Geometry Coordinate Plane

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This is the Geometry course; the very first lesson is on the coordinate plane, which should be somewhat of a review.0002

So, make sure to check out the other free lessons of the syllabus.0008

Let's begin: the coordinate system: the coordinate plane is part of the coordinate system.0011

This here is called the coordinate plane; right here, this is the x-axis, and this is the y-axis.0020

And these two make up the four quadrants of the coordinate plane.0034

If we were to label these, this is I, II, III, and so on; we know that if we go this way, we are going to be going negative; positive and negative.0042

Now, these axes make up four quadrants; the four sections of the coordinate plane are known as quadrants.0063

Here, this is the first quadrant, so this is quadrant I; around this side, we have quadrant II; this is quadrant III, and quadrant IV.0073

So, it starts here, and it goes this way: I, II, III, IV.0085

And for quadrant I, we have a positive; we are only dealing with the positive x-axis and the positive y-axis.0091

For quadrant II, we have a negative x-axis, and then the positive y-axis.0100

For quadrant III, it is negative x and negative y; quadrant IV is positive x and negative y.0104

Those are quadrants; make sure that you remember that there are four of them.0115

The origin is right there: this is known as the origin.0120

The origin is (0,0): the x is 0, and the y is 0--right where they meet, that is the origin.0128

And this is an example of an ordered pair: an ordered pair is when you have the x-coordinate paired with a y-coordinate.0138

And together, it is called an ordered pair.0157

So, if I have a point, (4,2), this would be an ordered pair; my x-coordinate is before, and then 2 would be my y-coordinate.0161

So, let's practice graphing using the coordinate plane: we are going to look for these points and write the coordinates.0177

Here is A, B, and C; for point A, we know that this is 0; this is x; this is y; here is 1, 2, 3, -1, -2, -3;0188

so for point A, we always start with the x-axis first; so the x-coordinate goes first, and that is 1;0211

and then what is my y? 1; that is my ordered pair.0224

For B: my x-coordinate for point B is -1, and my y is going to be -2; so here is (-1,-2).0230

And for C, it is 2 for my x and -1 for my y.0246

We are going to graph each of the points on the coordinate plane.0258

For A, I have (4,2); this is my x and this is my y; this is my x and this is my y.0280

So, I am going to go to positive 4 on this side: 1, 2, 3, 4; and 2 on my y is +2, which is there; (4,2) is going to be right there.0289

I am going to label that point A.0305

For B, (-3,0): x is -3, which is 1, 2, -3...and my y is 0; that means I do not go up or down anything--I stay right there.0309

OK, this is where y is 0; and this part right here is going to be B.0322

C: it will be 1, 2, and 1 right there for C; and then D is (-4,-2); OK.0329

OK, there are all my points on the coordinate plane.0354

One more thing to go over: collinear points are points that lie on the same line.0359

When we have points that line up--you can draw a line through those points--those points will be collinear.0366

And let's see if we have any collinear points here.0377

Well, if you remember from algebra, for slope, we have rise over run;0380

all of those have to do with a line and points, when we are graphing lines on the coordinate plane.0387

So, here we have A and C--we know that those two, or any two, points will be collinear,0393

because you can draw a line through any two points.0401

Here, I know that those three points, A, C, and D, are collinear, because they will be on the same line.0406

They lie on the same line, so A, C, and D are collinear.0419

If you want to double-check this, you can use what you learned from algebra; you can count rise over run.0425

Find your slope from D to C, and then from C to A; and it should be the same, and also from C to A and D to A.0433

OK, so points A, C, and D are collinear.0441

Write the coordinates and quadrants for each point: let's look at point A.0455

Point A: this is -1, and my y is -1, -2, and -3; so for A, -1 is my x-coordinate, and -3 is my y-coordinate.0464

And this is quadrant III, because it goes I, II, and III; so this is in quadrant III.0484

For B: my x is +1, and my y is +2; and that is in quadrant I.0494

C is 3, and my y is -1; and that is quadrant IV; D is -3, and 3; quadrant II.0507

OK, let's do another example: Name two points in each of the four quadrants.0530

OK, we have quadrant I; now quadrant I, I know, is here; quadrant II, quadrant III, and quadrant IV.0536

Quadrant I is going to be positive, and then my y-coordinate is going to be positive.0549

Quadrant II: my x (x is always first), we know, is negative; and then y is positive.0558

Quadrant III is negative for the x-coordinate and negative for my y.0568

Quadrant IV is positive and negative.0575

They should all be different; their signs will be different for each of the quadrants.0578

So, I can name any point; as long as my x-coordinates and my y-coordinates are both positive, they are going to be from Quadrant I.0583

I can just say (1,2) and then maybe (3,4); those are two points from Quadrant I.0592

Quadrant II will be...we have to have a negative x-coordinate and a positive y-coordinate, so what about (-1,2) and (-3,4).0602

Now, you can use your own numbers; you can use the same numbers.0619

As long as you have a negative x and a positive y, they are from Quadrant II.0625

Quadrant III: x and y are both negative, so (-1,-2) and (-3,-4) will be from Quadrant III.0630

And then, Quadrant IV: we have a positive x and a negative y, so (1,2) and (3,-4).0642

Those are two points from each of the four quadrants.0655

The next example: Graph each point on the same coordinate plane.0659

Let me do these: the first one, point A, is (0,3).0663

Now, be careful--this 0 is my x; that means, on my x-axis, I am going to be at 0, which is right there.0685

And then, for my y (I'll just write out a few of these numbers: 1, 2, 3, 4, 1, 2, 3...OK, let me erase that...-3 and -4; OK)...0698

again, it is 0 for my x, and then 3 on my y; so there is 3 on my y.0725

And that is going to be my point A.0735

For point B, I'll go to -2 on my x and -1 on my y; there is B.0740

C is -5; there is -5 on my x and 0 on my y; that means I am not going to move up or down; I am going to stay there; there is C.0748

And then, D will be 4, and then -6 is all the way down here; so there is point D.0758

And my final example: Point A is (3,1) and B is (0,-5); they both lie on the graph y = 2x - 5.0771

Determine whether each point is collinear with points A and B.0781

OK, if I have my coordinate plane, my x- and my y-axis, my point A is going to be (3,1); there is A.0785

B is going to be (0,-5), right there; there is B.0802

They both lie on the graph y = 2x - 5; so if I draw a line through these points, that is going to be the line for this equation of y = 2x - 5.0812

And you are just going to determine whether each point is collinear with the points A and B.0832

Now, "collinear" means that they are going to be on the same line.0837

So, we are just going to see if these three points (since we know that points A and B are on this line) are going to also be on the line.0841

And if they are, then they will be collinear with the points.0852

For point C, instead of graphing the line and seeing if the point lies on the line, you can just...0858

since you know that this is x and this is your y, you can just plug it into the equation and see if it works.0866

y = 2x - 5: you are just going to plug in -1 for x and 4 for y.0874

So, 4 = 2(-1) - 5: here, this is 4 = -2 - 5; do we know that...since we don't know that these are equal...does 4 equal -7?0880

No, it does not; so this point does not lie on this line; that means that point C is not collinear--this says no.0906

OK, for point D, I am going to also plug in: 7 is my y; 7 = 2(6) - 5.0918

OK, I am going to put a question mark over my equals sign, just because I am not sure if it does yet--I can't see if it equals.0940

This is 12 - 5; 7 = 7, so this is a yes--they are collinear points.0947

And then, my last point, point E: -15 = 2(-5) - 5): put a question mark again.0958

-10 - 5...-15 does equal -15, so this is also a yes; OK.0972

Points D and E are collinear with points A and B, since they are all on the line y = 2x - 5.0984

That is it for this lesson; thank you for watching Educator.com!0998