2 CHAPTER 1 Graphs Rectangular Coordinates We locate a point on the real number line by assigning it a single real number, called the coordinate of the point . For work in a two-dimensional plane, we locate points by using two numbers. Begin with two real number lines located in the same plane: one horizontal and the other vertical. The horizontal line is called the x-axis , the vertical line the y-axis , and the point of intersection the origin O. See Figure 1. Assign coordinates to every point on these number lines using a convenient scale. In mathematics, we usually use the same scale on each axis, but in applications, different scales appropriate to the application may be used. The origin O has a value of 0 on both the x -axis and the y -axis. Points on the x -axis to the right of O are associated with positive real numbers, and those to the left of O are associated with negative real numbers. Points on the y -axis above O are associated with positive real numbers, and those below O are associated with negative real numbers. In Figure 1, the x -axis and y -axis are labeled as x and y, respectively, and an arrow at the end of each axis is used to denote the positive direction. The coordinate system described here is called a rectangular or Cartesian * coordinate system . The x -axis and y -axis lie in a plane called the xy-plane , and the x -axis and y -axis are referred to as the coordinate axes . Any point P in the xy -plane can be located by using an ordered pair x y , ( ) of real numbers. Let x denote the signed distance of P from the y -axis ( signed means that if P is to the right of the y -axis, then x 0, > and if P is to the left of the y -axis, then < x 0); and let y denote the signed distance of P from the x -axis. The ordered pair x y , , ( ) also called the coordinates of P, gives us enough information to locate the point P in the plane. For example, to locate the point whose coordinates are 3, 1 , ( ) − go 3 units along the x -axis to the left of O and then go straight up 1 unit. We plot this point by placing a dot at this location. See Figure 2, in which the points with coordinates 3,1, 2, 3, 3, 2, ( ) ( ) ( ) − − − − and 3, 2 ( ) are plotted. The origin has coordinates 0, 0 . ( ) Any point on the x -axis has coordinates of the form x, 0 , ( ) and any point on the y -axis has coordinates of the form y 0, . ( ) If x y , ( ) are the coordinates of a point P, then x is called the x-coordinate, or abscissa , of P, and y is the y-coordinate , or ordinate , of P. We identify the point P by its coordinates ( ) x y , by writing P x y , . ( ) = Usually, we will simply say “the point x y , ( ) ” rather than “the point whose coordinates are x y , . ( ) ” The coordinate axes partition the xy -plane into four sections called quadrants , as shown in Figure 3. In quadrant I, both the x -coordinate and the y -coordinate of all points are positive; in quadrant II, x is negative and y is positive; in quadrant III, both x and y are negative; and in quadrant IV, x is positive and y is negative. Points on the coordinate axes belong to no quadrant. Now Work PROBLEM 7 1.1 Graphing Utilities; Introduction to Graphing Equations Now Work the ‘Are You Prepared?’ problems on page 10. • Algebra Essentials (Section A.1, pp. A1–A10) PREPARING FOR THIS SECTION Before getting started, review the following: Figure 1 xy -Plane x y –4 –2 2 4 2 –2 4 –4 O OBJECTIVES 1 Graph Equations by Plotting Points (p. 4) 2 Graph Equations Using a Graphing Utility (p. 6) 3 Use a Graphing Utility to Create Tables (p. 8) 4 Find Intercepts from a Graph (p. 9) 5 Use a Graphing Utility to Approximate Intercepts (p. 9) Figure 2 x y –4 4 1 3 2 2 3 3 2 4 (3, 2) 3 (–3, 1) (–2, –3) (3, –2) O Figure 3 x y Quadrant I x > 0, y > 0 Quadrant IV x > 0, y < 0 Quadrant II x < 0, y > 0 Quadrant III x < 0, y < 0 *Named after René Descartes (1596–1650), a French mathematician, philosopher, and theologian.
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