Slope Intercept Form | Graphing and Writing Equations

Welcome to Math with Mr. J. In this video, I'm going to cover how to graph a line in slope intercept form and I'm going to cover how to write an equation in slope intercept form from a graph. We will start with graphing and then move on to writing equations. We'll go through four examples of each. 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 three. 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, positive2. 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 -3 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. So there's how to graph a line in slope intercept form. Let's move on to writing equations. Here's our section on how to write an equation in slope intercept form from a graph. Now, all we need to do here is find the slope and y intercept. Then, we plug those in to our equation for m and b, and we have our equation. The slope and y intercept give us all the information we need for the equation of a line. So looking at number one, let's start by finding the y intercept. Then we will find another point on the line and determine what the slope is. And for this example, you can see the two points that we are going to use. They've already been marked. I plotted points there. So for the y intercept, where does this line cross the y ais? Well, it looks like this line crosses the y ais right here at four. So the coordinates are 04. So our y intercept is four. Now we find our slope here by going from one point on the line to another point on the line. And we're going to work from left to right here. So we're going to start at that y intercept and then work our way to that other point. Remember slope equals rise over run. So we need to determine how far up or down. And then how far over we need to go? Well, we need to go up one unit, two units in order to align with the other point. That's our rise. And since we went up, that's a positive two. So, our rise is two. And then we need to go to the right one unit, 2 units, three units to get to our other point. So our run is three, a positive three since we went to the right. So rise of two over our run of three. So our slope equals 2/3. Now we plug 2/3 in for m and 4 for b. So, our equation for this line is y = 2/3 x + 4. And that's it. We're done. That's our equation. Let's move on to number two. Taking a look at number two here, let's find our slope and y intercept. Now for number one, our line was going upward as it moved from left to right. It had a positive slope. Now for number two, we can see that the line is moving downward as it moves from left to right. This line has a negative slope. Let's start with the y intercept and we will go from there. This line crosses the y ais at 1. So our y intercept here is one. Now, let's use another point on the line, which I have one marked right here, and we are going to figure out the slope. So, we'll work our way from the left point to the right. So, let's start with the rise. We need to go down one unit two units in order to align with the other point. And since we went down, this is going to be a two. Our rise is -2. Then we need to go over one unit to the right to get to our other point. So our run is one and a positive one since we went to the right. So our slope here is rise of -2 over our run of 1 which we can write this as just -2. So we can simplify that fraction to -2. Now all we need to do is write our equation. So we plug in -2 for m and 1 for b. So y = -2x + 1. And we're done. That's our equation for that line. Let's move on to number three. Moving on to number three. Here we have a line with a positive slope. It's moving upward as it moves from left to right. Now, as far as this example, I don't have the points that we're going to use clearly marked on the line. I don't have the points plotted like I did for numbers 1 and two. But we're going to do the exact same thing here. We just need to find those points. Our y intercept is down here at -5. So, let's use this point. And then we need to find another point on the line that we can easily read that we can easily use to find the slope. We need to look for something at a grid intersection. We want a point where we can clearly see what the coordinates are. A point where we can clearly find the rise and run. So let's use this point right here. And now let's find our slope. We'll start with the rise. We need to go up one unit, two units, three units, four units, five units, six units. And since we went up, that is a positive. So our rise is six. And then our run, well, we need to go over one unit. And that's to the right, so it's positive. Our run is 1. So our slope here we have a rise of six over a run of one. So 6 over 1 which can be simplified to just six. And then our y intercept is -5. Now we can write our equation. So we plug in six for m and -5 for b. So y = 6x + -5. Now is this equation correct? Yes, it is correct. But we can simplify it here. So we don't have that double sign of addition and a negative. We can rewrite adding a negative as just subtraction. So we can simplify this to y equ= 6x - 5. Adding -5 and subtracting five give us the same thing. So either of these equations will work. Now a lot of times we will see slope intercept form simplified to show a negative y intercept as subtraction. So this is something to be aware of. we have this equation right here where we are adding -5 or we can simplify that and rewrite it as subtraction. So again just something to keep in mind when we have negative y intercepts. Let's move on to number four. Lastly taking a look at number four here we have a line with a negative slope. And as far as the y intercept that's going to be right here at -2. Now, we need to look for another point on our line here so we can find the slope. Let's use this point right here. Now, let's find the slope. And we're going to work from the left point to the right point. So, starting with the rise, we need to go down one unit, 2 units, 3 units, four units, five units. And since we went down, that's negative. So our rise is -5. And then as far as our run here, we need to go to the right one unit, 2 units, 3 units, four units. And since we went to the right, that's positive. So our run is four. Our slope is -5 over 4. So - 54s. And then our y intercept is -2. So let's write our equation here. y = 54s x plus -2. Now just like we talked about with number three, we can simplify this. So let's rewrite this using subtraction. So we can simplify this equation and rewrite it as 54s x - 2. So here's our equation with adding -2 or we can simplify that and rewrite it with subtraction. So there you have it. There's how to graph and write equations in slope intercept form. I hope that helped. Thanks so much for watching. Until next time, peace.