Algebra 1 Rectangular Coordinate System

Welcome back to www.educator.com.0000

In this lesson we are going to take a look at the rectangular coordinates system.0002

It will help us out as we get into more of the graphing process with some of equations later.0006

Some of the specific things that we will look at as we are looking at our Cartesian coordinate system are just some of the parts of it.0012

We will learn all about the x axis, y axis, origin, quadrants and how we can plot an ordered pair.0018

When we get into plotting more things such as the graph we will look at looking at a table to do that graph.0026

And again how you can find the x and y intercepts of that line once on the graph.0032

The Cartesian coordinate system is formed by taking 2 number lines and putting them together after 0.0041

You will notice that one is in the up and down direction and one is completely horizontal.0047

Now the horizontal one, the one that goes completely flat, this guy right here, we will call this one our x axis.0053

The one that go straight up and down that will be known as our y axis.0064

The 0s of each of them when it both connects, we will also give that one a special name, we will call this point right here the origin.0071

The little tick marks that you see usually represent how far you have to go on each of these number lines.0087

You can see that it breaks down into these number lines into so much larger parts of this graph.0095

We will give each of these squares a name, we will call them quadrants.0103

We usually number these quadrants starting in the upper right corner, say quadrant 1 and move in a counterclockwise direction.0110

This entire space here would be quadrant 2 but I have quadrant 3 and quadrant 4.0118

Now in each of these quadrants we can plot a point.0126

A point is an ordered pair and we look at it as a pair of the x and y put together.0131

If I'm looking at a point say out here, I have identify what it is x and y value are0138

by looking at how may tick marks I have to go over on the x axis and this would be 2.0148

How may tick marks I have to go in the y direction, 1, 2, 3, 4, I can plot out that point.0154

Keep in mind that with all of these points that you see, the first one is the x and the second one is y.0163

Let us just do a real quick example of how we know little bit more about the parts of the graph and see if we can plot out some of these ordered pairs.0177

In all of these, the first one will represent an x value and the second will represent our y value.0185

We will start at the origin with each of these and moving the x and then the y direction.0191

Starting with first one, we need to plot 2, 4.0198

I’m going to start at the origin and I'm going to move 2 units to the right and 4 units up.0201

This point right here will be the point 2, 4.0213

The next point has a negative value in it and it is a negative in the x.0222

I actually move left 1, 2, 3 and then I will go up 7, that point is located right here, 4, -3, 7.0227

As long as you can keep track of which one was x and which ones what is y, not a bad task at all.0244

The next one is -1 and -5, at we will move in the x direction -1 and then down 5.0251

1, 2, 3, 4, 5, so there is our point -1, -5.0258

Just a couple of more to do, the next one is 3, -2, that is in the positive direction 1, 2, 3 and then down 1, 2.0269

And one last one to go, this one is 0, 4.0286

What should you do if 0, should you go right or should you go left?0293

Since it is at 0, we do not go right or left in the x direction, we actually stay at the origin and then we will move in the y direction 4.0297

Up 1, 2, 3, 4 that point is right here 0, 4.0304

Now that we have identified some points on here, we can actually say which quadrants each of these are in.0313

If we look at our first point up here, our 2, 4 it is in quadrant number 1 because it is in that first square.0319

The next point -3, 7 is in quadrant number 2.0331

-1, -5 quadrant number 3.0338

3, -2 that one is in quadrant number 4.0345

The last point 0, 4 as the tricky one is not in quadrant 1, it is not in quadrant 2, it is actually right on the y axis.0351

Some more things about this rectangular coordinate system, when we get into actually trying to graph an equation on here0370

we can look at it as a special relation between its x values and its y values.0377

There are a lot of good things we can say about that equation.0383

To actually visually see what that relationship is and of course put it on the graph,0386

we will go ahead and create a table of values and see how they are related.0390

Usually this is done by picking a lot of different values for one of the variables and see what the other corresponding other values need to be.0395

Here are very important terms you want to keep in mind.0402

Where the graph of that equation crosses the x axis that will be known as its x intercept.0404

In a similar fashion, if it crosses the y axis, we will call that the y intercept.0415

Just draw a general graph so this is not a line since it is all curvy but it still crosses the x axis in some spot so I will call this the x intercept over there.0422

It still crosses the y axis so it has a y intercept.0440

You want to be able to identify both of these points.0446

A little bit more about generating those points from the table.0453

I said that a line is a special relationship between its x values and its y values and to see that relationship,0457

we want to be able to create a little chart and put in some values for either x or y.0463

Let us look at one I have 5x - 3y =12.0476

If I want to go ahead and try and visualize what this equation looks like, I need to know what values go on my graph.0486

What point should be there?0493

A good way to figure out if the point is on the graph or not, is to see if it simply satisfies the equation.0495

If I’m just picking some point like 3, 1, I can substitute the 3 in for x and 1 for y and see if it does satisfy it.0502

And see if 3, 1 is on the graph or not.0513

(5 × 3) – (3 × 1) does that equal 12, let us find out.0516

That will be 15 - 3 = 12 or 12 = 12.0524

What that tells me is that 3, 1 is one of the points on my graph, 1, 2, 3 on the x axis up one.0532

Let us pick another point, 0, -4.0546

But when I plugged this one in, I get 5 × 0 which is 0 - 3 × - 4 = 12.0554

I know that one is on the graph as well, 0, -1, 2, 3, 4.0570

In practice, we usually do not pick up points out of thin air and then test to see if they work or not.0583

Usually we end up picking a lot of different things for one variable then see what it has to be for the other variable.0588

Let us get a little bit of space in here and see how we would do this more in practice rather than just picking things out of thin air.0595

Let us pick out some more values for x.0603

For example what if x was 2, what with that 4y to be, we can find out by putting in the 2 for x and actually solving for y.0606

That will be 10 - 3y = 12, I will subtract the 10 from both sides and get y = -2/3.0620

I know sure enough that is another point that I can put on my graph, 2 in the x direction then down 2/3.0632

What you may notice is I start to do more and more points over here on the graph.0641

You will get a better sense of what the entire graph looks like.0645

I have done about 3 points here but I’m going to draw lines for all of them to say that if I had even more points they would all line up along there.0650

Let us get into some more examples and see what other parts of the graph we can identify0665

and get a little bit more into the nuts and bolts of using that chart to graph out an entire line.0669

In this example we just simply want to find the x intercepts and the y intercept of the line.0676

Remember, this is where it crosses the x axis and where crosses the y axis.0681

My x axis is horizontal and I can see it crosses right at that point.0686

That would be at -1, 2, 3, 4, 5 in the x direction and 0 in the y direction.0691

x intercepts is at -5, 0 it crosses the y axis as well right up here at 4.0701

For that one we would not go in the x direction whatsoever but we would go out 4, so its y intercept is at 0, 4.0717

With these ones we want to determine whether the point is on the line or not.0735

We can figure this out by substituting the values in for x and y.0739

The first one will always be an x and the next one a y.0746

-2 + 3y = 21, let us put in the -3 for x and 5 for y.0753

-2 × -3 = 6, 3 × 5 =15, for a total of 21, 6 + 15 = 21.0768

It looks like this one checks out since it makes the equation true, we can say that -3, 5 is on the line.0783

Let us do the same thing for the next one, my x is 1 and my y is 7.0794

Let see if this one rings true.0811

Plugging in 1 for x, 7 for y and simplify 3 + 4 is 7 and 5 × 7 is 35, these things are definitely not equal.0815

What is that tells us that 1, 7 is not on the line.0836

You can take a point, plug it into the equation and see if it satisfies it and makes it true.0845

Let us get into more of the graphing process.0852

Here is an entire equation and I want to know what its entire graph looks like.0854

Rather than just picking points out of thin air and testing them, we are going to try and create a few values of our own by generating them.0858

We will generate them using a nice little table.0867

I will pick some values for x and we will see what makes y.0870

To make this processing a little bit easier, we will take our equation and we will solve for y first.0875

Our equation is 2x + 3y =12, to solve this for y I would move the 2x to the other side by subtracting a 2x and then divide it by 3, I will get -2/3x + 4.0881

This is the same equation, I just manipulated it a little bit so I can work with it a little bit easier.0905

We are going to pick some values for x, plug them in and see what they make y.0912

It does not matter what values you pick but I do recommend choosing some negative values or may be throw in 0 and also choose some positive values.0918

Let us choose -3, -2, -1, 0, 1, 2 and 3.0927

With each of these, imagine what happens if you take -3 plug it in into the equation, what result do you get.0938

If you do some scratch work that is okay, y = -2/3 we multiply that by -3, add 4 to it and see what the result is.0944

-2/3 × -3 that would give us 2, 2 + 4 =6.0955

You put that on the other side, I know that -3, 6 is one of the points on our graph.0966

-3, 1, 2, 3, 4, 5, 6 there is a point right up there.0972

Let us take up the -3 and try this again, this time we will plug in -2.0982

This would give us 4/3 + 4 which should be the same as 5 and 1/3.0991

It is okay if you get fractions with these just the way it turns out since you are choosing all kinds of different values for x.1001

-2 up 1, 2, 3, 4, 5 and just a third more there is that point right there.1009

Let us take out the -2 and put in -1 and let us see if this works out.1019

negative × negative would be a positive, 2/3 + 4 = 4 2/3.1032

-1 up 1, 2, 3, 4 and 2/3 right above there.1043

I have not even graphed all of the points yet and you can see that they all seem to be following along in a straight line.1049

We will do one more then will go ahead and connect the dots.1058

Let us see what happens if x =0, 0 × -2/3 would be 0, 0 + 4 = 4.1066

We will put that point on there as well right up here at 4 and now we will just connect all of our dots and this would be our graph of the equation.1078

Using this table we can definitely see a lot of points even sometimes we do not need to graph all the points,1094

we should graph enough of them that you have a good sense of what it looks like.1100

Also if your graphing a line and you are going through this table and you got one that is a little off from all the rest, it is okay.1104

But if it is off from all the rest then we will check your work on now to make sure that you have done it correctly.1110

In this next one, we want to graph the equation again using a table of values.1121

Like last time, I will go ahead and solve this for y so we have a little bit easier of the time working with it.1126

I will move that 5x together just by adding 5x to both sides then I will go ahead and divide by 2.1133

This is the equation that I will be using it is the same as the original.1149

I have just manipulated it a little bit so I will have an easier time working with it.1153

Let us pick some values to put in there for x so we can see what y needs to be.1158

We will pick some nice ones like -4, -2, 0, 2 and 4.1165

This one I’m picking values that are multiples of 2 since all of them we have to multiply by 5/2 first.1172

That will make our lives a little bit easier.1179

y =5/2, let us plug back in -4 then we will add 10.1184

-4 × 5/2 =-10 when we add 10 to that we get 0 so our first point is -4, 0.1194

Let us try another one, this one is -2, I will plug that one in there -2 × 5/2 =-5.1212

I will add 10 to that and get a total of 5 so this point -2, 5.1229

What happens if we go ahead and put in a 0, 0 × 5/2 = 0 + 10 = 10.1250

I have another point for my chart here and this one is off the chart 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 way up here.1263

We can already see that after doing a bunch of these, the rest of my graph is going to go completely off of my graph paper.1275

We will go ahead and do the other 2 just so you can see what their value will be but we already have plenty of points so we go ahead and graph these line for 1, 2.1286

If I put in 2 I will have 5 +10 = 15.1293

If we put in 4, 4 × 5/2 = 10 + 10 =20.1305

Now we have plenty of points, let us go ahead and graph this out.1325

I will go ahead and connect the dots.1328

The graph of this equation looks like this and with this one we have taken it definitely farther and mark out its x intercept.1332

This one is at -4, 0 and its y intercept way up here will be at 0, 10 and see they show up in the chart here and here when one of the variables is 0.1342

We will look at more next time with being able to do some shortcuts for graphing these lines1362

But with many types of equations you can often use a table to generate some values and put on some points.1367

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