Chapter3

# Chapter3 - Pre-Calculus Chapter 3A Chapter 3A Rectangular...

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© : Pre - Calculus - Chapter 3A Chapter 3A - Rectangular Coordinate System Introduction : Rectangular Coordinate System Although the use of rectangular coordinates in such geometric applications as surveying and planning has been practiced since ancient times, it was not until the 17th century that geometry and algebra were joined to form the branch of mathematics called analytic geometry. French mathematician and philosopher Rene Descartes (1596-1650) devised a simple plan whereby two number lines were intersected at right angles with the position of a point in a plane determined by its distance from each of the lines. This system is called the rectangular coordinate system (or Cartesian coordinate system). y x x-axis y-axis origin (0, 0) Points are labeled with ordered pairs of real numbers x , y , called the coordinates of the point, which give the horizontal and vertical distance of the point from the origin, respectively. The origin is the intersection of the x -and y -axes. Locations of the points in the plane are determined in relationship to this point 0,0 . All points in the plane are located in one of four quadrants or on the x -or y -axis as illustrated below. To plot a point, start at the origin, proceed horizontally the distance and direction indicated by the x -coordinate, then vertically the distance and direction indicated by the y -coordinate. The resulting point is often labeled with its ordered pair coordinates and/or a capital letter. For example, the point 2 units to the right of the origin and 3 units up could be labeled A 2,3 . Quadrant I Quadrant II Quadrant III Quadrant IV (-, +) (+, +) (-, -) (+, - ) (a, 0) (0, b) (0,0) Notice that the Cartesian plane has been divided into fourths. Each of these fourths is called a quadrant and they are numbered as indicated above. © : Pre - Calculus

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© : Pre - Calculus - Chapter 3A Example 1 : Plot the following points on a rectangular coordinate system: A 2, 3 B 0, 5 C 4,1 D 3,0 E 2, 4 Solution: -5 -4 -3 -2 -1 0 1 2 3 4 5 - 5 - 4- 3 - 2- 1 12345 C(-4,1) E(-2,-4) B(0,-5) A(2,-3) D(3,0) Example 2 : Shade the region of the coordinate plane that contains the set of ordered pairs  x , y x 0 . [The set notation is read “the set of all ordered pairs x , y such that x 0”.] Solution: This set describes all ordered pairs where the x -coordinate is greater than 0. Plot several points that satisfy the stated condition, e.g., 2, 4 , 7,3 , 4,0 . These points are all located to the right of the y -axis. To plot all such points we would shade all of Quadrants I and IV. We indicate that points on the y -axis are not included x 0 by using a dotted line. Example 3 : Shade the region of the coordinate plane that contains the set of ordered pairs  x , y x 1, 2 y 3 .
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Chapter3 - Pre-Calculus Chapter 3A Chapter 3A Rectangular...

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