Converting a plane into a Cartesian plane QI QIII QIV QII Converting a plane into a Cartesian plane Construct two real number lines perpendicular to each other at the 0 point of each line. Label this point of intersection as O for Origin. This divides the plan into four quadrants, labeled QI, QII, QIII and QIV. The horizontal number line is called the x-axis. The vertical number line is called the y-axis. X-axis Y-axis O (Left Arrow then left click to start animation.)

QI QIII QIV QII Converting a plane into a Cartesian plane A grid is formed from the whole numbers on each number line. Thus, establishing a 1-1 mapping between the points in the plane and ordered pairs of real numbers (x, y) X-coordinate is often called the abscissa Y-coordinate is often called the ordinate X-axis Y-axis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 O 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 Left click to continue -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2

To find the coordinates of a point. QI QIII QIV QII 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 At the beginning, you will only be working with whole numbers to make the learning easier. However, all ordered pairs of real numbers are valid values. Notice the point labeled A. To determine the ordered pair of real numbers that is mapped to this point see below. (Left Click to see animation.) (Left Arrow / Left Click to rerun.) X-axis Y-axis (6, 4) A O To find the coordinates of a point. Draw a line from the point perpendicular to the x-axis. The point where the line intersects the x-axis is the x-coordinate. 2. Draw a line from the point perpendicular to the y-axis. The point where the line intersects the y-axis is the y-coordinate.

QI QIII QIV QII (2, 8) (4, 3) (11, 5) (14, 11) (9, 13) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 (2, 8) (4, 3) (11, 5) (14, 11) (9, 13) Using the procedure shown on the previous page find the coordinates for each point highlighted in QI. (Left click to see the lines drawn.) (Left click again to see the coordinates.) X-axis Y-axis What do you notice about the coordinators of all these points in QI? (Left click to see the answer.) O The coordinators have positive values. All points in Q1 always have positive values for their x-coordinate and y-coordinate.

QI QIII QIV QII (6, -11) (10, -6) (3, -3) (-1, -9) (-10, -7) (-3, -2) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 (6, -11) (10, -6) (3, -3) (-1, -9) (-10, -7) (-3, -2) (-5, 4) (-9, 8) (-12, 11) X-axis Y-axis Find the coordinates for each point highlighted. (Left click to see the lines drawn.) (Left click again to see the answer.) (Left click to continue.) O

QI QIII QIV QII (6, -11) (10, -6) (3, -3) (-1, -9) (-10, -7) (-3, -2) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 (6, -11) (10, -6) (3, -3) (-1, -9) (-10, -7) (-3, -2) (-5, 4) (-9, 8) (-12, 11) All points in QIV have x-coordinates that are positive and y-coordinates that are negative. All points in QIII have x-coordinates that are negative and y-coordinates that are negative. All points in QII have x-coordinates that are negative and y-coordinates that are positive. Left click to continue. X-axis Y-axis Write three rules that describes the numbers that make up the x- and y-coordinates for points in QII, QIII and QIV? (Left Click to see the answers.) O

QI QIII QIV QII x < 0 and y > 0 x > 0 and y > 0 (0, -4) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 x < 0 and y > 0 x > 0 and y > 0 Give the coordinates of the points plotted on this graph. (Left click for the coordinates`.) To which quadrant, if any, does these points belong? (Left click on the answer.) X-axis Y-axis (0, -4) (0, -9) (11, 0) (3, 0) (0, 1) (0, 5) (-7, 0) (-14, 0) O Points on the x-axis have a y-coordinate of 0. Points on the y-axis have a x-coordinate of 0. So, points on either axes belong to no quadrant. (Left click to continue.) x < 0 and y < 0 x >0 and y < 0

QI QIII QIV QII x < 0 and y > 0 x < 0 and y > 0 The Origin 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 x < 0 and y > 0 x < 0 and y > 0 The Cartesian plane is composed of six sets of points: QI, QII, QIII, QIV, x-axis, and y-axis. Therefore, the x-axis and y-axis are not part of any quadrant. (Left click to continue.) X-axis Y-axis The Origin O The sets do not have any points in common, except for 1 point that is in two of the sets. What point is that? (Left click to see the answer.) (Left click to mark important information.) (Left click to continue.) x < 0 and y < 0 x >0 and y < 0

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Now, we turn our attention to plotting the graph of a ordered pair of coordinates. Plot the graph of the point (7, 9). To find this point. (Remember the ordered pair is (x, y), so 7 is the x-coordinate and 9 is the y-coordinate.) Starting at the origin go right (positive direction) 7 units on the x-axis. From this point go up (positive direction) 9 units along the grid line. Mark the point and label (First left click shows the animation.) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 (7, 9) (Left click again to continue.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 (-4, 6) Now, we turn our attention to plotting the graph of a ordered pair of coordinates. Plot the graph of the point (-4, 6). To find this point. (Remember x<0 and y>0, so the point will be in QII.) Starting at the origin go left (negative direction) 4 units on the x-axis. From this point go up (positive direction) 6 units along the grid line. Mark the point and label (First left click shows the animation.) (Left click again to continue.)

(9, -9) Plot the graph of the point (9, -9). (Since x>0 and y<0 the point is in QIV.) Starting at the origin go right (positive direction) 9 units on the x-axis. From this point go down (negative direction) 9 units along the grid line. Mark the point and label (First left click shows the animation.) (Left click again to continue.) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 (9, -9)

(-10, -5) Plot the graph of the point (-10, -5). (Since x<0 and y<0 the point is in QIII.) Starting at the origin go left (negative direction) 10 units on the x-axis. From this point go down (negative direction) 5 units along the grid line. Mark the point and label (First left click shows the animation.) (Left click again to continue.) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 (-10, -5)

(8, 8) (-4, -5) (6, -5) Plot the point having coordinates (6, -5). (First left click shows the animation.) (Left click again for next one.) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 Plot the point having coordinates (-4, -5). (First left click shows the animation.) (Left click again for next one.) (8, 8) (-4, -5) (6, -5) Plot the point having coordinates (8, 8). (First left click shows the animation.) (Left click again for next one.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 If you are confident that you know this material, left click to exit. Or, you can click on this text to repeat this part.