Coordinates in Four Quadrants

11 Key excerpts on "Coordinates in Four Quadrants"

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  Precalculus

  A Prelude to Calculus

  • Sheldon Axler(Author)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  These lines are called the coordinate axes. • The intersection point of the coordinate axes is called the origin; it receives a label of 0 on both axes. • On the horizontal axis, pick a point to the right of the origin and label it 1. Then label other points on the horizontal axis using the scale determined by the origin and 1. • Similarly, on the vertical axis, pick a point above the origin and label it 1. Then label other points on the vertical axis using the scale determined by the origin and 1. The same scale has been used on both axes in the figure here. However, sometimes it is more convenient to have different scales on the two axes. A point in the plane is identified with its coordinates. The coordinates are written as an ordered pair of numbers surrounded by parentheses, as described below. The plane with this system of labeling is often called the Cartesian plane in honor of the French mathematician René Descartes (1596–1650), who described this technique in his 1637 book Discourse on Method. Coordinates • The first coordinate indicates the horizontal distance from the origin, with positive numbers corresponding to points right of the origin and negative numbers corresponding to points left of the origin. • The second coordinate indicates the vertical distance from the origin, with positive numbers corresponding to points above the origin and negative numbers corresponding to points below the origin. The coordinates of the origin are (0, 0). The coordinate axes divide the plane into four pieces, which are called quadrants. The next example shows one point in each of the four quadrants. Section 1.2 The Coordinate Plane and Graphs 51 Example 1 Locate on a coordinate plane the following points: (a) (2, 1); (b) (−1, 2.5); (c) (−2.5, −2.5); (d) (3, −2). solution (a) The point (2, 1) can be located by starting at the origin, moving 2 units to the right along the horizontal axis, and then moving up 1 unit; see the figure below.

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  Years 9 - 10 Maths For Students

  • (Author)
  • 2015(Publication Date)
  • For Dummies

    (Publisher)

  Whenever 0 is a coordinate within the ordered pair, the point must be located on an axis. Table 12-1 gives you the names of the quadrants, their positions in the Cartesian plane, and the characteristics of coordinate points in the various quadrants. Table 12-2 describes what’s happening on the axes as they radiate out from the origin. Table 12-1 Quadrants Quadrant Position Coordinate Signs How to Plot Quadrant I Upper‐right side (positive, positive) Move right and up Quadrant II Upper‐left side (negative, positive) Move left and up Quadrant III Lower‐left side (negative, negative) Move left and down Quadrant IV Lower‐right side (positive, negative) Move right and down Table 12-2 Axes Position Coordinate Signs How to Plot Right axis (positive, 0) Move right and sit on the x‐axis Left axis (negative, 0) Move left and sit on the x‐axis. Upper axis (0, positive) Move up and sit on the y‐axis. Lower axis (0, negative) Move down and sit on the y‐axis. 272 Part IV: Applying Algebra and Understanding Geometry Actually Graphing Points To plot a point, look at the coordinates — the numbers in the parentheses. The first number tells you which way to move, horizontally, from the origin. Place your pencil on the origin and move right if the first number is positive; move left if the first number is negative. Next, from that position, move your pencil up or down — up if the second number is positive and down if it’s negative. The following points are graphed in Figure 12-3. The letters serve as names of the points so you can compare their coordinates. A (9, 0) B (7, 4) C (3, 8) D (0, 7) E (−2, 2) F (−8, 0) G (−5, −3) H (0, −3) J (3, −2) K (8, −7) Figure 12-3: Points A through K graphed in the Cartesian plane. If you have an eagle eye, you may have noticed that I skip from H to J in Figure 12-3. When labelling points on a graph, try to avoid using the letters I and O — these letters are easily mistaken for 1 and 0.

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  Mathematics for Information Technology

  • Alfred Basta, Stephan DeLong, Nadine Basta, , Alfred Basta, Stephan DeLong, Nadine Basta(Authors)
  • 2013(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  A collection of ordered pairs is called a relation , and a relation in which all the points have distinct x -coordinates (that is, there are no two points in the collection having the same x -coordinate) is called a function . Let’s put the ideas of plotting points together with the concept of the quad-rants. In the first quadrant, what can we say about the values of x and y ? Due to the convention of choice for the positive direction on the number lines, points in quadrant I are such that both the x - and the y -coordinates of the point must be positive (Figure 4.5). In the second quadrant, we observe that points have negative x -coordinates and positive y -coordinates. Similar observations can be made for points lying in the other two quadrants, and this nearly puts to rest the interrelationship of coordinate signs and the quadrants. The only remaining question is about those 4 3 2 1 2 1 3 4 H11002 4 H11002 3 H11002 2 H11002 1 H11002 1 H11002 2 H11002 3 H11002 4 x -axis y -axis Quadrant IV Quadrant I Quadrant II Quadrant III 4 3 2 1 2 1 3 4 H11002 4 H11002 3 H11002 2 H11002 1 H11002 1 H11002 2 H11002 3 H11002 4 x > 0 y > 0 x < 0 y > 0 x < 0 y < 0 x > 0 y < 0 Figure 4.4 The Cartesian plane with quadrants numbered Figure 4.5 Note that, according to the numeration of the axes, in quadrant I both x and y are positive © Cengage Learning 2014 © Cengage Learning 2014 Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Straight-Line Equations and Graphs 119 points that happen to not fall within any of the particular quadrants.

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  Coordinate Systems and Map Projections

  • D.H. Maling(Author)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  There is a sign convention to be observed in the use of rectangular coordinates. This states that the X-axis is reckoned positive towards the right and the Y-axis is positive towards the top of the page. In other words, a point in the top right-hand quarter of a graph illustrated by Fig. Fourth quadrant First quadrant - X -i -y 30 Coordinate Systenns and M a p Projections tFrequent reference will be made in this book to the labours of the United Kingdom Working Group on Terminology and to the preparation of the Glossary of Technical Terms in Cartography, published by the Royal Society in 1966. The definitions in that work were subsequently combined with other national contributions to the Multilingual Dictionary of Technical Terms in Cartography, published by ICA in 1973. The preferred terms relating to map projections which appear in those works are used throughout the book. Definitions which are those used in the Glossary are prefaced with the symbol *. 2.01 is defined by positive values of χ and y, whereas a point in the bottom left-hand quarter has negative values for χ and y. The quarters are termed quadrants and these are numbered 1-4 in a clockwise direction com-mencing with the top right quadrant. Hence the sign convention is: 1st quadrant -l-x, + y 2nd quadrant + x, — y 3rd quadrant -x , -y 4th quadrant — x, -f y The map grid as an example of plane rectangular coordinates A grid has been defined in the Glossary of Technical Terms in Cartography (Royal Society, 1966) as * 'a cartesian reference system using distances measured on a chosen projection In the first edition of this book the author disagreed with the last seven words in this definition, but as a major contributor to the Glossary felt a certain loyalty to the deliberations of the working group, limiting himself to making only a mild criticism of this particular definition.

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  A Concise Handbook of Mathematics, Physics, and Engineering Sciences

  • Andrei D. Polyanin, Alexei Chernoutsan(Authors)
  • 2010(Publication Date)
  • CRC Press

    (Publisher)

  Chapter M4 Analytic Geometry M4.1. Points, Segments, and Coordinate Plane M4.1.1. Cartesian and Polar Coordinates on Plane ◮ Rectangular Cartesian coordinates in the plane. A rectangular Cartesian coordinate system consists of two mutually perpendicular directed lines, called coordinate axes , each treated as a number line (see Subsection M6.1.1). The point of intersection of the axes is called the origin and usually labeled with the letter O , while the axes themselves are called the coordinate axes . As a rule, one of the coordinate axes is horizontal, directed from left to right, and called the abscissa axis . The other axis is vertical, directed upwards, and called the ordinate axis . The two axes are usually denoted by X or OX and Y or OY , respectively, and the coordinate system itself is denoted by XY or OXY . The two coordinate axes divide the plane into four parts, which are called quadrants and numbered I, II, III, and IV counterclockwise as shown in Fig. M4.1. Figure M4.1. A rectangular Cartesian coordinate system. Each point M in the plane is uniquely defined by a pair of real numbers ( x 0 , y 0 ), called its coordinates , which specify its projections onto the X -and Y -axes. The numbers x 0 and y 0 are called, respectively, the abscissa and the ordinate of the point M . Remark. Strictly speaking, the coordinate system introduced above is a right rectangular Cartesian coordinate system . A left rectangular Cartesian coordinate system can, for example, be obtained by changing the direction of one of the axes. A right rectangular Cartesian coordinate system is usually called simply a Cartesian coordinate system. If A and B are two points in the plane, then the length of the segment AB will be denoted | AB | . ◮ Polar coordinates. A polar coordinate system is determined by a point O called the pole , a ray OA issuing from this point, which is called the polar axis , a scale segment for measuring lengths, and the positive sense of rotation around the pole.

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  Handbook of Mathematics for Engineers and Scientists

  • Andrei D. Polyanin, Alexander V. Manzhirov(Authors)
  • 2006(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  As a rule, one of the coordinate axes is horizontal and the right sense is positive. This axis is called the abscissa axis and is denoted by the letter X or by OX . On the vertical axis, which is called the ordinate axis and is denoted by Y or OY , the upward sense is usually positive (see Fig. 4.2 a ). The coordinate system introduced above is often denoted by XY or OXY . ( ) c ( ) a ( ) d ( ) b O O O O A A A II I Left Right Upper half-plane Lower half-plane III IV half-plane half-plane X Y X X X X Y Y Y Y Figure 4.2. A rectangular Cartesian coordinate system. 4.1. P OINTS , S EGMENTS , AND C OORDINATES ON L INE AND P LANE 79 The abscissa axis divides the plane into the upper and lower half-planes (see Fig. 4.2 b ), while the ordinate axis divides the plane into the right and left half-planes (see Fig. 4.2 c ). The two coordinate axes divide the plane into four parts, which are called quadrants and numbered as shown in Fig. 4.2 d . Take an arbitrary point A on the plane and project it onto the coordinate axes, i.e., draw perpendiculars to the axes OX and OY through A . The points of intersection of the perpendiculars with the axes are denoted by A X and A Y , respectively (see Fig. 4.2 a ). The numbers x = OA X , y = OA Y , (4. 1 . 2 . 1 ) where OA X and OA Y are the respective values of the segments −→ OA X and −→ OA Y on the abscissa and ordinate axes, are called the coordinates of the point A in the rectangular Cartesian coordinate system. The number x is the fi rst coordinate, or the abscissa , of the point A , and y is the second coordinate, or the ordinate , of the point A . One says that the point A has the coordinates ( x , y ) and uses the notation A ( x , y ). Example 1. Let A be an arbitrary point in the right half-plane. Then the segment −→ OA X has the positive sense on the axis OX , and hence the abscissa x = OA X of A is positive.

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  Precalculus

  Functions and Graphs, Enhanced Edition

  • Earl Swokowski, Jeffery Cole(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  We shall now show how to assign an ordered pair of real numbers to each point in a plane. Although we have also used the notation to denote an open interval, there is little chance for confusion, since it should always be clear from our discussion whether represents a point or an interval. We introduce a rectangular, or Cartesian,* coordinate system in a plane by means of two perpendicular coordinate lines, called coordinate axes, that intersect at the origin O , as shown in Figure 1. We often refer to the horizon-tal line as the x -axis and the vertical line as the y -axis and label them x and y , respectively. The plane is then a coordinate plane, or an xy -plane. The coor-dinate axes divide the plane into four parts called the first, second, third, and fourth quadrants, labeled I, II, III, and IV, respectively (see Figure 1). Points on the axes do not belong to any quadrant. Each point P in an xy -plane may be assigned an ordered pair , as shown in Figure 1. We call a the x -coordinate (or abscissa ) of P , and b the y -coordinate (or ordinate ). We say that P has coordinates and refer to the point or the point . Conversely, every ordered pair deter-mines a point P with coordinates a and b . We plot a point by using a dot, as illustrated in Figure 2. We may use the following formula to find the distance between two points in a coordinate plane. a , b P a , b a , b a , b a , b a , b a , b a , b FIGURE 1 FIGURE 2 y x 1 1 a b II I IV III P ( a , b ) O y x 1 1 ( 5, 3) ( 4, 0) ( 4, 3) (0, 5) (0, 3) (5, 3) (5, 2) (0, 0) O Distance Formula The distance between any two points and in a coordinate plane is d P 1 , P 2 x 2 x 1 2 y 2 y 1 2 . P 2 x 2 , y 2 P 1 x 1 , y 1 d P 1 , P 2 *The term Cartesian is used in honor of the French mathematician and philosopher René Descartes (1596–1650), who was one of the first to employ such coordinate systems. Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

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  Calculus: An Applied Approach, Brief, International Metric Edition

  • Ron Larson(Author)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Each point in the plane corresponds to an ordered pair ( x , y ) of real numbers x and y , called coordinates of the point. The x -coordinate represents the directed distance from the y -axis to the point, and the y -coordinate represents the directed distance from the x -axis to the point, as shown in Figure 1.2. ( x , y ) The notation ( x , y ) denotes both a point in the plane and an open interval on the real number line. The context will tell you which meaning is intended. Plotting Points in the Cartesian Plane Plot the points ( -1, 2 ) , ( 3, 4 ) , ( 0, 0 ) , ( 3, 0 ) , and ( -2, -3 ) . SOLUTION To plot the point ( -1, 2 ) imagine a vertical line through -1 on the x -axis and a horizontal line through 2 on the y -axis. The intersection of these two lines is the point ( -1, 2 ) . The other four points can be plotted in a similar way and are shown in Figure 1.3. Checkpoint 1 Worked-out solution available at LarsonAppliedCalculus.com Plot the points ( -3, 2 ) , ( 4, -2 ) , ( 3, 1 ) , ( 0, -2 ) , and ( -1, -2 ) . -3 -4 1 3 2 4 -1 -3 -2 -4 1 2 3 4 -1 -2 x -axis Vertical real number line Horizontal real number line Origin Quadrant II Quadrant I Quadrant III Quadrant IV y -axis The Cartesian Plane FIGURE 1.1 Directed distance from x -axis Directed distance from y -axis x -coordinate y -coordinate x -axis x y ( x , y ) y -axis FIGURE 1.2 x (3, 0) (0, 0) (3, 4) ( -1, 2) ( -2, -3) -3 -4 1 3 2 4 -1 -3 -2 -4 1 3 4 -1 -2 y FIGURE 1.3 In Exercise 29 on page 9, you will use a line graph to estimate the Dow Jones Industrial Average. iStockphoto.com/Gradyreese Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 Using a rectangular coordinate system allows you to visualize relationships between two variables. In Example 2, data are represented graphically by points plotted in a rectangular coordinate system.

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  The Shape of Algebra in the Mirrors of Mathematics

  A Visual, Computer-Aided Exploration of Elementary Algebra and Beyond (With CD-ROM)

  • Gabriel Katz, Vladimir Nodelman;;;(Authors)
  • 2011(Publication Date)
  • WSPC

    (Publisher)

  The idea of coordinates goes back to René Descartes (1596-1650), the great French mathematician and philosopher. By now, his idea is so common that it is difficult to appreciate its truly revolutionary nature. By drawing a grid on a blank page (by converting it into a sheet of math paper) we attach a pair of real numbers ( x A , y A ) to every point A of the page. Moreover, any A can be uniquely located if you know its coordinates! This profound observation of Descartes is a crucial bridge between geometry and algebra. In this chapter, we will present some basic examples of two-dimensional modular spaces, and will discuss alternative models for a variety of mathematical objects, objects that can be characterized by two parameters. Maps, Models and the Coordinate Plane 3 The pair of numbers may be the length and the height of a rectangle, the size of the two axes of an ellipse, the x - and y -intercepts of a line in a Cartesian plane, or any other ordered pair of numerical characteristics of some mathematical object. Then the corresponding point in the plane is no longer just a point, but a model for a rectangle, an ellipse, a line. We explore the conceptual benefits of such nonstandard interpretations. The most important of them is the focus on families of like-objects rather than on individual objects. In this context, such objects can be organized into a mathematical universe, the structure of the universe reflecting the common properties of the objects. This fundamental shift of investigation focus from a prototypical object to a universe of like objects, a modular space, is a characteristic of modern science. Deep mathematical insights can emerge from investigating the linkage between different two-dimensional representations of the same mathematical universe. The linkage can be expressed as a map from one modular space to the other.

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  A Collection of Problems in Analytical Geometry

  Three-Dimensional Analytical Geometry

  • D. V. Kletenik, W. J. Langford, E. A. Maxwell(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  P A R T II Three-Dimensional Analytical Geometry This page intentionally left blank C H A P T E R 6 ELEMENTARY PROBLEMS OF THREE-DIMENSIONAL ANALYTICAL GEOMETRY § 27. Rectangular Cartesian Coordinates in Three-dimensional Space A rectangular Cartesian coordinate system in three-dimen-sional space is defined when a unit of length for measuring lengths and three mutually perpendicular, concurrent axes are given, the axes being considered ordered. The point of intersection of the axes is called the coordinate origin, and the axes themselves are called the first, second, and third, coordin-ate axes. The origin is denoted by the letter O and the three axes by Ox, Oy, Oz respectively. Let M be an arbitrary point in space, and let M x , M y , Μ ζ be its projections on to the coordinate axes (Fig. 38). Then the coordinates of the point M in the given coordinate system are the numbers x = OM x , y = OM y , z = OM z where OM x is the measure on the interval OM x on the x-axis, OM y the measure of the interval OM y on the j-axis, and OM z the measure of the interval OM z on the z-axis. The number x is called the first coordinate of the point M, or its x-cgordinate, the number y the second coordinate or the j-coordinate, the number z the third coordinate or the z-coordinate. The co-ordinates are written in round brackets after the letter designat-ing the point: M{x, y, z). 3 4 THREE-DIMENSIONAL ANALYTICAL GEOMETRY The plane Oyz divides the whole space into two half-spaces, and the one which contains the positive direction of the x-axis will be referred to as the near half-space, and the other as the remote half-space. The plane Ozx similarly divides space into two half-spaces, and the one which contains the positive direction of the >>-axis is referred to as the right half-space, and the other as the left half-space. And, finally, the plane Oxy M 2 Mr M / V Fro. 38 K m W/ / / / '/> L / X / 0 2 / 1 , / ' 1 λ -Jf V / y FIG.

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  Fundamental Maths

  For Engineering and Science

  • Mark Breach(Author)
  • 2017(Publication Date)
  • Red Globe Press

    (Publisher)

  20 Coordinate systems What this chapter is about In this chapter you will investigate spatial relationships in a two-dimensional plane. By the end of this chapter you will be able to calculate distance and orientation in a two-dimensional plane, the areas of straight-sided figures, find the intersection of lines, and calculate parameters and formulae of circles. You will be able to change from Cartesian to polar coordinates and vice versa and to apply Pythagoras’ theorem in three dimensions. 20-01 Cartesian coordinate system applications The Cartesian coordinate system was introduced in Chapter 6 and it is now appropriate to look at some of its implications. The first is that not everyone uses it in the same way as mathematicians. In the Cartesian coordinate system the primary reference direction is the x -axis and orientation is measured anticlockwise from it. Geographers, and particularly land and engineering surveyors, use a system in which the primary direction is north and orientation, such as a compass bearing, is measured clockwise from it. Although the fundamental mathematical relationships are not affected by this difference, their applications may be. 20-02 Distance and orientation between two points Two points in a coordinate frame may be joined by a straight line. The length of that line will be the distance between the two points. The orientation of the line is measured with 209 Figure 20.1 north east α P ( E P , N P ) x -axis y -axis α P ( x P , y P ) respect to the x -direction within the Cartesian coordinate system and with respect to the north direction in the geographical frame. An application of Pythagoras’ theorem can be used to find the length of the line, and the tangent of its orientation relates the sides of the right angled triangles in Figure 20.2.

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