The Coordinate Plane
成绩单
Coordinate geometry. One of the most elegant ideas in all of mathematics is the idea of
坐标平面。其他名称包括x-y平面、直角坐标平面和
the Cartesian plane, and that final name is in honor of the person who discovered it, the French mathematician Rene Descartes.
笛卡尔的绝妙想法始于简单地把两条数线彼此成直角。 So of course, we know a number line has whole numbers on it. It has fractions. It has decimals. And it goes on forever in both the positive direction and the negative direction.
And so what we have here really are just two number lines crossing. The horizontal number line is called the x-axis. The vertical number line is called the y-axis. And of course, each one of them goes on forever. Each one of them contains positive whole numbers, negative whole numbers, positive fractions and decimals, negative fractions and decimals, the whole nine yards.
轴相交的点，在每个轴上为零，叫做原点，它被认为是整个平面的中心。 当然，这允许我们通过点的x和y坐标来指示平面上任何点的位置。 So for example, we look at this particular point. This particular point is, the point is vertically above x equals five, so the x coordinate has to be five.
It is on the same horizontal line as y equals 4, so its hori, so its y-coordinate is 4, and its position is given by 5,4. That is the ordered pair that denotes the exact position of that point. As you may remember, 5,4 is an ordered pair, with an x-coordinate followed by a y-coordinate, so they are in alphabetical order. First the x-coordinate then the y-coordinate.
Every one of the infinite number of points in the plane can be indicated by a unique ordered pair. So that's amazing fact number one. You could go to any position in the plane, an infinite number of points in the plane, every single one will have a unique ordered pair, a unique x-y coordinate denoting its exact location.
On the test, given an ordered pair, you need to be able to locate that point, and given a point, you need, picture of a point, you need to be able to figure out what the coordinates for that point are. So that is an absolutely essential skill. Here's a very simple practice problem. Pause the video and then we'll talk about this.
Okay, so this is actually much easier than anything you'll see on the test. It may be that this would be part of another problem on the test. But we want to know what are the coordinates of this point. Well, first of all, notice that we're to the left of the y axis, we're on the left side of the x-y plane. And so this would be where that horizontal number line is negative.
And so the x-axis, because we're to the left of zero, we're in the negative part of that axis so this is gonna have a negative x-coordinate. 所以我们再也忍t backwards one, two, three, four, five, six, seven, and then we count up, one, two, three, four. So that means that the x-coordinate is negative seven, the y-coordinate is positive four, and the coordinates of that point are negative 7,4.
That is the unique ordered pair which gives the exact location of that point. 轴将整个平面划分为四个区域，称为象限。这些象限从右上角顺时针表示为I， II, III, and IV. And they're almost always denoted with four Roman numerals like this.
如果我们知道一个点的象限，我们立刻知道x坐标和y坐标的正负号。 So for example, in the first quadrant, both coordinates are positive. 在第二个qu，象限中，x是负的，但是y是正的。在第三象限，两个坐标都是po， 在那一点上是阴性的。
They're, everything is negative in the third quadrant. In the fourth quadrant, the x's are positive, but the y's are negative. It's also important to note, that any point that is exactly on the x-axis, or exactly on the y-axis, or certainly the origin, these are not in any of the four quadrants. So the four quadrants are only for points that are off the axes.
Here's a practice problem. Pause the video and then we'll talk about this. 好吧，这个问题可能会出现在测试中，因为， it's a little less straightforward and requires a little bit of visualization. Point M is the midpoint of segment AB.
If A equals 2, negative 3 and M is on the negative x-axis, in what quadrant is B? So let's visualize this. We have A here, we don't know where M is, but M is going to be on the negative x-axis, the negative x axis here. So let's just pick a random point.
We could even pick one relatively close to the origin right there. So if A goes to M, well, then B would have to be up here. And it turns out, no matter where we put M on that axis, we could move it back and forth, B is always going to wind up in the second quadrant. So really, the answer to this question is quadrant number II. In summary, you need to know the terms, origin, x-axis, y-axis, x-coordinate, and y-coordinate.
这些是测试将使用的术语，您需要能够识别它们并知道它们的含义。 重要的是要认识到，平面上的每一个点，平面上的无穷多个点，每一个点都可以用一个唯一的有序对来表示， a unique set of x and y coordinates. And finally, the axes divide the plane into four quadrants, the quadrants of a point determines the positive and negative signs of its x and y coordinates, and the test likes to ask about quadrants.
Read full transcript
笛卡尔的绝妙想法始于简单地把两条数线彼此成直角。 So of course, we know a number line has whole numbers on it. It has fractions. It has decimals. And it goes on forever in both the positive direction and the negative direction.
And so what we have here really are just two number lines crossing. The horizontal number line is called the x-axis. The vertical number line is called the y-axis. And of course, each one of them goes on forever. Each one of them contains positive whole numbers, negative whole numbers, positive fractions and decimals, negative fractions and decimals, the whole nine yards.
轴相交的点，在每个轴上为零，叫做原点，它被认为是整个平面的中心。 当然，这允许我们通过点的x和y坐标来指示平面上任何点的位置。 So for example, we look at this particular point. This particular point is, the point is vertically above x equals five, so the x coordinate has to be five.
It is on the same horizontal line as y equals 4, so its hori, so its y-coordinate is 4, and its position is given by 5,4. That is the ordered pair that denotes the exact position of that point. As you may remember, 5,4 is an ordered pair, with an x-coordinate followed by a y-coordinate, so they are in alphabetical order. First the x-coordinate then the y-coordinate.
Every one of the infinite number of points in the plane can be indicated by a unique ordered pair. So that's amazing fact number one. You could go to any position in the plane, an infinite number of points in the plane, every single one will have a unique ordered pair, a unique x-y coordinate denoting its exact location.
On the test, given an ordered pair, you need to be able to locate that point, and given a point, you need, picture of a point, you need to be able to figure out what the coordinates for that point are. So that is an absolutely essential skill. Here's a very simple practice problem. Pause the video and then we'll talk about this.
Okay, so this is actually much easier than anything you'll see on the test. It may be that this would be part of another problem on the test. But we want to know what are the coordinates of this point. Well, first of all, notice that we're to the left of the y axis, we're on the left side of the x-y plane. And so this would be where that horizontal number line is negative.
And so the x-axis, because we're to the left of zero, we're in the negative part of that axis so this is gonna have a negative x-coordinate. 所以我们再也忍t backwards one, two, three, four, five, six, seven, and then we count up, one, two, three, four. So that means that the x-coordinate is negative seven, the y-coordinate is positive four, and the coordinates of that point are negative 7,4.
That is the unique ordered pair which gives the exact location of that point. 轴将整个平面划分为四个区域，称为象限。这些象限从右上角顺时针表示为I， II, III, and IV. And they're almost always denoted with four Roman numerals like this.
如果我们知道一个点的象限，我们立刻知道x坐标和y坐标的正负号。 So for example, in the first quadrant, both coordinates are positive. 在第二个qu，象限中，x是负的，但是y是正的。在第三象限，两个坐标都是po， 在那一点上是阴性的。
They're, everything is negative in the third quadrant. In the fourth quadrant, the x's are positive, but the y's are negative. It's also important to note, that any point that is exactly on the x-axis, or exactly on the y-axis, or certainly the origin, these are not in any of the four quadrants. So the four quadrants are only for points that are off the axes.
Here's a practice problem. Pause the video and then we'll talk about this. 好吧，这个问题可能会出现在测试中，因为， it's a little less straightforward and requires a little bit of visualization. Point M is the midpoint of segment AB.
If A equals 2, negative 3 and M is on the negative x-axis, in what quadrant is B? So let's visualize this. We have A here, we don't know where M is, but M is going to be on the negative x-axis, the negative x axis here. So let's just pick a random point.
We could even pick one relatively close to the origin right there. So if A goes to M, well, then B would have to be up here. And it turns out, no matter where we put M on that axis, we could move it back and forth, B is always going to wind up in the second quadrant. So really, the answer to this question is quadrant number II. In summary, you need to know the terms, origin, x-axis, y-axis, x-coordinate, and y-coordinate.
这些是测试将使用的术语，您需要能够识别它们并知道它们的含义。 重要的是要认识到，平面上的每一个点，平面上的无穷多个点，每一个点都可以用一个唯一的有序对来表示， a unique set of x and y coordinates. And finally, the axes divide the plane into four quadrants, the quadrants of a point determines the positive and negative signs of its x and y coordinates, and the test likes to ask about quadrants.
