什么是平面直角坐标系---**Chapter Information:** * Chapter Number: 11 * Chapter Title: 平面直角坐标系 (Plane Rectangular Coordinate System) **Section Information:** * 11.1: 平面内点的坐标 (Coordinates of points in a plane) * 11.2: 图形在坐标系中的平移 (Translation of figures in the coordinate system) **Introductory Text:** 在现实生活中, 我们常常需要确定物体的位置, 例如, 学生在教室听课、观众在电影院里看电影、都有确定的座位等. 类似地, 在数学中也要研究如何确定平面内点的位置. 本章将学习平面内确定点的位置的方法和坐标系中图形的平移. (English Translation: In real life, we often need to determine the position of objects, for example, students attending class in a classroom, audience watching a movie in a cinema, all have determined seats, etc. Similarly, in mathematics, we also need to study how to determine the position of points in a plane. This chapter will study the method of determining the position of points in a plane and the translation of figures in the coordinate system.) **Diagram Description:** * **Type:** Plane Rectangular Coordinate System with two quadrilaterals plotted. * **Coordinate Axes:** * Horizontal X-axis labeled 'x', with markings at intervals of 2 (e.g., -4, -2, 0, 2, 4). Grid lines are present, suggesting unit increments. * Vertical Y-axis labeled 'y', with markings at intervals of 2 (e.g., -2, 0, 2, 4, 6, 8). Grid lines are present, suggesting unit increments. * Origin (0,0) is at the intersection of the axes. * **Figures:** Two quadrilaterals are shown. * **Quadrilateral 1:** Vertices labeled A, B, C, D. * A: (1, 7) * B: (-2, 5) * C: (1, 0) * D: (4, 2) * **Quadrilateral 2:** Vertices labeled A1, B1, C1, D1. * A1: (-3, 7) * B1: (-6, 5) * C1: (-3, 0) * D1: (0, 2) * **Lines/Segments:** The vertices of each quadrilateral are connected by straight lines to form the quadrilaterals ABCD and A1B1C1D1. * **Relationship:** Quadrilateral A1B1C1D1 appears to be a translation of quadrilateral ABCD. The translation vector from A to A1 is (-3 - 1, 7 - 7) = (-4, 0). This indicates a shift 4 units to the left. This translation vector also applies to other corresponding vertices (B to B1, C to C1, D to D1). **Introductory Text:** 我们知道, 建立数轴后, 数轴上的点与实数是一一对应的, 数轴上每一个点都对应一个实数, 这个实数叫做这个点在数轴上的坐标. 那么, 怎样确定一个点在平面内的位置呢? **Question Stem:** 问题 图 11-1 是某教室学生座位位的平面图, 你能描述 吴小明和王健同学座位的位置吗? **Diagram 11-1 Description:** * Type: Grid/Layout diagram representing classroom seating. * Title: 图 11-1. * Main Elements: A rectangular grid representing seats. * Axes/Labels: * Vertical labels on the left indicate "排" (Row) with numbers 1, 2, 3, 4, 5, 6 from bottom to top. * Horizontal labels on the top indicate "列" (Column) with numbers 1, 2, 3, 4, 5, 6, 7, 8 from left to right. * "讲台" (Podium) is located at the bottom of the grid. * Specific Points/Labels: * The seat labeled "吴小明" is located at Row 5, Column 2. * The seat labeled "王健" is located at Row 3, Column 5. **Explanatory Text (about Coordinate Systems, referring to Figure 11-2):** 数学中, 为了确定平面内一个点的位置, 我们先在平面内画两条互相垂直并且原点重合的数轴, 如图 11-2. 水平的数轴叫做x轴或横轴 (x-axis), 取向右为正方向; 垂直的数轴叫做y轴或纵轴 (y-axis), 取向上为正方向; 两轴交点O为原点, 这样就建立了平面直角坐标系 (rectangular coordinate system), 这个平面叫做坐标平面. 有了平面直角坐标系, 平面内的点就可以用一对实数来表示了, 例如, 在图 11-2 中, 点 P 可以这样来表示: 由点 P 向 x轴作垂线, 垂足 M 在x轴上的坐标是-2; 由点P向y轴... (Text is cut off) **Textual Information:** 作垂线,垂足N在y轴上坐标是3,于是,我们说点P的横 坐标是-2,纵坐标是3,把横坐标写在纵坐标的前面,记作 (-2,3), (-2,3)就叫做点P在平面直角坐标系中的坐标, 简称点P的坐标表示为P(-2,3). 图 11-2 **Chart/Diagram Description:** * **Type:** Cartesian Coordinate System (Grid). * **Main Elements:** * **Coordinate Axes:** X-axis and Y-axis intersecting at the origin (0,0). The positive direction of the X-axis is to the right, labeled 'x'. The positive direction of the Y-axis is upwards, labeled 'y'. * **Grid:** A grid is shown with horizontal and vertical lines corresponding to integer values on the axes. * **Scales:** * X-axis is marked with integers from -6 to 6. * Y-axis is marked with integers from -5 to 5. * **Points:** * Point P: Shown in the second quadrant, indicated by a red dot. * Point M: Located on the X-axis at approximately x = -2. * Point N: Located on the Y-axis at y = 3. * **Lines:** * A vertical dashed line connects Point P to Point M on the X-axis. * A horizontal dashed line connects Point P to Point N on the Y-axis. * **Labels:** The points are labeled P, M, and N. The axes are labeled x and y. The figure is labeled "图 11-2". 图 11-4 x 轴和 y 轴把坐标平面分成四个部分，分别叫做第一、 二、三、四象限，各象限内的点的坐标符号分别为 (+, +)、 (-, +)、(-, -)、(+, -)，如图 11-5。**坐标轴上的点， 也就是 x 轴、y 轴上的点不属于任何一个象限。** 图 11-5 通过直角坐标系的建立，我们把平面内的点与有序实数 一一对应起来。即对于坐标平面内任意一点 P，都有唯一的 一个有序实数对 (x, y) 和它对应；反之，对于任意一个有序实 **Chart/Diagram Description:** * **Type:** Cartesian Coordinate System (Coordinate Plane). * **Main Elements:** * **Coordinate Axes:** * Horizontal axis labeled 'x'. * Vertical axis labeled 'y'. * Origin (0, 0) is marked at the intersection of the axes. * Scales are marked on both axes. The x-axis is numbered from -4 to 4, and the y-axis is numbered from -4 to 4. * **Quadrants:** The plane is divided into four regions by the axes. * Top right region is labeled "第一象限" (First Quadrant) and annotated with "(+, +)" and "x>0, y>0" (written in red). * Top left region is labeled "第二象限" (Second Quadrant) and annotated with "(-, +)" and "x<0, y>0" (written in red). * Bottom left region is labeled "第三象限" (Third Quadrant) and annotated with "(-, -)" and "x<0, y<0" (written in red). * Bottom right region is labeled "第四象限" (Fourth Quadrant) and annotated with "(+, -)" and "x>0, y<0" (written in red). * **Labels and Annotations:** Labels for the axes (x, y), quadrant names, coordinate sign conventions for each quadrant, and inequality conditions for x and y in each quadrant are present. The text below the title "图 11-4" explains the division into quadrants and the signs of coordinates in each quadrant. The underlined sentence emphasizes that points on the axes do not belong to any quadrant. The text below the chart explains the one-to-one correspondence between points in the plane and ordered pairs (x, y). Here is the extracted content from the image: **English Text:** of the story are really _______ and sad. Lily 9. _______ to help. With th of the unicorn. they search **Mathematical Text:** 数对 (x, y), 在坐标平面内都有唯一的点 P 和它对应. 练习 1. 在平面直角坐标系中描出下列各点, 并指出它们分别在哪个象限或哪条坐标轴上: [First point is partially visible, likely A with coordinates cut off] B(4, -6) [Annotation: 四 (Quadrant IV)] C(0, -1) D(-5, 3) [Annotation: 二 (Quadrant II)] **Interpretation/Description of Mathematical Content:** - The text states that an ordered pair (x, y) has a unique corresponding point P in the coordinate plane. - It is followed by an "Exercise" section. - Question 1 asks the user to plot the given points in a Cartesian coordinate system and indicate which quadrant or coordinate axis each point is located on. - The points provided are: - A: Coordinates are cut off. - B: (4, -6). An annotation "四" (meaning four or Quadrant IV) is present near this point. - C: (0, -1). - D: (-5, 3). An annotation "二" (meaning two or Quadrant II) is present near this point.