y=mx+b | Graphing Lines in Slope-Intercept Form

Welcome to Math with Mr. J. [music] In this video, I'm going to cover how to graph lines in slope intercept form. We will go through eight example problems together and then I'll give you some practice problems that you can try on your own. Remember, slope intercept form is a way to write the equation of a line. So, we can write and graph equations of lines using this form. We have slope intercept form right here. y equ= mx + b with m being the slope and b being the y intercept. Remember slope measures the steepness of a line and the y intercept is where the line crosses the yaxis. The slope and y intercept give us all the information we need to graph a line. Let's jump into number one where we have y = 2x + 3. So we need to graph this line. Our slope is 2. So the slope equals 2 and then our y intercept is 3. So the y intercept equals 3. Now, with our slope being a whole number here, I'm going to write it in fractional form because remember slope equals rise over run. And writing this in fractional form will make our rise and run more clear. All we need to do here is put 2 over 1. And now we are in fractional form. So two is the rise and one is the run. Now we graph and we start with the y intercept which is three. So again when graphing we start with the y intercept and that's because we know exactly where that point is. Then we use the slope to go from there. We need to plot a point at three on the y ais. So at 03 that's right here. And that's our y intercept. Now we go from here using our slope which is 2 or 2 over 1. So rise of two positive two. So we go up two units and then a run of one a positive one. So we go to the right one unit. So up two and over one. And that's our second point on the line. Now we can use that slope to plot as many points as we would like. But once we have two, we can draw our line through the two points. And we are done. Our line is graphed. And since we have a positive slope here, notice the line is going upward as it moves from left to right. Let's move on to number two. Taking a look at number two, we have y = 1/4 x - 6. Our slope here is 1/4 and our y intercept is -6. So we have a slope of 1/4 and a y intercept of -6. Now you may be thinking why don't we have plus b here plus our y intercept. That's because this equation has been simplified. So we don't have a double sign. So we don't have plus a negative. Meaning this can be rewritten as y equ= 1/4 x + -6. And that does make the negative a little more clear. We can clearly see that our y intercept is -6. But equations will a lot of times be simplified and just show that as subtraction. So again, we don't have that double sign of the addition sign and then the negative. Subtracting six gives us the same thing as adding -6. So both of these equations will work. So this is something to be aware of when it comes to negative y intercepts. And you can always rewrite the equation using addition if you prefer that. Now let's graph and we'll start with the y intercept which is -6. So we need a point right here at 06. Now we use our slope 1/4. So rise of one and run of four. Positive 1 and positive4. So we go up one to the right four. And our second point on the line is right here. And now we draw our line through the two points. And we are done. Our line is graphed. Let's move on to number three. Moving on to number three. Here we have y = -3x + 5. Our slope here is -3 and our y intercept is 5. So a slope of -3 and a y intercept of five. Now as far as our slope, let's write that in fractional form by putting it over one. So our rise is -3 and our run is one, a positive one. Now let's graph and we'll start with the y intercept at five. So we need a point at 05 which is right here. Now we go from that y intercept using our slope. So rise of -3. Since that's a ne3 we go down three units. And then we have a run of one, a positive one. So we go to the right one unit. So down three to the right one. And here is our second point on the line. Now we can draw our line through those points. And that's it. That's our line for number three. Now keep in mind since we have a negative slope here, our line is going downward as it moves from left to right. Let's move on to number four. Lastly, let's take a look at number four here where we have y = -54 x - 1. So, our slope is -54s and our y intercept is -1. So, our slope here is -54s. And I'm writing that negative sign in the numerator, the top number. That's going to help us make things more clear when it comes to the rise and the run. So we have a rise of -5 here and a run of four. Then our y intercept is -1. Now we can graph. So starting with our y intercept of -1 here we need a point on the y ais at -1. So we go to 01 which is right here. And then we go from that y intercept using our slope. So we have -54s -5 over4. Since we have a rise of -5, we need to go down five units. And then we have a run of four, a positive four. So we go to the right four units. So down five to the right four. And here is our second point. Now we can draw our line through those two points. And that's it. That's our line for number four. And we have another example here with a negative slope. So the line is going downward as it moves from left to right. Let's move on to number five. Here's number five. And for this one, we have a line with a slope of zero. For our example here, we have y = 7, which for slope intercept form, you may be thinking we should have something that looks like y = mx + b. We should have y equ= our slope next to the x and then our y intercept. But here we just have y = 7. So how do we graph this and what do we use for the slope? Well, let me rewrite this to make this a little more clear. We actually have y equals 0x + 7. Now we can see that our slope is 0 and our y intercept is 7. So our slope is zero and our y intercept is 7. Since we have a slope of 0, we have 0x. Multiplying any number by 0 always gives us zero. So no matter what x is here, we get zero. So basically this is just zero and not changing the value of the equation at all. So we can simplify the equation and write our slope intercept form without 0x. That means we can just write this as y = 7. Now, can we write our slope intercept form equation with 0x to show that slope of zero? Sure. Use whatever works best for you. But just be aware of what this means when we have something like this example. Now, let's graph. We start with the y intercept, which is at 7. So we plot a point at 07 which is just at 7 on the y ais. So right here. Now we go from that y intercept using our slope. Well the slope is zero. So what's our rise and run here? Well a slope of zero gives us a horizontal line. But thinking of that rise and run, we can write zero in fractional form rise over run by putting zero over any nonzero number that still gives us that slope of zero. So for example 0 equ= 0 over and let's put 0 over one. So a rise of zero and a run of one. So is that still equal to that slope of 0? Yes. 0 over 1 0 / 1 equals 0. So going from our y intercept rise of 0. So we don't go up or down and then a run of one a positive one. So we go one unit to the right and we have another point. Again, we can put zero over any nonzero number because that will still be equal to that slope of zero. So for another example, we can put 0 over -1. So rise of 0, run of -1. Now, we don't need to plot any more points here, but let's go from the y intercept using the rise of zero and run of -1. Rise of zero, so we don't go up or down. And then the run of -1 means we go to the left one unit. So we have another point right here. What about 0 over three? Would that work? Well, 0 over three. So rise of zero, run of three. That still equals that slope of zero as well. So let's do one more point here. Rise of zero. So we don't go up or down. and then a run of three, a positive three. So we go to the right and we can go from any point on our line using that. But let's go from the y intercept. So we have another point right here. Now again this can be any nonzero number meaning we can't use zero because a denominator of 0 dividing by 0 is undefined. So that won't work here. But any other number is going to give us a point on this horizontal line. Now we can draw our line. And that's it. We can see we have a horizontal line that crosses the y-axis at 7. Whatever the x coordinate is here, the ycoordinate is 7. Therefore, we have that horizontal line. Let's move on to number six. So here's number six. And for this one, we have a line with a y intercept of zero. Now, for our example, we have y equals -2x. So, where's our y intercept here? Do we not have one? Now, it's not that we don't have a y intercept. Our y intercept is actually zero. This equation has been simplified. So, plus 0 adding 0 is not shown. That's because adding zero doesn't change the value of something. It's not changing the value of that side of the equation. So, it's not necessary to write it. So, we can write our slope intercept form here as just y = - 1/2x. Now, if we want to make the y intercept more clear, we can always rewrite this with + 0. So y = -2x + 0. So we have our slope of -12 and then our y intercept of 0. So whenever you see something like this example here, the y intercept is zero. And feel free to rewrite the equation with plus 0 to help see what that y intercept is. So our slope again is - one/2 and then our y intercept is zero. Now you'll notice I wrote that negative in the numerator the top number that helps us when it comes to the rise and the run. So we need to put that negative in the numerator or denominator. Just don't put it in both the numerator and the denominator because a negative over a negative a negative / a negative gives us a positive. Now let's graph. We will start with our y intercept which is at 0. So we need a point at 0 0 which is right here. And now we go from the y intercept using our slope which is one2. So our rise here over our run. Our rise is -1. So we go down one unit. And then our run is two. A positive two. So we go to the right two units. So down one to the right two. And our second point is right here. And now all we need to do is draw a line through our two points. And that's it. Our line is graphed. We can see that it crosses the y ais at zero. And keep in mind since we have that negative slope, our line is going down as it moves from left to right. Let's move on to number seven. So here's number seven. And for this one, we have a line with a slope of 1. For our example, we have y = x + 2. So, we need to figure out our slope and y intercept here. Now, we're going to start with the slope. And this is where it's common for people to get held up because we just have an x here. We don't see any slope in front of it. Normally, we have m x. So, the slope and then x. But again, we don't have any slope in front of our x here. That's because there's an implied one in front of that x. In other words, it's understood that there is just one of that x when we see it by itself with no coefficient, no number in front. So again, this is one x. And we can even rewrite this with the one if we would like. So y = 1 x + 2. So our slope is 1 and our y intercept is two. Now let's graph and we'll start with the y intercept. We need to plot a point at 2 on the y ais. So at 02 which that is right here. Now we need to go from that y intercept using our slope which is 1. Now in terms of rise over run, let's write one in fractional form. That way the rise and run are more clear. So let's rewrite one in fractional form by putting one over one. One is our rise and one is our run. Using that slope we go from our y intercept now. So rise of positive one means we go up one unit and then a run of one of positive one means we go to the right one unit. So up one over to the right one. And here is our second point. Once we have those two points we can draw our line through the points. And that's it. That's our line. Let's move on to number eight. Lastly, here's number eight. And for this one, we have a line with a slope of -1. For our example, we have y = -x - 4. So, we need to figure out our slope and y intercept here. Now, we'll start with the slope. And this is where it's common for people to get held up because we just have ax. we don't see any slope, any number in front of it. Well, when we just see an x with no number written in front, there's an implied one in front of that x. In other words, it's understood that there's just one of that x. And in this case, we have ax. So, this is actually -1x. And we can even rewrite this with the -1 in front which we will do here in a second. So our slope is -1 and then our y intercept is -4. Now you may be thinking shouldn't we have + b plus the y intercept? Well instead of writing plus -4 so adding -4 the equation has been simplified. So we don't have a double sign here. Addition and then the negative. We can just write this as subtraction. Now we do have the option of rewriting this with addition. That can help make the y intercept a little more clear. So let's rewrite this as y = 1x + -4. So we included that -1 and then adding -4 and subtracting four give us the same thing. So either will work here again slope of -1 and y intercept of -4. Now for our slope let's actually write this in fractional form. That way we can see our rise and run. And all we need to do here is put -1 over 1. So -1 over 1. And now we can clearly see that our rise is -1 and our run is pos1. Now let's graph. Starting with the y intercept, we need to plot a point at -4 on the y ais. So at 0 -4, which is right here. Now we need to go from that y intercept using our slope. So rise of -1 that means we need to go down one unit and then our run is pos1 which means we go to the right one unit. So down one and then to the right one. Our second point is right here. Once we have those two points, we can draw our line through the points. And that's our line. And you'll notice since we have a negative slope here, the line is moving downward as it's moving from left to right. So now that we've gone through some example problems together, it's time for you to try some practice problems on your own. So have paper and a pencil ready. Here are your four practice problems. Graph each line. And for these, just draw the x and y axis the best you can. You can go off of what I drew. I went up to positive five and down to negative five on both the x and y axis. Then you can graph each line. I'll give you 5 minutes and then we will go over the answers. Feel free to pause the video if you need more time or jump ahead to the answers whenever you are ready. Go ahead and start. Are you Heat. Heat. Number Okay, so that was five minutes. Let's go over the answers. So here are our answers. For number one, our y intercept is -2. So we should have a point there. And then our slope is five. So we can go up five units and then to the right one unit for our second point. For number two, the y intercept is one. So we should have a point there on the y ais. And then our slope is - one2. So we can go down one unit and then to the right two units for our second point. Moving on to number three. Our y intercept is -4. So we should have a point at -4 on the y ais. And then our slope is one. So up one unit and then to the right one unit for our second point. And lastly for number four, our y intercept is zero. So we should have a point at zero on the y ais. And then the slope is 3/4s. So we can go up three units and then to the right four units for our second point. So there you have it. There's how to graph a line in slope intercept form. I hope that helped. Thanks so much for watching. Until next time. Peace.