### Slope & Graphing

- The slope of a line is a measure of its steepness. It can be calculated by looking at the change in the rise over the run of the line.
- A horizontal line (y = a) has a slope of zero, and a vertical line (x = a) has an undefined slope.
- When the equation of a line is written in slope-intercept form (y = mx + b), m stands for slope, and b stands for the y-intercept. This form is especially handy for graphing lines.
- When the equation of a line is written in standard from (Ax + By = C), you can substitute zero in for y and x and quickly find the x and y intercepts. This form is also very good for graphing.
- You can switch back and forth between slope-intercept and standard form, by re-writing the equation.

### Slope & Graphing

- slope = [(y
_{2}− y_{1})/(x_{2}− x_{1})] - slope = [( − 3 − ( − 6))/(10 − ( − 2))]
- slope = [3/12]

- slope = [(y
_{2}− y_{1})/(x_{2}− x_{1})] - slope = [( − 7 − ( − 9))/( − 11 − 15)]

- slope = [(y
_{2}− y_{1})/(x_{2}− x_{1})] - slope = [( − 3 − 4)/( − 12 − ( − 8))]
- slope = [( − 7)/( − 4)]

- slope = [(y
_{2}− y_{1})/(x_{2}− x_{1})] - − [3/2] = [( − 4 − ( − 6))/( − 13 − x)]
- − [3/2] = [2/( − 13 − x)]
- 2(2) = − 3( − 13 − x)
- 4 = 39 + 3x
- 4 − 39 = 39 + 3x − 39
- − 35 = 3x

- slope = [(y
_{2}− y_{1})/(x_{2}− x_{1})] - 2 = [(5 − y)/( − 2 − 4)]
- 2 = [(5 − y)/( − 6)]
- 2( − 6) = 5 − y
- − 12 = 5 − y
- − 12 − 5 = 5 − y − 5
- − 17 = − y

- slope = [(y
_{2}− y_{1})/(x_{2}− x_{1})] - − [1/2] = [( − 1 − y)/(3 − ( − 6))]
- − [1/2] = [( − 1 − y)/9]
- − 1(9) = 2( − 1 − y)
- − 9 = − 2 − 2y
- − 9 + 2 = − 2 − 2y + 2
- − 7 = − 2y

- Use two points along the line to find the slope
- (Points can vary along the line)

_{2}− y

_{1})/(x

_{2}− x

_{1})] = [(3 − 0)/(1 − 0)] = 3

- Use two points along the line to find the slope
- (Points can vary along the line)

_{2}− y

_{1})/(x

_{2}− x

_{1})] = [(3 − 0)/(0 − ( − 18 ))] = − [1/6]

- Use two points along the line to find the slope
- (Points can vary along the line)

_{2}− y

_{1})/(x

_{2}− x

_{1})] = [(3 − 0)/(0 − ( − [3/4] ))] = 4

- Use two points along the line to find the slope
- (Points can vary along the line)

_{2}− y

_{1})/(x

_{2}− x

_{1})] = [(25 − 0)/(0 − ( − 75))] = [1/3]

- y = mx + b

slope = [(y_{2}− y_{1})/(x_{2}− x_{1})] - slope = [(8 − 4)/(0 − 5)] = [4/( − 5)]

- y = mx + b

slope = [(y_{2}− y_{1})/(x_{2}− x_{1})] - slope = [( − 6 − 6)/(0 − ( − 3))]
- slope = [( − 12)/3]
- slope = − 4

- y = mx + b
- m = − 4

b = ? - − 8 = − 4( − 10) + b
- − 8 = 40 + b
- b = − 48

- y = mx + b
- m = − 1

b = ? - 15 = − 1(7) + b
- 15 = − 7 + b
- 22 = b
- y = − 1x + 22

- slope = m = [(y
_{2}− y_{1})/(x_{2}− x_{1})] - m = [(6 − 1)/(0 − 4)]
- m = [5/( − 4)]
- y = mx + b

- y = mx + b
- m = − 7

b = ? - 11 = − 7( − 8) + b
- 11 = 56 + b
- − 45 = b

- m = [(y
_{2}− y_{1})/(x_{2}− x_{1})] - m = [( − 2 − ( − 3))/(4 − ( − 2))]
- m = [1/6]
- y = mx + b
- − 3 = [1/6]( − 2) + b
- − 3 = − [2/6] + b
- − 2[4/6] = b

- m = [(y
_{2}− y_{1})/(x_{2}− x_{1})] - m = [(6 − ( − 4))/( − 1 − 5)]
- m = − [10/6] = − [5/3]
- 6 = − [5/3]( − 1) + b
- 6 = 1[2/3] + b

- Identify slope
- m = 3
- Identify intercept
- b = − 5

- Identify slope
- m = [1/2]
- Identify intercept
- b = 3

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Slope & Graphing

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Objectives 0:11
- Slope and Graphing 0:48
- Standard Form
- Example 1 2:24
- Slope and Graphing Cont. 4:58
- Slope, m
- Slope is Rise over Run
- Don't Mix Up the Coordinates
- Example 2 9:39
- Slope and Graphing Cont. 14:26
- Slope-Intercept Form
- Example 3 16:55
- Example 4 18:00
- Slope and Graphing Cont. 19:00
- Rewriting an Equation in Slope-Intercept Form
- Rewriting an Equation in Standard Form
- Slopes of Vertical & Horizontal Lines
- Example 5 22:49
- Example 6 24:09
- Example 7 25:59
- Example 8 26:57

### Algebra 1 Online Course

### Transcription: 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 line*0012

*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 xâ€™s and yâ€™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 xâ€™s and yâ€™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 Ã— 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 Ã— 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 Ã— 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 xâ€™s and yâ€™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 Iâ€™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 0 *0557

*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

*Iâ€™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 Iâ€™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 Iâ€™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

*Iâ€™m going to switch which ones Iâ€™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 â€“ 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 xâ€™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 â€“ 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 xâ€™s and yâ€™s on the same side.*1215

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

*Only xâ€™s and yâ€™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 Iâ€™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 Iâ€™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 Iâ€™m going to do is try and work to get my xâ€™s and yâ€™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 Â½.*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 Ã— -x would be x and -1 Ã— 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

*Iâ€™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

*Thanks for watching www.educator.com.*1676

1 answer

Last reply by: Professor Eric Smith

Fri Aug 26, 2016 6:55 PM

Post by Kevin Zhang on July 17 at 11:18:21 PM

Professor eric,

On example 8, you said that the x value will always be 2, i think you meant -2.

Otherwise great lesson!

1 answer

Last reply by: Professor Eric Smith

Fri Aug 26, 2016 6:57 PM

Post by Kevin Zhang on July 17 at 11:01:43 PM

Professor Eric,

I think on the slide about standard form, you wrote in positive but inteded to write is positive.

Otherwise, I love your lessons!

Thanks

1 answer

Last reply by: Professor Eric Smith

Mon Mar 2, 2015 8:14 PM

Post by Arvind Ganesh on March 2, 2015

Nice Lecture Sir! It helped me study for my test! AND I GOT 100!!! Thank you so much!

1 answer

Last reply by: Professor Eric Smith

Sat Jan 3, 2015 9:06 PM

Post by Ardeshir Badr on January 3, 2015

if x value has more than one possible y, we no longer have a function - am I wrong?

1 answer

Last reply by: Professor Eric Smith

Mon Jun 16, 2014 3:12 PM

Post by patrick guerin on June 13, 2014

Oh, I see. When I reviewed the lesson, I found that you said that the x term is the only one that can't be negative. Thanks again for the video!

2 answers

Last reply by: Professor Eric Smith

Wed Jan 6, 2016 1:47 PM

Post by patrick guerin on June 13, 2014

On example 6 I got a little confused. I thought that you said there couldn't be a negative on standard form, but you ended up with a negative 6. Why is that okay?