D的线条?

Plot the following points on a graph and identify which quadrant each point lies in:
::在图表上绘制以下点,并标明每个点的象限:

(4, 2)

(-3, 5.5)

(4, -4)

(-2, -3)

Without graphing the following points, identify which quadrant each lies in:
::在不绘制下列点图的情况下,标明每个象限的大小:

(5, 3)

(-3, -5)

(-4, 2)

(2, -4)

Review (Answers)
::回顾(答复)

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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