Plane coordinate geometry has manyuses in the practical world. The architectsand the engineers, the pilot and thenavigators, the businessmen and theaccountants, the artist and the sculptors,and even the carpenters make use ofcoordinate geometry in their respective fieldof work.
7.1 Cartesian Coordinate System To fully understand the mathematicalcharacteristics of lines and curves represented bytheir equations, it is important to picture them in acoordinate plane. This coordinate plane used incoordinate geometry is attributed to Renẻ Descartes,the father of Modern Mathematics, who bridged thegap between Algebra and Geometry. The Cartesian coordinate system consist oftwo number lines whish are perpendicular to eachother. One is vertical and the other is horizontal. Thehorizontal number line is called the x-axis, while thevertical number line is called the y-axis. These twoaxes intersect at a point (0, 0) called the origin.
Look at the figure below. II 6 I 4 2 -6 -4 -2 0 2 4 6 -2 -4 III -6 IV
The x-axis and y-axis divide the plane into fourquadrants, properly labeled in the figure as I, II, III, IV.They are also referred to as FIRST QUADRANT,SECOND QUADRANT, THIRD QUADRANT, FOURTHQUADRANT . Coordinates to the right of the y-axis are positive,while those to the left of y-axis are negative.Coordinates upward form the x-axis are positive andthose downward from the x-axis are negative. Any point in a plane is identified by an ordered pair of numbers denoted as (x, y) where x and y are called the coordinate of a point. The x-coordinate is the abscissa and the y-coordinate is the ordinate. The abscissa is always the first coordinate in the ordered pair, and the ordinate is always the second coordinate in the ordered pair.
There is one-to-one correspondence between the setof the ordered pairs and the points in the plane. Y 8 B(2 ,7) C(-4 ,6) 4 2 A(3 ,2) D(-7 ,1) X -8 -4 -2 0 2 4 8 E(-3 ,-2 F(-6 ,-3) G(4, -4) 4 H(9, -5) 6 8
The abscissa of the x-coordinate represents the distance ofa point from the y-axis,And its sign indicates whether it is to the left or to the right ofthe y-axis .The ordinate or the y-coordinate represents thedistance of the point from the x-axis, and its sign indicateswhether it is above or below the x-axis. Remember!!! Each ordered pair of number corresponds to exactly one point in the coordinate plane. Each point in the coordinate plane corresponds to exactly one ordered pair of numbers.
7.1.1 Slope Of a LineHighways, roads, and bridges have different degrees ofsteepness. This steepness is also called Slope. Slope isthe ratio of the rise to the run,written as rise runConsider the following Illustrations:A. B. 3 (rise) 6 (rise) 12 (run) 12 (run)Car A goes through a road w/slope 3 or 1 , while car B 12 4goes through a road w/slope 6 or 1 . 12 2 Next Page!!!!!!!!!
Lines have slopes, too!!!!... L m 6 (run) 2 (rise) 4 (rise) 6 (run) The slope of line l is: The slope of line m is: Rise = 2 = 1 Rise = -4 = -2 Run 6 3 Run 6 3Consider the graph of y=2x. Observe that the 2Slope of the non-vertical line is found by 1 4 u ni tscomparing the vertical change (rise) to theHorizontal Change (run). Consider the points (-1,-2) and (1,2) . The vertical change is 3 2 1 0 1 2 3 4 4 unit while the horizontal change is 2 unit. 1 2 2 u n it s
Illustrative Example:A.Rise= 2-(-2)=4=2 The Slope is Positive when the lineRun 2-0 2 rises from left to rightB.Rise=2-0= 2= -2 The slope is negative when the lineRun 0-3 -3 3 falls from left to rightC. 2-2 = 0 = 0 3-(-1) 4 The slope of a horizontal line is Zero .It does not rise or fallD. 3-(-3) = 6 (Undefined)-2-(-2) o The slope of a vertical line is Undefined
Remember!!!The slope m of a non vertical line containingtwo points with coordinates (x,y) and (x,y) is given by the formula M= Y2-Y1 X2-X1The slope of a horizontal line is Zero “0”.The slope of a vertical line is Undefined.
7.1.2 Linear Equation Plot the points in the following table of values. Then connect the points with a straight line.x -3 -2 -1 0 1 2y -4 -2 0 2 4 6