Video transcript
We are asked to graph y is equal to 1/3x minus 2. Now, whenever you see an equation in this form, this is called slope-intercept form. And the general way of writing it is y is equal to mx plus b, where m is the slope. And here in this case, m is equal to 1/3-- so let me write that down-- m is equal to 1/3, and b is the y-intercept. So in this case, b is equal to negative 2. And you know that b is the y-intercept, because we know that the y-intercept occurs when x is equal to 0. So if x is equal to 0 in either of these situations, this term just becomes 0 and y will be equal to b. So that's what we mean by b is the y-intercept. So whenever you look at an equation in this form, it's actually fairly straightforward to graph this line. b is the y-intercept. In this case it is negative 2, so that means that this line must intersect the y-axis at y is equal to negative 2, so it's this point right here. Negative 1, negative 2, this is the point 0, negative 2. If you don't believe me, there's nothing magical about this, try evaluating or try solving for y when x is equal to 0. When x is equal to 0, this term cancels out and you're just left with y is equal to negative 2. So that's the y-intercept right there. Now, this 1/3 tells us the slope of the line. How much do we change in y for any change in x? So this tells us that 1/3, so that right there, is the slope. So it tells us that 1/3 is equal to the change in y over the change in x. Or another way to think about it, if x changes by 3, then y would change by 1. So let me graph that. So we know that this point is on the graph, that's the y-intercept. The slope tells us that if x changes by 3-- so let me go 3 three to the right, 1, 2, 3-- that y will change by 1. So this must also be a point on the graph. And we could keep doing that. If x changes by 3, y changes by 1. If x goes down by 3, y will go down by 1. If x goes down by 6, y will go down by 2. It's that same ratio, so 1, 2, 3, 4, 5, 6, 1, 2. And you can see all of these points are on the line, and the line is the graph of this equation up here. So let me graph it. So it'll look something like that. And you're done.
Video transcript
So you may or may not already know that any linear equation can be written in the form y is equal to mx plus b. Where m is the slope of the line. The same slope that we've been dealing with the last few videos. The rise over run of the line. Or the inclination of the line. And b is the y-intercept. I think it's pretty easy to verify that b is a y-intercept. The way you verify that is you substitute x is equal to 0. If you get x is equal to 0-- remember x is equal to 0, that means that's where we're going to intercept at the y-axis. If x is equal to 0, this equation becomes y is equal to m times 0 plus b. m times 0 is just going to be 0. I don't care what m is. So then y is going to be equal to b. So the point 0, b is going to be on that line. The line will intercept the y-axis at the point y is equal to b. We'll see that with actual numbers in the next few videos. Just to verify for you that m is really the slope, let's just try some numbers out. We know the point 0, b is on the line. What happens when x is equal to 1? You get y is equal to m times 1. Or it's equal to m plus b. So we also know that the point 1, m plus b is also on the line. Right? This is just the y value. So what's the slope between that point and that point? Let's take this as the end point, so you have m plus b, our change in y, m plus b minus b over our change in x, over 1 minus 0. This is our change in y over change in x. We're using two points. That's our end point. That's our starting point. So if you simplify this, b minus b is 0. 1 minus 0 is 1. So you get m/1, or you get it's equal to m. So hopefully you're satisfied and hopefully I didn't confuse you by stating it in the abstract with all of these variables here. But this is definitely going to be the slope and this is definitely going to be the y-intercept. Now given that, what I want to do in this exercise is look at these graphs and then use the already drawn graphs to figure out the equation. So we're going to look at these, figure out the slopes, figure out the y-intercepts and then know the equation. So let's do this line A first. So what is A's slope? Let's start at some arbitrary point. Let's start right over there. We want to get even numbers. If we run one, two, three. So if delta x is equal to 3. Right? One, two, three. Our delta y-- and I'm just doing it because I want to hit an even number here-- our delta y is equal to-- we go down by 2-- it's equal to negative 2. So for A, change in y for change in x. When our change in x is 3, our change in y is negative 2. So our slope is negative 2/3. When we go over by 3, we're going to go down by 2. Or if we go over by 1, we're going to go down by 2/3. You can't exactly see it there, but you definitely see it when you go over by 3. So that's our slope. We've essentially done half of that problem. Now we have to figure out the y-intercept. So that right there is our m. Now what is our b? Our y-intercept. Well where does this intersect the y-axis? Well we already said the slope is 2/3. So this is the point y is equal to 2. When we go over by 1 to the right, we would have gone down by 2/3. So this right here must be the point 1 1/3. Or another way to say it, we could say it's 4/3. That's the point y is equal to 4/3. Right there. A little bit more than 1. About 1 1/3. So we could say b is equal to 4/3. So we'll know that the equation is y is equal to m, negative 2/3, x plus b, plus 4/3. That's equation A. Let's do equation B. Hopefully we won't have to deal with as many fractions here. Equation B. Let's figure out its slope first. Let's start at some reasonable point. We could start at that point. Let me do it right here. B. Equation B. When our delta x is equal to-- let me write it this way, delta x. So our delta x could be 1. When we move over 1 to the right, what happens to our delta y? We go up by 3. delta x. delta y. Our change in y is 3. So delta y over delta x, When we go to the right, our change in x is 1. Our change in y is positive 3. So our slope is equal to 3. What is our y-intercept? Well, when x is equal to 0, y is equal to 1. So b is equal to 1. So this was a lot easier. Here the equation is y is equal to 3x plus 1. Let's do that last line there. Line C Let's do the y-intercept first. You see immediately the y-intercept-- when x is equal to 0, y is negative 2. So b is equal to negative 2. And then what is the slope? m is equal to change in y over change in x. Let's start at that y-intercept. If we go over to the right by one, two, three, four. So our change in x is equal to 4. What is our change in y? Our change in y is positive 2. So change in y is 2 when change in x is 4. So the slope is equal to 1/2, 2/4. So the equation here is y is equal to 1/2 x, that's our slope, minus 2. And we're done. Now let's go the other way. Let's look at some equations of lines knowing that this is the slope and this is the y-intercept-- that's the m, that's the b-- and actually graph them. Let's do this first line. I already started circling it in orange. The y-intercept is 5. When x is equal to 0, y is equal to 5. You can verify that on the equation. So when x is equal to 0, y is equal to one, two, three, four, five. That's the y-intercept and the slope is 2. That means when I move 1 in the x-direction, I move up 2 in the y-direction. If I move 1 in the x-direction, I move up 2 in the y-direction. If I move 1 in the x-direction, I move up 2 in the y-direction. If I move back 1 in the x-direction, I move down 2 in the y-direction. If I move back 1 in the x-direction, I move down 2 in the y-direction. I keep doing that. So this line is going to look-- I can't draw lines too neatly, but this is going to be my best shot. It's going to look something like that. It'll just keep going on, on and on and on. So that's our first line. I can just keep going down like that. Let's do this second line. y is equal to negative 0.2x plus 7. Let me write that. y is equal to negative 0.2x plus 7. It's always easier to think in fractions. So 0.2 is the same thing as 1/5. We could write y is equal to negative 1/5 x plus 7. We know it's y-intercept at 7. So it's one, two, three, four, five, six. That's our y-intercept when x is equal to 0. This tells us that for every 5 we move to the right, we move down 1. We can view this as negative 1/5. The delta y over delta x is equal to negative 1/5. For every 5 we move to the right, we move down 1. So every 5. One, two, three, four, five. We moved 5 to the right. That means we must move down 1. We move 5 to the right. One, two, three, four, five. We must move down 1. If you go backwards, if you move 5 backwards-- instead of this, if you view this as 1 over negative 5. These are obviously equivalent numbers. If you go back 5-- that's negative 5. One, two, three, four, five. Then you move up 1. If you go back 5-- one, two, three, four, five-- you move up 1. So the line is going to look like this. I have to just connect the dots. I think you get the idea. I just have to connect those dots. I could've drawn it a little bit straighter. Now let's do this one, y is equal to negative x. Where's the b term? I don't see any b term. You remember we're saying y is equal to mx plus b. Where is the b? Well, the b is 0. You could view this as plus 0. Here is b is 0. When x is 0, y is 0. That's our y-intercept, right there at the origin. And then the slope-- once again you see a negative sign. You could view that as negative 1x plus 0. So slope is negative 1. When you move to the right by 1, when change in x is 1, change in y is negative 1. When you move up by 1 in x, you go down by 1 in y. Or if you go down by 1 in x, you're going to go up by 1 in y. x and y are going to have opposite signs. They go in opposite directions. So the line is going to look like that. You could almost imagine it's splitting the second and fourth quadrants. Now I'll do one more. Let's do this last one right here. y is equal to 3.75. Now you're saying, gee, we're looking for y is equal to mx plus b. Where is this x term? It's completely gone. Well the reality here is, this could be rewritten as y is equal to 0x plus 3.75. Now it makes sense. The slope is 0. No matter how much we change our x, y does not change. Delta y over delta x is equal to 0. I don't care how much you change your x. Our y-intercept is 3.75. So 1, 2, 3.75 is right around there. You want to get close. 3 3/4. As I change x, y will not change. y is always going to be 3.75. It's just going to be a horizontal line at y is equal to 3.75. Anyway, hopefully you found this useful.