Three ways to graph a line

let's go and explore all the different ways that we can graph a line now we're going to be graphing a line on what we call the cartisian coordinate grid in that case we're going to have a y- axis as well as an x- axis now I'm assuming that you have a basic idea of how these number lines are kind of orientated right we have the x-axis going along horizontally we have the positive direction going to the right negative Direction going to the left and then the y- axis is basically just the exact same number line but flip vertically where going up is going to be positive and going down is going to be negative so when we're first learning how to graph a line one of the first things that we learn how to do is use point slope form and what we're going to Simply do is go ahead and identify where the point is on the coordinate grid and then we're going to use the slope to find our next point now a lot of times slope gets confusing when we have a negative because if you write the negative like most textbooks it's not going to be in the numerator or it's not going to be in the denominator but guess what it doesn't matter where it's at now when we're graphing a line all we need is two points but for the sake of this video and for this instruction I'm I'm going to go ahead and find at least a couple points especially at least for the first two first thing you want to do is be able to identify the point where exactly does that land because if you don't know how to graph a single point then you might need to check out another video or just make sure you can practice and understand that so we label our points As an X and a y coordinate so the first number tells you where to go on the x- axis so therefore we have a positive one so I'm going to go one unit to the right the y-coordinate is going to tell you where you're going on the y- axis since that's a NE 1 I'm going to go down one unit now one thing we're doing is we're creating one single point so you don't go plot a point here and then plot a point on each axis right because that' be two points so what you need to do when you're plotting point is you need to take into consideration the X and the y-coordinate together so I'm going to go over one unit on the X and then down one unit on the Y that is going to be my point all right so we got one point now we need to find at least another point to be able to create our line so then we looked to our slope now here's the big thing that I was tell my students if you have a negative 2 3 that is the same thing as a -2 / 3 which is the exact same thing as a pos2 / a -3 but it is not the same thing as a -2 over a -3 and just use Simple numbers like think of this instead of 2/3 let's think about is as a 8 over 4 right so if I say -8 over 4 we know that answer is -2 if I said8 / 2 that's -4 if I said POS 8 / -2 that's -4 however 8 divided by -2 that is equal to a positive4 right because the negatives are going to divide out so work this way or make sure you can use either one of these options doesn't really matter now the difference between these two is how we understand what our slope is the slope is going to be the relationship between any two points on our coordinate grid is basically telling you how to navigate from one point to the next so a lot of students remember this as like the rise over the run or the change in the Y over the change in the X however your instructor however makes sense to you that's perfectly fine I like to use like the change in the y- coordinates over the change in the x coordinates so what that means is let's go and look at this example first so when you have a point slope form you're always going to start from the point that you have so here's our point this is our starting point so let's go and use this example first it's saying the change in the y-coordinates between any two points on our graph is going to be -2 well again what does the -2 for our change in our y that means we're going to be going down right CU on the Y AIS if up is positive down is negative so what I'm going to do is I'm going to go down two units now you don't have to create this slope triangle but I think when you're first learning I always like using and understanding this so I'm going to go down two units from my initial point then the change in the x coordinates or what we call the run is going to be a positive three so what that means is on the x coordinate positive means to the right so now I'm going to go over from the slope triangle 1 2 3 that is going to be my next point I don't know why different colors that's going to be where's my red there we go postive 3 so you can see from one point to the next -2 over three and now I can go ahead and connect my line going to the right hand side now you can continue this line going to the left but yes let's use the next definition the next definition says the change in the rise or run is positive two so now what I'm going to do is I'm going to go up two and then it says the change in the run or the change in the x coordinat is going to be neg3 so let's go to left three 1 2 3 and that' be my next point over there which is right here and if I wanted to connect what do you know it follows this exact same line and again you can continue this pattern right you can continue the slope triangle if you need but you only need ladies and gentlemen two points to be able to create a line I just wanted to go a little bit extra to help you out with that now let's get into slope intercept now slope intercept is a preferred method for graphing especially when you have a line that's already in slope intercept form slope intercept form is basically y = a mx + b where B represents your Y intercept that is going to be where the graph crosses the Y AIS and and M is going to represent your slope which is your change in your rise over your run so let's go ahead and set up our X and Y axis here and let's go ahead and get started with graphing on this one now the difference is here's the point right they gave you a point here and you plotted them on the XY grid the reason why this one's important is because two represents that Y intercept which technically is really just saying our point is going to be a 0 comma 2 right if it lies on the Y AIS then we know that the x value is going to be zero and the Y value is going to be a two so now I'm just going to go up to this point zero on my y AIS there's my x axis and I'm just going to go to two now I'm going to go ahead and follow my slope now a lot of students get confused with slope intercept because they Negative they don't know it should go in the top or the bottom they get it confused and they put in both or whatever they just make a mistake the next way you can make a mistake using slope is when you just have an integer remember guys the integer can always be Rewritten over one right cuz anything divided by one is just anything whatever it originally was so if I want to find the slope I can see the change in the y or the rise is going to be three so I can go up three units 1 two three and then my change in my run is going to be a positive 1 so it's going to be 3 over to the one therefore that is going to be in the positive direction and then I know just like we did over here guys remember 3 over 1 is the same thing as a -3 over Nega 1 right so I could go in positive direction but if I wanted to find a point in the negative Direction I would just go down three 1 2 3 and to the left one and voila that is how you apply with slope intercept form now the last one is one that I don't love to use all the time but there is definitely times that we want to be able to use the intercept method and especially when you have an equation that is in what we call our standard form which is ax plus b y is equal to C and you could obviously take it in standard form and rewrite it in slope intercept form and then graph it in this way completely a useful strategy to be able to graph a line however when you notice that the coefficients of A and B evenly divided to your C sometimes if you just need to identify like the orientation of the graph or where the intercepts are intercept method is very very important so how can we understand or use the intercept method well the important thing is understanding what exactly are the intercepts remember slope intercept form says when this value two that was The Intercept what did we know about the point the x value was Zero the Y value was two so if I want to find the Y intercept one thing I know about the Y intercept is that point lies on my y AIS and also my x value is going to be zero so to find the Y intercept just replace x with zero so getting a little lost on space but Y is going to equal four so I know that coordinate point here is going to be a0 comma four so let's go to here and then if the Y intercept is when X is equal to zero well what do you think the x intercept is going to be the x intercept you guessed it is going to be when Y is equal to zero so now I can just go and plot that point on the x-axis