Beg Algebra: Writing Point-Slope Form Equations

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WRITING AN EQUATION IN POINT-SLOPE FORM
Okay, so we see that we can find the equation of a line using the slope intercept form, y = mx + b, and this is a fantastic tool, because it always works, and it allows us to quickly find the equation for a line. Great. But you know what? There’s even a faster way in some cases, and I want to tell you about that method. This is called the point-slope form. Now, point-slope form has got to be the greatest way to write a line. This is my favorite way. Point-slope form is a form where you’re given a point and the slope, hence its name, point-slope form. This is the general form of point-slope form for a line, and let me tell you all the characters here. Well, m you’ve already met; m is the slope of the line. And then this point here, which looks just like sort of x and y, but they’re not. In fact, this is just a particular point on the line. So that’s a particular point that’s actually on the line. Now, let’s think about this. So if I just have the slope and a point that’s on the line, I should be able to determine that line uniquely. Let’s think about that for a second, because remember we talked about some time ago that slope just tells me pitch, so there I get the pitch down, but then, of course, there’s a lot of ways that that pitch could play out. But if I had a point that I know it goes through, that will solidify the line for sure. So if I just know a point and I know the slope, I should be able to report the equation, and here’s the formula.
Now, where does that come from? Well, if you think about it, slope is just change in y over change in x, so what I know is that slope will have to equal the change in y over the change in x. And if you just multiply both sides by x - x1, just multiply both sides by this. Look what you see. You see exactly this. Because when I multiply on this side by x - x 1 they cancel, and when I multiply this side by x - x1, I get this. So, in fact, this just comes from literally the very definition of slope. There’s nothing to remember here. Just remember that slope equals the change in y divided by the change in x.
But the power of this is absolutely amazing. Suppose that I tell you that I’m thinking of a line, and it passes through the point -1, 3, and it has slope equal to -¼. What’s the line? It turns out this is absolutely no big deal if you use point-slope form, because all I do is report this into the formula, where this is x1, this is y1, and this is the slope. So what I would see is y - y1, which is 3, equals the slope, minus ¼(x) - x1. Now, that’s a -1, but a minus and a minus combine to give a plus 1. So notice I just plugged in the -1 here, x - -1, x + 1, and I plugged a three in to here, that’s y - 3. The slope into here, and all of a sudden this is the equation for that line. Look how easy that was. That really was easy. I didn’t have to find a y intercept at all.
Let’s go backwards. Suppose I tell you I’m thinking of a line and it looks like this. 2/3(x + 1). What can you tell me about that line? Well, you can tell me a lot of stuff about that line. For example, you can immediately tell me the slope. The slope is 2/3. and, in fact, you can tell me a point that that line must pass through, namely, well, you have to be careful--this +1 doesn’t quite fit this formula, so I have to write this as 2/3(x - -1). It always has to be in the form, x minus the value. So here I’d see -1, 2 as a point that the line contains. So the slope of this line is 2/3 and it passes through the point -1, 2. So, in fact, we could actually give a very quick sketch of the graph of this, just given the formula
Let’s see how this would go. So I know that this thing passes through -1, 2. So I go -1 in the x direction and 2 up. So I know it passes through that point, and its slope is 2/3. That means I go 3 units over and 2 units up. So 1, 2, 3 units over; 1, 2 units up. There’s another point on the line. There’s an extremely accurate graph of the line, just given this equation in point-slope form.
So it’s really a fantastic form to write a line in if you’re just given any old point and you’re just given a slope. No problem. It is by far my favorite way to write lines. I even like it better than slope-intercept form, even though slope intercept form is the thing I grew up with. When I was a kid that’s all they were teaching us--slope intercept form--and I loved it because x = mx + b. It sounds so good. But, when you start thinking about it, and if you were to go into calculus, boy, point-slope form, definitely the way to go. So buy some point-slope form futures now.