Lecture Note
University
Duke UniversityCourse
Data Science Math SkillsPages
4
Academic year
2022
KatrCrayon
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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 thevertical axis. The point here is really simple. These are called coordinate axes. Eachof them has an origin point (this point here) from where they begin counting theirvalues as positive or negative numbers in either direction. It is often denoted R2, just as the real number was denoted R, and this notationrepresents two pieces of information: the x-coordinate and the y-coordinate. Thecorrespondence between ordered pairs of numbers and points in this plane is knownas 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 howthey work. If we plot A, whose x-coordinate is 2 and whose y-coordinate is 3, this firstnumber tells us to move right 2 units; so let's walk 1 unit and then another 2 units.Now, from there, let's walk three units up. So three units up, we got up one unit, wegot 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 theleft; now five up. One unit, two units and three units – four units and five units - that'sB. C tells us to go four units to the right. One, two, three, and four, and then half aunit down.And remember, these instructions can really be any real numbers, so we could'vegone 4.1, 4.2, and so on. Here we want (4, -5), and D, walk 5 units to the left, 2 3 45, 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.
Let's formally define the x-axis. The x-axis is a line in the Cartesian plane thatcontains all points whose y values are zero. So, obviously, this is the x axis. Andsimilarly, let's formally define the y-axis. The y-axis is also a line in the Cartesianplane that contains all points whose x values are zero. In other words, if you start atthe origin, then go either right or left until you reach zero (in either direction), then goup from there, then you'll be on the y-axis (at some point). For instance, one point onthe y-axis could be (0,1). One point on the x-axis could be (-5, 0). The Cartesian plane is divided into four quadrants, which are defined by thecoordinate axes. The first quadrant consists of points (x,y) whose x-coordinate ispositive 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.
Well, let's give a real world example. So over here, let's draw our plane again. A wayof plotting a table of data, where each object or person corresponds to two differentnumbers. Consider the example of three people, each with a height measured incentimeters 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'velooked up on the internet is 177 cm and 88.8 kg. If we plot A over here, we have tofigure out a scale. But if we go all the way over here and say that's about 177, andup 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, 164cm, which is currently the standard in America. The kilograms would be 74.7. Youwill notice that the average American woman is shorter and weighs less than theaverage American male; therefore, point B is going to be down this way—about hereand about here—and there's B. We see this visually, as the fact that the woman hasto reach downwards and to the left means that the average American woman isshorter and lighter than the average American man.
Now notice that if we move along this horizontal line, what are we doing? If we takeany other person who sits on this line, that person will have the same averageweight. If we move up, that person will get heavier; if we move down, that person willget lighter. You can probably figure out where to put yourself on this. The same istrue as we move from the average American female (represented by the box on theleft) and move say along this horizontal line: what would we be doing? Any personalong that line would have the same weight as the average American female. If wemove left, it should be getting shorter; if we move right, it should be getting taller."
Plotting Points on the Cartesian Plane: A Step-by-Step Guide
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