Algebra 1 Slope & Graphing

Welcome back to www.educator.com.0000

In this lesson we will be taking a look at more about the slopes of lines and how we can use that to better graph.0002

Specifically some of the information that we went up looking at is how we can first determine the slope of line0012

whether we are just given a couple of points or whether we have the entire graph of that line.0017

Once we know more about slope we will be able to learn how to graph the entire thing using its slope and its y-intercept.0026

This will bring about many different forms that you can represent a line.0032

We will learn about switching back and forth between these many forms.0035

We will learn about some very special lines, the ones that are vertical and horizontal.0040

Look for those equations so you can easily recognize them in the future.0043

When it comes to a line, there is actually many ways that you can go about graphing or representing that line.0051

You can write it in standard form, slope intercept form and point slope form.0056

For now I will be mainly concerned with these first 2 forms, the standard form and the slope intercept form.0062

We will get more into the point slope form in another lesson.0068

When it comes to standard form, it looks a lot like this ax + by = c.0074

The way you want to recognize a standard form if you ever come across it, is that both of your x’s and y’s will be on the same side of the equation.0082

We like to usually put them on the left side.0089

There are no fractions or decimals present in the equation so the a, b, and c, those are nice whole numbers in there.0093

The x term here will be positive that is the a value.0102

We do not want to be a -6 or -7, we usually like to be 3, a nice positive value.0106

The reason why this form of the line will be so important is that if you have a line in standard form, it usually easy to graph.0114

The way we go about graphing something in standard form is we use its intercepts.0122

That is where it crosses the x and y axis.0127

The reason why the intercepts are so nice for our standard form is because when it does cross one of those axis, one of the values either x or y will be 0.0130

It will be making a table, but it would not be that big.0140

Let us do a quick example of something in standard form to see how easy it is to graph.0145

In this one I have 7x + 2y = 14.0151

You can see that it is in a standard form because I have both of my x’s and y’s on the same side.0154

I do not see any fractions, no decimals and the coefficient of x here is positive, a nice 7.0160

In order to graph this, I will find its x and y intercepts.0169

I will make my chart here rather than picking a lot of different points, I will look at where x = 0 and where y = 0.0174

Watch what this does to the equation.0184

If I use 0 for x, then it is going to get rid of the entire x term.0187

Since 7 × 0 right here, all of that is just going to go away.0196

I just have to solve the nice simple equation 2y = 14 which I can do by dividing both sides by 2.0201

I know that one of my intercepts, my y-intercept is at 0, 7, nice and simple.0212

Let us do the same thing by putting in a 0 for y.0219

7x + 2 × 0 = 14.0222

You will see that 0 is in there and again it is going to get rid of this term entirely since 2 × 0 is 0.0229

Then I have 7x = 14, divide both sides by 7 and we will get x = 2.0237

Now that I have both of these points, we will put them on our graph.0249

0-7,0, 1, 2, 3, 4, 5, 6, 7 way up here and the other 1, 2, 0 over here.0253

Connect those intercepts and I will get the graph of the entire line.0273

One disadvantage with using just the intercepts to graph a line is if you make a mistake on one of them, it is often hard to catch.0277

If you want to get around that problem, it might not be a bad idea to actually put in an additional point to see what happens with the graph.0284

That way if you do make a mistake with one of them you be able to quickly see that they are not all in a straight line.0292

To understand some of the other forms like slope intercept form, you have to understand a lot more about slope.0301

What exactly is this slope?0307

If I had to describe it, it is a measure of the steepness of a line, how steep is a line, if it is shallows or more steep?0310

Can you attach a number to that steepness?0319

What we do and we call it the slope.0322

In many equations we will use the letter m to represent that slope.0324

How do we attach a number to the steepness?0329

What we will do is we take 2 points from the line and we end up looking at the difference in the y values over the difference in the x values.0331

This gives us a nice equation for figuring out the slope of a line.0340

You will see these whole numbers in here and you can interpret that as each of these points.0345

These x’s and y’s come from point number 1 and this other one, these values come from point number 2.0351

In our work later, it is often a good idea to label one of your points as .2, and one of this is .1, just we do not mix up things in the slope formula.0360

You may have also heard of other ways to describe slope.0373

One of the most common is to call slope the rise of the line divided by the run of the line or simply rise over run.0376

That is actually a good way to remember how it is related to its steepness.0384

Its change in the rise would be the y direction and this change in the run would be the x direction.0388

One thing that may throw you off is that sometimes there is a negative sign in the slope.0401

You can interpret that negative line sign as whether you are going up in your rise or down, or do you need to go left or right in your run.0405

Let me go ahead and pick this is apart.0414

If you see a positive sign in the top then think of going up on your rise.0417

If you see a negative sign then think of going down.0424

With the run if it is positive, you will end up going right and if it is negative, then go left.0431

To make it a little bit more sense once we start seeing some more lines.0439

Maybe I can give me a quick example right here.0445

A line like this, I can say that maybe the rise is 5 and run over here is 4 so the slope will be equal 5/4.0450

If I had a different line like this, I can see them going down and then I have to go right.0466

Since I’m going down, I will mark this one as -3 maybe this one over here as 5, then I will have a slope of -3/5.0477

In practice 1, if I do have a negative sign in my slope, I usually just give it to the top part of the fraction.0488

That way I only have to remember about going up and down.0492

I do not even have to worry about left and right since the bottom is positive.0496

Another word of warning, be careful not to mix up all your points.0503

If you have your y values from point 2 being first then take the x values from point 2 being first as well.0507

I also use the sign of this to give you a little bit more intuition as to which direction that line should be facing.0516

There are only a few instances of what your line can look like.0523

It could be going from the lower left to the upper right and that is an indication that your slope is going to be positive,0527

since both your rise and your run are going to be positive numbers.0534

If your line is going from the upper left to the lower right and you are going to have a negative slope.0543

This will be because your rise is negative, but your run is positive.0551

The other 2 special cases that you have to come and watch out for is what happens when your slope is 00557

and what happens when it is undefined.0562

In these 2 instances, you either have the alignment as completely horizontal.0565

This is when you have a 0 slope and it is completely up and down if you have an undefined slope.0570

Let us get into our examples and see how we can start finding our slope just from a couple of points.0580

With these ones we will use the formula for the slope of the line to pick it apart.0587

I’m going to try and keep things a little bit together by marking these out as point number 2 and I will mark the other one out as point number 1.0593

In the formula, here is what we are looking at.0610

Our slope should take our y values and subtract them all over our x values and subtract those.0613

Notice how I’m keeping things all lined up, I have both of my x and y from point 2 over here and both my x and y from point 1over here.0622

Let us put in some values.0636

I need the y value from point number 2, that is a 4 then we will minus our y value from point number 1 that is -1.0638

All over x value from point number 2, -2 and x value from point 1, -1.0654

Notice I’m subtracting these even though they already have a negative sign in them.0668

Be very careful on your signs with that.0672

Looking at the top, 4 - -1 is the same as 4 +1 = 5.0675

-2 - -1is the same as -2 + 1= -1.0683

It looks like this slope for the first one 5/-1 I will just get a slope of -5 between these 2 points.0690

It does not matter which one you will label as point number 2 or point number 1 just as long as you keep them straight.0702

I’m going to switch which ones I’m calling point number 2 and point number 1 just to highlight this, but you will get the same either way.0712

Let us start off, I want to subtract my y2 from my y1.0721

Y2 is 6 – y1 6, my x2 is -5 and my x1 is 3.0727

On the top of this, I have 6 - 6 giving us 0.0752

On the bottom I have -5 - 3 - 8 and 0 divide by anything is 0.0757

This indicates that our slope is 0.0764

This is one of our special cases where we have a horizontal line, it is completely horizontal.0767

Let us do one more, point number 2 and point number 1.0772

The slope for this I will take my y value from the second point I will subtract the y value from the first one.0789

Then to our x’s, x from our second point minus x from our first point.0804

Now we will work to simplify, 5 - -4 is the same as 5 + 4 =9, 3 – 3 =0.0814

Be very careful with this one, we can not divide by 0.0826

Whatever that mean, it will get us around the bottom.0839

This is indicating that our slope is undefined.0841

It is not that there is not a line there, there is a line.0845

The line is just completely straight up and- own, it is our vertical line.0847

Just to make this a little more clear, I will say slope is undefined or sometimes we might say that there is no slope.0851

In order to know a lot more about slope, we will get a little bit more into the slope intercept form.0868

Slope intercept form looks like this, y = mx + b.0875

The way you can recognize this form is that the y will be completely alone on one side of the equation.0880

Usually I like to put it on the left side.0886

Our slope will usually be represented by that m and we will put it right next to the x.0889

The b in this equation stands for our y-intercept of the line.0896

That is where it crosses the y axis on our Cartesian coordinate system.0900

The reason why that this form is usually everyone's favorite is because it makes graphing a nice and simple process.0906

The way it makes a graphing so nice for us, is we start at the y intercept.0913

Just rely from the graph by whatever that d value is.0919

What we do is we use the slope as directions on how to get to another point on our graph.0922

Let me give you a real quick example of how this would work in practice.0929

I have 1/7x + 3.0934

The very first thing that I would do is I would look over here at the y intercept and I would take its value.0945

I know that this particular line crosses the y axis 3 and I would put a point on the y axis right at 3.0952

Starting at that y-intercept, I would use the slope as directions to get to another point.0967

Keep in mind that it is the same as rise over run.0976

Starting at that y-intercept, I will go up one into the right 7, up 1 to the right 1, 2, 3, 4, 5, 6, 7.0980

Now that I have 2 points, I can go ahead and connect them and make the entire graph of this equation.0995

One you start with the y-intercept and two you use the slope as directions to get to a second point.1006

Let us see this in action by actually graphing out some linear equations.1016

They said before let us start here on the nth with our y-intercept and make that our first point that we put on the graph.1022

This one is -5, it crosses the y axis down here at -5, it looks good.1029

In terms of our slope, we want to think of this as rise over run so starting at that y intercept, we go up 3 and to the right 4.1038

1, 2, 3 to the right, 1, 2, 3, 4 and now we have a second point.1050

Now that we have 2 points, simply connect them to a nice solid line and there is your entire graph.1063

It makes the graphing process much easier.1073

You do not even have to worry about the table and doing all of those values.1075

Let us try this one.1082

Graph the line using the y intercept and the slope.1083

On the back in here, I see that my y-intercept is 3 and that will be the first point on my graph.1087

My slope is -2, how will that work with rise over run?1099

It does not look like a fraction like in some of my other examples.1107

It does not feel free to turn it into a fraction by simply putting it over 1.1111

This tells me that I need to go down 2 since that is negative and to the right 1, down -2 and right 1, now I have a second point right there.1116

I can draw the entire graph, very nice.1132

We have both of these forms under our standard form and our slope intercept form,1142

you may be curious which one should you be using most of the time.1152

Both of them are good for graphing.1159

And what I often recommend is if you have to graph something and it is already in standard form, just go ahead and use the intercept.1163

It is usually one of the quickest ways to do it.1169

If it is already in slope intercept form then use its y intercept and slope as a direction to the second point and graph it that way.1171

That is usually the quickest.1178

If you do have to switch back and forth between these 2 maybe you are more comfortable with slope intercept form, then feel free to do so.1179

If you have something that is in some other form and you want to get in the slope intercept form then the process is pretty quick.1188

What you should do is simply solve for y and get it all by itself on one side of the equation.1195

In doing so, you will be able to better see what it slope is and the y-intercept.1200

Not very many people go the other direction, but potentially you could end up rewriting something into standard form.1206

There is a lot more criteria that go in there.1212

One of the first things is you should get both of your x’s and y’s on the same side.1215

Then you want to make sure that your constant is on the other side.1221

Only x’s and y’s on one, constant on the other.1226

Try and clear out your fractions by multiplying by a common denominator.1231

a, b, and c should not be fractions.1233

Then look at the coefficient in front of x and it should be positive.1237

If that is not positive, then multiply it by -1 and make it positive.1243

More practice switching back and forth so we can see how this process works.1250

Even though we have our standard form and our slope intercept form.1259

You want to be aware that there are 2 special cases and we seen them come up but once before.1262

We have some lines that are completely vertical and some that are horizontal.1268

The vertical ones are straight up and down, and the horizontal ones are left and right.1275

The way you can recognize their equations are they are simply x equals a number or y equals a number.1281

You may see something like x = 2, maybe this is like y = 15.1289

For the one it says x equals these are your vertical lines.1296

For the one that says y equals these are your horizontal lines.1304

The way that they can keep in track of which one should be horizontal and which one should be vertical is the way you interpret them.1312

If you have an equation like x = 2, what that is trying to tell you is that the x value no matter where you are in that line is always 2.1318

If my line looks something like that and I decide to pick up some individual points,1330

no matter what point I pick out I can be sure that the x value will be 2, no matter where I am on that line.1335

In a similar fashion, if I’m looking at y = 15, no matter where I am on that line I should end up with the y value being 15.1346

Watch for these 2 special cases to come up in my examples.1364

Let us first work on switching back and forth between these 2 different forms.1371

What I have here is a line in standard form and we want to put it into slope intercept form and we want to put it into slope intercept form.1375

I want to actually go through the process of graphing it and I’m more familiar with slope intercept.1383

When I already did take this and put it in to that other form, I need to solve for y.1388

Let us start by moving the 7x to the other side, 2y = -7x + 14.1395

And then we will divide everything by 2 and that should get our y completely by itself.1407

Notice on the right side there, I have to divide both of those terms by 2, -7/2 x + 7.1416

The most important part about writing it in this new form is now I can easily identify what my y intercept is.1427

It looks like it 7 and I can more easily identify what my slope is.1435

It has a slope of -7/2 and I know it is facing down from the upper left to the lower right.1441

Let us go to the other direction.1450

Let us take a line that is written in slope intercept form and put it into standard form.1452

It requires a little bit more work but we can do it.1458

The first thing I’m going to do is try and work to get my x’s and y’s on the same side.1461

I will subtract 1/2x on both sides, -1/2x + y = 3.1466

I want to make sure that my constant is on the other side, that is the 3 and it is already there.1481

I want to get rid of all fractions so I need to get rid of that ½.1487

I can do this if I multiply both sides of the equation by a common denominator and in this case that would be 2.1492

-x + 2y = 6, we are almost there.1500

You can see that it certainly look a lot more like that standard.1507

The last thing we need to make sure is that the coefficient in front of x is positive.1510

Right now looks like it is negative, I already fixed that.1515

I will multiply everything through by -1.1518

-1 × -x would be x and -1 × 2y = -2y, equals -6.1528

This form is in standard form.1544

They might be looking at it and say what good is that? Why would you want it in standard form?1547

Remember that you can graph it in standard form now by simply looking at its intercepts plugging in at 0 for x and 0 for y.1551

Let us get into some very special cases.1560

We want to graph the line y =5.1563

There is not much of the equation to look at.1568

What should be the slope? What should be the y intercept?1570

This is one of our special cases, y equals a number.1574

Since it is y over here, this is going to help me indicate that this is going to be a horizontal line.1578

1, 2, 3, 4, 5 would be one point.1584

I will just make a giant horizontal line with all points y = 5 and double checking this make sense.1588

If I was to pick a point on a line at random, in this case it is 1, 2, 3, 4, 5, 6 its y value is 5.1596

If I pick something over here, its y value is still 5 no matter where you go on this line its y value will always be 5.1606

One last special case, this is x = -2.1618

This will be a vertical line straight up and down because we are dealing with x over here.1623

I’m at x = -2 and we will make it straight up and down.1631

The reason why this makes sense is no matter what point you choose on line, as a way up here at 1, 2, 3, 4, 5, 6, 7 the x value will always be 2.1641

With the slope of this one, remember that its slope is undefined.1663

There you have it some have very nice techniques you can use for graphing lines and now 2 forms that you can use to represent lines.1668

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