Algebra: Match Point-Slope Equations to Graphs

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Relations and Functions
-Equations of a Line
-Matching a Slope-Intercept Equation with Its Graph

Well, let’s see all these graphic techniques now in action and see if I just give you an equation for a line, if we can just graph it. So how would you do that? Well, here’s the one that I want us to take a look at. How about y + 1 = 2(x +1)? Well, how would you graph that? Well, of course, you could always plot points and make a little table. But look, this is actually in point-slope form. So 2 is the slope and a point that the line must contain is 1, 1. Well, that’s not quite right, actually, because remember, we always have to write this in the form of a minus, so we have to write this as y - (-1) = 2(x - -(1)). Now that it’s written that way, we can really see what’s going on. The slope is still 2, but now a point that’s on the line is actually -1, -1. So we have to be careful not to make that mistake for real. And now, what would that line look like? Well, goes through -1, -1, slope is 2, so I have -1, -1, goes through there, and the slope is 2. That means that I go 1 unit over, 2 units up. 1 unit over, 2 units up. So if I connect those things what I see is 1 unit over and 2 units up. There’s the graph. One unit over, 2 units up, 1 unit over, 2 units up. So there’s the line; really simple.
Okay, let’s try another one. How about this? y= 3. Well, you’re saying, “Gee, let’s see, what form can I think of that in?” Well, it might be sort of hard to see, but if you think about it I could write it in slope-intercept form--0x + 3. Then you can see it a little easier. The slope is 0 and the y intercept is 3. So what does that look like? The slope is 0 and the y intercept is 1, 2, 3, and the slope is 0. That means the line is horizontal. These are all the places where y = 3. It’s the horizontal line three units up above the x-axis. Piece of cake.
Let’s try another one. How about x = 1? Well, what form can we write this in? Well, actually, this is something that I don’t quite know the form of. x = 1. It’s the formula that produces just x = 1 for any value of y. So what would that be? Well, in fact, this line would be what? Well, x = 1. Well, where does x = 1? x = 1 here, but also x = 1 here--because y-something, but x is still 1, x is still 1, x is still 1. So, in fact, this is the formula for a vertical line. Remember, a vertical line has undefined slope, but if you want to right the equation of a vertical line, we can always do it by saying x = the value. Because notice that every single point on this red line has the feature that x = 1. So if I say to you what’s the equation of this line? Well, it’s vertical, so it has undefined slope, but notice that every single point, what does x equal? It’s always the same. 1, 2, 3, 4, 5. So on this line, at every single point, x = 5, so we say that the equation of this line is x = 5. So that’s how we actually write down the equation for vertical lines.
How about this one. Here’s one that’s going to look sort of freaky. What’s the graph of this look like? -2x - 5 = 7. Well, you may say, “Well, there’s no y’s there. What do I do?” Well, you can sort of solve this for x, I guess, if you wanted to by bringing the 5 over to the other side by adding 5 to both sides. I would see -2x = and this would be a positive 12. If I divide both sides by the -2 I’d see that x equals… Well, -6. So this weird, complicated thing--you might think this was a slope or something here, but there’s no y’s there. Right? y= mx + b. y - y1 = m(x - x1). So there’s no y’s there. Don’t be fooled by that. If you solved this out you’d see x = -6. So what does that mean in English? It means in English or any other language--1, 2, 3, 4, 5, and if I go 6, one more unit, it’s going to be a vertical line. Six units over in the negative direction from the origin. So if you have x equals a number, it’s always a vertical line.
All right. Let’s try another one. How about this? y = x - 3. Well, that’s actually all right in a nice form. It’s actually in slope-intercept form. So I see the intercept is -3, the y intercept is -3, and I see the slope is--oh, maybe the slope is 0 because there’s nothing in front of it. No, the slope is 1. There’s always an invisible 1 that follows us wherever we go--not 0, but 1. So this has slope 1, y intercept minus 3. What does that look like? Slope 1, y intercept minus 3. So I go down to -3 and y. 1, 2, 3, there’s minus 3, and slope 1. One over, one up, one over, one up, one over, one up. So you get the idea. There’s the equation for the line. Presto! Look how easy this is.
Let’s try one last one together. Suppose that I give you the line in standard form. Remember, that’s the really stupid form that no one ever actually uses. 2x - 3y = 1. Well, how would you graph that? Well, what I would do probably is just solve for y and write it in slope intercept form. So if I solve this for y, let’s see… Let me bring the -3y to the other side. So I see 2x = 1 + 3y, then I’ll subtract 1 from both sides, and I see 2x - 1 = 3y. And if I divide both sides by 3, I would see that 2/3x - 1/3 = y or y = 2/3x - 1/3. By the way, you might say, “Wait a minute. How come I’ve got those two 3’s there? If I divide both sides by 3 and then I just see one big 3 under the whole thing.” Well, that’s true. But remember how fractions get added or subtracted. I can just subtract the tops and keep the bottoms the same. So, in fact, that would be 2/3x - 1/3 here, and if you combine them, the bottoms are the same, so it’s combined, and I get this again.
But now, in this form, it actually is very easy to see what this equation looks like. It’s y = 2/3x - 1/3. That means that the y intercept is -1/3 and the slope is 2/3. So that means rise over run is 2/3. So if I look at the graph of this, the y intercept--now watch this--is going to be -1/3. Well, this is -1, so I’m actually only going to -1/3, so the y intercept is actually a third of the way down. And what’s the slope? The slope is 2/3. So that means 3 over in the x, and then 2 up. So I start here and I go 3 over--1, 2, 3, and now 2 up--1, 2. And so there’s another point that this line contains, and if I connect those points--let me use actually a really small line this time so you can really see the detail. If you connect them very carefully you’ll see that is the graph of the line. Because notice it has y intercept -1/3, and the slope is 2/3, which means I go 2 units up--1, 2, and then 3 over--1, 2, 3, and there’s another point. So to be really careful there I’m drawing a very thin line and not the usual thick red one just to indicate that this is really sort of sitting right in that grid in a very delicate way. But the point is, no matter what the form of the line that we’re given is in terms of the equation, we can always graph is by just manipulating it into a form we know, or recognizing it as a form that we know and then using the information--y = mx + b, m is the slope, b is the y intercept. Or y - y1 = m(x - x1), in which case we know that the slope is m, and it contains the point x1, y1, point slope form. Okay, see if you can graph these things and enjoy.