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The Coordinate Plane
Coordinate geometry. One of the most elegant ideas in all of mathematics is the idea of the coordinate plane. Other names include the x-y plane, the rectangular coordinate plane, and the Cartesian plane, and that final name is in honor of the person who discovered it, the French mathematician Rene Descartes.
Descartes’ brilliant idea began by simply putting two number lines at right angles to each other. 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.
The point where the axes cross, zero on each axis, is called the origin, and that’s considered the center of the entire plane. Of course, this allows us to indicate the position of any point on the plane by the x and y coordinate of the point. 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. So we count 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. The axes divide the entire plane into four regions known as quadrants. These quadrants are denoted, clockwise from the upper right, as I, II, III, and IV. And they’re almost always denoted with four Roman numerals like this.
If we know the quadrant of a point, we immediately know the positive or negative sign of both the x-coordinate and the y-coordinate. So for example, in the first quadrant, both coordinates are positive. In the second qu, quadrant, the x’s are negative, but the y’s are positive. In the third quadrant, both the coordinates are po, are negative at that point.
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. Okay, this is a problem that actually could appear on the test, because it’s, 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.
Those are terms the test will use, and you need to be able to recognize them and know what they mean. It’s important to appreciate that every single point in the plane, infinite number of points in the plane, every single one can be noted by a unique ordered pair, 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.
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