Cartesian Plane – Plotting Points
The Cartesian plane is a two-dimensional coordinate system that has two axes, X and Y. The X axis represents the horizontal axis, and the Y axis represents the vertical axis. The point here is really simple. These are called coordinate axes. Each of them has an origin point (this point here) from where they begin counting their values as positive or negative numbers in either direction. It is often denoted R2, just as the real number was denoted R. and this notation represents two pieces of information: the x-coordinate and the y-coordinate. The correspondence between ordered pairs of numbers and points in this plane is known as a Cartesian plane. First, a special kind of point is called the origin. It’s at the place where two lines meet. The origin’s coordinates are (0, 0). We’ll just plot some more points and see how they work. If we plot A, whose x-coordinate is 2 and whose y-coordinate is 3. this first number tells us to move right 2 units; so let’s walk unit and then another 2 units. Now, from there, let’s walk three units up. So three units up, we got up one unit, we got up two units and we went up three units. About there is the point (2, 3). that’s A = (2, 3). B, look at the first coordinate which is -1. That tells us to move one unit to the left; now five up. One unit, two units and three units – four units and five units – that’s B. C tells us to go four units to the right. One, two, three, and four, and then half a unit down. And remember, these instructions can really be any real numbers, so we could’ve gone 4.1, 4.2, and sO on. Here we want (4, -5), and D, walk 5 units to the left. 2 3 4 5. and about 5 units down. down to about here. And there’s D which is equal to (-5, -5). And that’s the entire idea of how to plot points in a plane. [Graph] Let’s formally define the x-axis. The x-axis is a line in the Cartesian plane that contains all points whose y values are zero. So, obviously, this is the x axis. And similarly, let’s formally define the y-axis. The y-axis is also line in the Cartesian plane that contains all points whose x values are zero. In other words, if you start at the origin, then go either right or left until you reach zero (in either direction), then go up from there, then you’ll be on the y-axis (at some point). For instance, one point on the y-axis could be (0,1). One point on the x-axis could be (-5, 0). [Graph] The Cartesian plane is divided into four quadrants, which are defined by the coordinate axes. The first quadrant consists of points (x,y) whose x-coordinate is positive and whose y-coordinate is positive. The second quadrant consists of points (x,y) whose x-coordinate is negative and whose y-coordinate is positive. [Graph] Well, let’s give a real world example. So over here, let’s draw our plane again. A way of plotting a table of data, where each object or person corresponds to two different numbers. Consider the example of three people, each with a height measured in centimeters and a weight measured in kilograms. We will call these individuals A, B, and C. Suppose A has the average height and weight for an American male, which we’ve looked up on the internet is 177 cm and 88.8 kg. If we plot A over here, we have to figure out a scale. But if we go all the way over here and say that’s about 177, and up a little less than that over here, and there’s about A, A = (177, 88.3). Now suppose B is the average height and weight of the American woman; thus, 164 cm, which is currently the standard in America. The kilograms would be 74.7. You will notice that the average American woman is shorter and weighs less than the average American male; therefore, point B is going to be down this way about here and about here and there’s B. We see this visually, as the fact that the woman has to reach downwards and to the left means that the average American woman is shorter and lighter than the average American man. Now notice that if we move along this horizontal line, what are we doing? If we take any other person who sits on this line, that person will have the same average weight. If we move up, that person will get heavier; if we move down, that person will get lighter. You can probably figure out where to put yourself on this. The same is true as we move from the average American female (represented by the box on the left) and move say along this horizontal line: what would we be doing? Any person along that line would have the same weight as the average American female. If we it move left, it should be getting shorter; if we move it right, it should be getting taller.” [Graph]
